Comptes Rendus
First principles homogenization of periodic metamaterials and application to wire media
Comptes Rendus. Physique, Volume 21 (2020) no. 4-5, pp. 367-388.

Here, we present an overview of a first principles homogenization theory of periodic metamaterials. It is shown that in a rather general context it is possible to formally introduce effective parameters that describe the time evolution of macroscopic (slowly-varying in space) initial states of the electromagnetic field using an effective medium formalism. The theory is applied to different types of “wire metamaterials” characterized by a strong spatial dispersion in the long wavelength limit. It is highlighted that the spatial dispersion may tailor in unique ways the wave phenomena in wire metamaterials leading to exotic tunneling effects and broadband lossless anomalous dispersion.

Nous présentons une revue d’une théorie ab-initio d’homogénéisation de métamatériaux périodiques. Nous montrons dans un cadre général qu’il est possible d’introduire formellement des paramètres effectifs qui décrivent l’évolution temporelle d’états initiaux (états à variation spatiale lente) du champ électromagnétique en faisant appel à un formalisme de type milieu effectif. Cette théorie est appliquée à différents types de métamatériaux constitués de réseaux périodiques de fils métalliques et qui ont par ailleurs la particularité d’être caractérisés par une forte dispersion spatiale dans la limite des grandes longueurs d’onde. Il est souligné que les effets de la dispersion spatiale peuvent mener à un contrôle sans précédent des phénomènes ondulatoires dans ces métamatériaux, notamment avec la possibilité d’effets tunnel exotiques ou encore de dispersion anormale sans perte et large bande.

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DOI: 10.5802/crphys.4
Keywords: Homogenization, Effective medium, Wire medium, Metamaterials, Spatial dispersion
Mot clés : Homogénéisation, Milieu effectif, Réseaux de fils métalliques, Métamatériaux, Dispersion spatiale

Sylvain Lannebère 1; Tiago A. Morgado 1; Mário G. Silveirinha 1, 2

1 Department of Electrical Engineering, University of Coimbra and Instituto de Telecomunicações, 3030-290 Coimbra, Portugal
2 University of Lisbon – Instituto Superior Técnico, Department of Electrical Engineering, 1049-001 Lisboa, Portugal
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Sylvain Lannebère; Tiago A. Morgado; Mário G. Silveirinha. First principles homogenization of periodic metamaterials and application to wire media. Comptes Rendus. Physique, Volume 21 (2020) no. 4-5, pp. 367-388. doi : 10.5802/crphys.4. https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.4/

[1] A. Sihvola Mixing rules, Metamaterials Handbook: Applications of Metamaterials (F. Capolino, ed.), CRC Press, 2009, 762 pages

[2] C. Kittel Introduction to Solid State Physics, John Wiley & Sons, Hoboken, NJ, 2004

[3] D. R. Smith; J. B. Pendry Homogenization of metamaterials by field averaging (invited paper), J. Opt. Soc. Am. B, JOSAB, Volume 23 (2006) no. 3, pp. 391-403 | DOI

[4] D. Sjöberg Dispersive effective material parameters, Microwave Optical Technol. Lett., Volume 48 (2006) no. 12, pp. 2629-2632 | DOI

[5] C. R. Simovski Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices, Metamaterials, Volume 1 (2007) no. 2, pp. 62-80 | DOI

[6] C. R. Simovski; S. A. Tretyakov Local constitutive parameters of metamaterials from an effective-medium perspective, Phys. Rev. B, Volume 75 (2007) no. 19, 195111

[7] M. G. Silveirinha Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters, Phys. Rev. B, Volume 75 (2007) no. 11, 115104

[8] M. G. Silveirinha Generalized Lorentz–Lorenz formulas for microstructured materials, Phys. Rev. B, Volume 76 (2007) no. 24, 245117

[9] G. P. Ortiz; B. E. Martínez-Zérega; B. S. Mendoza; W. L. Mochán Effective optical response of metamaterials, Phys. Rev. B, Volume 79 (2009) no. 24, 245132

[10] M. G. Silveirinha Nonlocal homogenization theory of structured materials, Metamaterials Handbook: Applications of Metamaterials (F. Capolino, ed.), CRC Press, 2009, Ch. 10

[11] J. T. Costa; M. G. Silveirinha; S. I. Maslovski Finite-difference frequency-domain method for the extraction of effective parameters of metamaterials, Phys. Rev. B, Volume 80 (2009) no. 23, 235124

[12] D. R. Smith Analytic expressions for the constitutive parameters of magnetoelectric metamaterials, Phys. Rev. E, Volume 81 (2010) no. 3, 036605

[13] C. R. Simovski On electromagnetic characterization and homogenization of nanostructured metamaterials, J. Opt., Volume 13 (2010) no. 1, 013001

[14] M. G. Silveirinha Time domain homogenization of metamaterials, Phys. Rev. B, Volume 83 (2011) no. 16, 165104

[15] C. Fietz; G. Shvets Current-driven metamaterial homogenization, Physica B: Condensed Matter, Volume 405 (2010) no. 14, pp. 2930-2934 | DOI

[16] A. V. Chebykin; A. A. Orlov; A. V. Vozianova; S. I. Maslovski; Y. S. Kivshar; P. A. Belov Nonlocal effective medium model for multilayered metal-dielectric metamaterials, Phys. Rev. B, Volume 84 (2011) no. 11, 115438

[17] A. V. Chebykin; A. A. Orlov; C. R. Simovski; Y. S. Kivshar; P. A. Belov Nonlocal effective parameters of multilayered metal-dielectric metamaterials, Phys. Rev. B, Volume 86 (2012) no. 11, 115420

[18] A. Alù First-principles homogenization theory for periodic metamaterials, Phys. Rev. B, Volume 84 (2011) no. 7, 075153

[19] A. D. Yaghjian; A. Alù; M. G. Silveirinha Homogenization of spatially dispersive metamaterial arrays in terms of generalized electric and magnetic polarizations, Photonics Nanostructures - Fundam. Appl., Volume 11 (2013) no. 4, pp. 374-396 | DOI

[20] A. D. Yaghjian; A. Alù; M. G. Silveirinha Anisotropic representation for spatially dispersive periodic metamaterial arrays, Transformation Electromagnetics and Metamaterials: Fundamental Principles and Applications (D. H. Werner; D.-H. Kwon, eds.), Springer London, London, 2014, pp. 395-457 | DOI

[21] V. Sozio; A. Vallecchi; M. Albani; F. Capolino Generalized Lorentz–Lorenz homogenization formulas for binary lattice metamaterials, Phys. Rev. B, Volume 91 (2015) no. 20, 205127 | DOI

[22] C. Simovski Composite Media with Weak Spatial Dispersion, Pan Stanford Publishing Pte Ltd, Singapur, 2018 | DOI

[23] D. Cioranescu; F. Murat A strange term coming from nowhere, Topics in the Mathematical Modelling of Composite Materials, Progress in Nonlinear Differential Equations and Their Applications (A. Cherkaev; R. Kohn, eds.), Birkhäuser, Boston, MA, 1997, pp. 45-93

[24] D. Felbacq; G. Bouchitté Homogenization of a set of parallel fibres, Waves Random Media, Volume 7 (1997) no. 2, pp. 245-256 | DOI | MR | Zbl

[25] C. G. Poulton; L. C. Botten; R. C. McPhedran; N. A. Nicorovici; A. B. Movchan Noncommuting limits in electromagnetic scattering: asymptotic analysis for an array of highly conducting inclusions, SIAM J. Appl. Math., Volume 61 (2001) no. 5, pp. 1706-1730 | MR | Zbl

[26] C. Poulton; S. Guenneau; A. B. Movchan Noncommuting limits and effective properties for oblique propagation of electromagnetic waves through an array of aligned fibres, Phys. Rev. B, Volume 69 (2004) no. 19, 195112 | DOI

[27] V. Zhikov On gaps in the spectrum of some divergent elliptic operators with periodic coefficients, St. Petersburg Math. J., Volume 16 (2005) no. 5, pp. 773-790 | DOI | Zbl

[28] A. Maurel; J.-J. Marigo Sensitivity of a dielectric layered structure on a scale below the periodicity: a fully local homogenized model, Phys. Rev. B, Volume 98 (2018) no. 2, 024306 | DOI

[29] J. B. Pendry Negative refraction makes a perfect lens, Phys. Rev. Lett., Volume 85 (2000) no. 18, pp. 3966-3969 | DOI

[30] C. Luo; S. G. Johnson; J. D. Joannopoulos; J. B. Pendry Subwavelength imaging in photonic crystals, Phys. Rev. B, Volume 68 (2003) no. 4, 045115

[31] P. A. Belov; Y. Hao; S. Sudhakaran Subwavelength microwave imaging using an array of parallel conducting wires as a lens, Phys. Rev. B, Volume 73 (2006) no. 3, 033108

[32] F. Capolino Applications of Metamaterials, CRC Press, Boca Raton, FL, 2009

[33] L. D. Landau; L. P. Pitaevskii; E. M. Lifshitz Electrodynamics of Continuous Media: Volume 8, Butterworth-Heinemann, Amsterdam u.a., 1984

[34] V. M. Agranovich; V. Ginzburg Crystal Optics with Spatial Dispersion, and Excitons, Springer Series in Solid-State Sciences, Springer-Verlag, Berlin Heidelberg, 1984 | DOI

[35] M. G. Silveirinha; N. Engheta Effective medium approach to electron waves: graphene superlattices, Phys. Rev. B, Volume 85 (2012) no. 19, 195413

[36] M. G. Silveirinha Effective medium theory of electromagnetic and quantum metamaterials, World Scientific Handbook of Metamaterials and Plasmonics (E. Shamonina; S. A. Maier, eds.) (World Scientific Series in Nanoscience and Nanotechnology), World Scientific, 2017, Ch. 2, pp. 37-86 | DOI

[37] M. G. Silveirinha; N. Engheta Metamaterial-inspired model for electron waves in bulk semiconductors, Phys. Rev. B, Volume 86 (2012) no. 24, 245302

[38] M. G. Silveirinha; N. Engheta Giant nonlinearity in zero-gap semiconductor superlattices, Phys. Rev. B, Volume 89 (2014) no. 8, 085205

[39] S. Lannebère; M. G. Silveirinha Effective Hamiltonian for electron waves in artificial graphene: a first-principles derivation, Phys. Rev. B, Volume 91 (2015) no. 4, 045416 | DOI

[40] B. Gralak; A. Tip Macroscopic Maxwell’s equations and negative index materials, J. Math. Phys., Volume 51 (2010) no. 5, 052902 | DOI | MR | Zbl

[41] M. G. Silveirinha Topological classification of Chern-type insulators by means of the photonic Green function, Phys. Rev. B, Volume 97 (2018) no. 11, 115146

[42] M. G. Silveirinha Modal expansions in dispersive material systems with application to quantum optics and topological photonics, Advances in Mathematical Methods for Electromagnetics (K. Kobayashi; P. D. Smith, eds.), IET, 2019

[43] G. Russakoff A derivation of the macroscopic Maxwell equations, Amer. J. Phys., Volume 38 (1970) no. 10, pp. 1188-1195 | DOI

[44] D. M. Pozar Microwave Engineering, Wiley, Hoboken, NJ, 2011

[45] M. G. Silveirinha Artificial plasma formed by connected metallic wires at infrared frequencies, Phys. Rev. B, Volume 79 (2009) no. 3, 035118

[46] W. Rotman Plasma simulation by artificial dielectrics and parallel-plate media, IRE Trans. Antennas and Propagation, Volume 10 (1962) no. 1, pp. 82-95 | DOI

[47] J. B. Pendry; A. J. Holden; W. J. Stewart; I. Youngs Extremely low frequency plasmons in metallic mesostructures, Phys. Rev. Lett., Volume 76 (1996) no. 25, pp. 4773-4776 | DOI

[48] S. I. Maslovski; S. A. Tretyakov; P. A. Belov Wire media with negative effective permittivity: a quasi-static model, Microw. Opt. Technol. Lett., Volume 35 (2002) no. 1, pp. 47-51 | DOI

[49] P. A. Belov; S. A. Tretyakov; A. J. Viitanen Dispersion and reflection properties of artificial media formed by regular lattices of ideally conducting wires, J. Electromagnetic Waves Appl., Volume 16 (2002) no. 8, pp. 1153-1170 | DOI

[50] P. A. Belov; R. Marqués; S. I. Maslovski; I. S. Nefedov; M. Silveirinha; C. R. Simovski; S. A. Tretyakov Strong spatial dispersion in wire media in the very large wavelength limit, Phys. Rev. B, Volume 67 (2003) no. 11, 113103

[51] I. S. Nefedov; A. J. Viitanen Wire Media, Metamaterials Handbook: Applications of Metamaterials (F. Capolino, ed.), CRC Press, 2009, Ch. 15

[52] C. R. Simovski; P. A. Belov; A. V. Atrashchenko; Y. S. Kivshar Wire metamaterials: physics and applications, Adv. Mater., Volume 24 (2012) no. 31, pp. 4229-4248 | DOI

[53] M. G. Silveirinha Nonlocal homogenization model for a periodic array of ε-negative rods, Phys. Rev. E, Volume 73 (2006) no. 4, 046612

[54] I. Nefedov; A. Viitanen; S. Tretyakov On reflection from interfaces with some spatially dispersive metamaterials, J. Magnetism Magnetic Mater., Volume 300 (2006) no. 1, p. e107-e110 | DOI

[55] P. A. Belov; C. R. Simovski; P. Ikonen Canalization of subwavelength images by electromagnetic crystals, Phys. Rev. B, Volume 71 (2005) no. 19, 193105

[56] P. A. Belov; M. G. Silveirinha Resolution of subwavelength transmission devices formed by a wire medium, Phys. Rev. E, Volume 73 (2006) no. 5, 056607

[57] P. A. Belov; Y. Zhao; S. Sudhakaran; A. Alomainy; Y. Hao Experimental study of the subwavelength imaging by a wire medium slab, Appl. Phys. Lett., Volume 89 (2006) no. 26, 262109

[58] P. A. Belov; Y. Zhao; S. Tse; P. Ikonen; M. G. Silveirinha; C. R. Simovski; S. Tretyakov; Y. Hao; C. Parini Transmission of images with subwavelength resolution to distances of several wavelengths in the microwave range, Phys. Rev. B, Volume 77 (2008) no. 19, 193108

[59] P. Ikonen; C. Simovski; S. Tretyakov; P. Belov; Y. Hao Magnification of subwavelength field distributions at microwave frequencies using a wire medium slab operating in the canalization regime, Appl. Phys. Lett., Volume 91 (2007) no. 10, 104102 | DOI

[60] G. Shvets; S. Trendafilov; J. B. Pendry; A. Sarychev Guiding, focusing, and sensing on the subwavelength scale using metallic wire arrays, Phys. Rev. Lett., Volume 99 (2007) no. 5, 053903 | DOI

[61] M. G. Silveirinha; P. A. Belov; C. R. Simovski Subwavelength imaging at infrared frequencies using an array of metallic nanorods, Phys. Rev. B, Volume 75 (2007) no. 3, 035108

[62] T. A. Morgado; M. G. Silveirinha Transport of an arbitrary near-field component with an array of tilted wires, New J. Phys., Volume 11 (2009) no. 8, 083023

[63] T. A. Morgado; J. S. Marcos; M. G. Silveirinha; S. I. Maslovski Experimental verification of full reconstruction of the near-field with a metamaterial lens, Appl. Phys. Lett., Volume 97 (2010) no. 14, 144102

[64] H. Latioui; M. G. Silveirinha Near-field transport by a bent multi-wire endoscope, J. Appl. Phys., Volume 120 (2016) no. 6, 063103 | DOI

[65] M. G. Silveirinha; C. A. Fernandes Homogenization of metamaterial surfaces and slabs: the crossed wire mesh canonical problem, IEEE Trans. Antennas and Propagation, Volume 53 (2005) no. 1, pp. 59-69 | DOI

[66] M. G. Silveirinha; C. A. Fernandes Nonresonant structured material with extreme effective parameters, Phys. Rev. B, Volume 78 (2008) no. 3, 033108

[67] M. G. Silveirinha Broadband negative refraction with a crossed wire mesh, Phys. Rev. B, Volume 79 (2009) no. 15, 153109

[68] M. G. Silveirinha Anomalous refraction of light colors by a metamaterial prism, Phys. Rev. Lett., Volume 102 (2009) no. 19, 193903

[69] T. A. Morgado; J. S. Marcos; M. G. Silveirinha; S. I. Maslovski Ultraconfined interlaced plasmons, Phys. Rev. Lett., Volume 107 (2011) no. 6, 063903

[70] T. A. Morgado; J. S. Marcos; S. I. Maslovski; M. G. Silveirinha Negative refraction and partial focusing with a crossed wire mesh: physical insights and experimental verification, Appl. Phys. Lett., Volume 101 (2012) no. 2, 021104

[71] J. T. Costa; M. G. Silveirinha Achromatic lens based on a nanowire material with anomalous dispersion, Opt. Express, OE, Volume 20 (2012) no. 13, pp. 13915-13922 | DOI

[72] T. A. Morgado; J. S. Marcos; J. T. Costa; J. R. Costa; C. A. Fernandes; M. G. Silveirinha Reversed rainbow with a nonlocal metamaterial, Appl. Phys. Lett., Volume 105 (2014) no. 26, 264101

[73] M. G. Silveirinha; C. A. Fernandes Homogenization of 3-D-connected and nonconnected wire metamaterials, IEEE Trans. Microw. Theory Tech., Volume 53 (2005) no. 4, pp. 1418-1430 | DOI

[74] A. Demetriadou; J. B. Pendry Taming spatial dispersion in wire metamaterial, J. Phys.: Condens. Matter, Volume 20 (2008) no. 29, 295222

[75] J. A. Bittencourt Fundamentals of Plasma Physics, Springer-Verlag, New York, 2004 | DOI | Zbl

[76] S. I. Maslovski; M. G. Silveirinha Nonlocal permittivity from a quasistatic model for a class of wire media, Phys. Rev. B, Volume 80 (2009) no. 24, 245101

[77] S. I. Maslovski; T. A. Morgado; M. G. Silveirinha; C. S. R. Kaipa; A. B. Yakovlev Generalized additional boundary conditions for wire media, New J. Phys., Volume 12 (2010) no. 11, 113047 | Zbl

[78] M. G. Silveirinha; S. I. Maslovski Radiation from elementary sources in a uniaxial wire medium, Phys. Rev. B, Volume 85 (2012) no. 15, 155125

[79] M. G. Silveirinha Poynting vector, heating rate, and stored energy in structured materials: a first-principles derivation, Phys. Rev. B, Volume 80 (2009) no. 23, 235120

[80] J. T. Costa; M. G. Silveirinha; A. Alù Poynting vector in negative-index metamaterials, Phys. Rev. B, Volume 83 (2011) no. 16, 165120

[81] M. G. Silveirinha Additional boundary conditions for nonconnected wire media, New J. Phys., Volume 11 (2009) no. 11, 113016

[82] V. V. Yatsenko; S. A. Tretyakov; S. I. Maslovski; A. A. Sochava Higher order impedance boundary conditions for sparse wire grids, IEEE Trans. Antennas and Propagation, Volume 48 (2000) no. 5, pp. 720-727 | DOI

[83] I. S. Nefedov; A. J. Viitanen; S. A. Tretyakov Electromagnetic wave refraction at an interface of a double wire medium, Phys. Rev. B, Volume 72 (2005) no. 24, 245113

[84] S. Pekar The theory of electromagnetic waves in a crystal in which excitons are produced, Sov. Phys. JETP, Volume 6 (1958), p. 785 | MR | Zbl

[85] M. G. Silveirinha Additional boundary condition for the wire medium, IEEE Trans. Antennas and Propagation, Volume 54 (2006) no. 6, pp. 1766-1780 | DOI | MR | Zbl

[86] A. B. Yakovlev; Y. R. Padooru; G. W. Hanson; A. Mafi; S. Karbasi A generalized additional boundary condition for mushroom-type and bed-of-nails-type wire media, IEEE Trans. Microw. Theory Tech., Volume 59 (2011) no. 3, pp. 527-532 | DOI

[87] G. W. Hanson; E. Forati; M. G. Silveirinha Modeling of spatially-dispersive wire media: transport representation, comparison with natural materials, and additional boundary conditions, IEEE Trans. Antennas and Propagation, Volume 60 (2012) no. 9, pp. 4219-4232 | DOI | MR | Zbl

[88] G. W. Hanson; M. G. Silveirinha; P. Burghignoli; A. B. Yakovlev Non-local susceptibility of the wire medium in the spatial domain considering material boundaries, New J. Phys., Volume 15 (2013) no. 8, 083018

[89] M. G. Silveirinha Boundary conditions for quadrupolar metamaterials, New J. Phys., Volume 16 (2014) no. 8, 083042

[90] A. D. Yaghjian Boundary conditions for electric quadrupolar continua, Radio Sci., Volume 49 (2014) no. 12, pp. 1289-1299 | DOI

[91] J. Shin; J.-T. Shen; S. Fan Three-dimensional electromagnetic metamaterials that homogenize to uniform non-Maxwellian media, Phys. Rev. B, Volume 76 (2007) no. 11, 113101 | DOI

[92] H. Latioui; M. G. Silveirinha Light tunneling anomaly in interlaced metallic wire meshes, Phys. Rev. B, Volume 96 (2017) no. 19, 195132 | DOI

[93] D. E. Fernandes; S. I. Maslovski; G. W. Hanson; M. G. Silveirinha Fano resonances in nested wire media, Phys. Rev. B, Volume 88 (2013) no. 4, 045130

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