Partial differential equations/Functional analysis
Electromagnetic wave propagation in media consisting of dispersive metamaterials
Comptes Rendus. Mathématique, Volume 356 (2018) no. 7, pp. 757-775.

We establish the well-posedness, the finite speed propagation, and a regularity result for Maxwell's equations in media consisting of dispersive (frequency dependent) metamaterials. Two typical examples for such metamaterials are materials obeying Drude's and Lorentz' models. The causality and the passivity are the two main assumptions and play a crucial role in the analysis. It is worth noting that by contrast the well-posedness in the frequency domain is not ensured in general. We also provide some numerical experiments using the Drude's model to illustrate its dispersive behaviour.

Nous montrons que les équations de Maxwell dans un milieu constitué de métamatériaux dispersifs (dépendant de la fréquence) forment un problème bien posé, à vitesse de propagation finie et satisfaisant un résultat de régularité. Deux exemples typiques de tels métamatériaux sont les matériaux régis par les modèles de Drude et de Lorentz. La causalité et la passivité sont les deux hypothèses principales ; elles jouent un rôle essentiel dans l'analyse. Il vaut la peine de remarquer qu'en revanche, rien n'assure, en général, le caractère bien posé dans le domaine des fréquences. Nous présentons également quelques résultats numériques utilisant le modèle de Drude, afin d'illustrer le comportement dispersif.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.05.012
Nguyen, Hoai-Minh 1; Vinoles, Valentin 1

1 Department of Mathematics, EPFL SB CAMA, Station 8, CH-1015 Lausanne, Switzerland
@article{CRMATH_2018__356_7_757_0,
     author = {Nguyen, Hoai-Minh and Vinoles, Valentin},
     title = {Electromagnetic wave propagation in media consisting of dispersive metamaterials},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {757--775},
     publisher = {Elsevier},
     volume = {356},
     number = {7},
     year = {2018},
     doi = {10.1016/j.crma.2018.05.012},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2018.05.012/}
}
TY  - JOUR
AU  - Nguyen, Hoai-Minh
AU  - Vinoles, Valentin
TI  - Electromagnetic wave propagation in media consisting of dispersive metamaterials
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 757
EP  - 775
VL  - 356
IS  - 7
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2018.05.012/
DO  - 10.1016/j.crma.2018.05.012
LA  - en
ID  - CRMATH_2018__356_7_757_0
ER  - 
%0 Journal Article
%A Nguyen, Hoai-Minh
%A Vinoles, Valentin
%T Electromagnetic wave propagation in media consisting of dispersive metamaterials
%J Comptes Rendus. Mathématique
%D 2018
%P 757-775
%V 356
%N 7
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2018.05.012/
%R 10.1016/j.crma.2018.05.012
%G en
%F CRMATH_2018__356_7_757_0
Nguyen, Hoai-Minh; Vinoles, Valentin. Electromagnetic wave propagation in media consisting of dispersive metamaterials. Comptes Rendus. Mathématique, Volume 356 (2018) no. 7, pp. 757-775. doi : 10.1016/j.crma.2018.05.012. http://www.numdam.org/articles/10.1016/j.crma.2018.05.012/

[1] Bécache, É.; Joly, P.; Vinoles, V. On the analysis of perfectly matched layers for a class of dispersive media and application to negative index metamaterials, 2016 | HAL

[2] Bonnet-Ben Dhia, A.S.; Chesnel, L.; Ciarlet, P. T-coercivity for scalar interface problems between dielectrics and metamaterials, ESAIM Math. Model. Numer. Anal., Volume 46 (2012), pp. 1363-1387

[3] Bonnetier, E.; Nguyen, H.-M. Superlensing using hyperbolic metamaterials: the scalar case, J. Éc. Polytech. Math., Volume 4 (2017), pp. 973-1003

[4] Burton, T.A. Volterra Integral and Differential Equations, Mathematics in Science and Engineering, vol. 167, Academic Press, Inc., Orlando, FL, USA, 1983

[5] Cassier, M. Étude de deux problèmes de propagation d'ondes, transitoires: 1) Focalisation spatio-temporelle en acoustique; 2) Transmission entre un diélectrique et un métamatériaux, Paris-Saclay University, 2016 (PhD thesis)

[6] Cassier, M.; Hazard, C.; Joly, P. Spectral theory for Maxwell's equations at the interface of a metamaterial. Part I: Generalized Fourier transform | arXiv

[7] Cassier, M.; Joly, P.; Kachanovska, M. Mathematical models for dispersive electromagnetic waves: an overview | arXiv

[8] Costabel, M.; Stephan, E. A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl., Volume 106 (1985), pp. 367-413

[9] Dautray, R.; Lions, J.-L. Mathematical Analysis and Numerical Methods for Science and Technology: Volume 5. Evolutions Problems I, Springer Science & Business, Media, 1992

[10] Evans, L.C. Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998

[11] Figotin, A.; Schenker, J.H. Spectral theory of time dispersive and dissipative systems, J. Stat. Phys., Volume 118 (2005), pp. 199-263

[12] Gralak, B.; Tip, A. Macroscopic Maxwell's equations and negative index materials, J. Math. Phys., Volume 51 (2010)

[13] Hecht, F. New development in FreeFem++, J. Numer. Math., Volume 20 (2012), pp. 251-265

[14] Jackson, J.D. Classical Electrodynamics, John Wiley & Sons, 1999

[15] Jacob, Z.; Alekseyev, L.V.; Narimanov, E. Optical hyperlens: far-field imaging beyond the diffraction limit, Opt. Express, Volume 14 (2006), pp. 8247-8256

[16] Kong, J.A. Theory of Electromagnetic Waves, Wiley–Interscience, New York, 1975

[17] Lai, Y.; Chen, H.; Zhang, Z.; Chan, C.T. Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell, Phys. Rev. Lett., Volume 102 (2009)

[18] Landau, L.D.; Lifshitz, E.M. Electrodynamics of Continuous Media, Pergamon Press, 1984

[19] Liu, Z.; Lee, H.; Xiong, Y.; Sun, C.; Zhang, Z. Far-field optical hyperlens magnifying sub-diffraction-limited objects, Science, Volume 315 (2007), p. 1686

[20] Mackay, T.G. Electromagnetic Anisotropy and Bianisotropy: A Field Guide, World Scientific, 2010

[21] Milton, G.W.; Nicorovici, N.A.; McPhedran, R.C.; Podolskiy, V.A. A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance, Proc. R. Soc. Lond. Ser. A, Volume 461 (2005), pp. 3999-4034

[22] Milton, G.W.; Nicorovici, N.A. On the cloaking effects associated with anomalous localized resonance, Proc. R. Soc. Lond. Ser. A, Volume 462 (2006), pp. 3027-3059

[23] Nguyen, H-M. Asymptotic behavior of solutions to the Helmholtz equations with sign changing coefficients, Trans. Amer. Math. Soc., Volume 367 (2015), pp. 6581-6595

[24] Nguyen, H-M. Cloaking via anomalous localized resonance for doubly complementary media in the quasistatic regime, J. Eur. Math. Soc. (JEMS), Volume 17 (2015), pp. 1327-1365

[25] Nguyen, H-M. Superlensing using complementary media, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 32 (2015), pp. 471-484

[26] Nguyen, H-M. Cloaking using complementary media in the quasistatic regime, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 33 (2016), pp. 1509-1518

[27] Nguyen, H-M. Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients, J. Math. Pures Appl., Volume 106 (2016), pp. 342-374

[28] Nguyen, H-M. Superlensing using complementary media and reflecting complementary media for electromagnetic waves, Adv. Nonlinear Anal. (2017) | DOI

[29] Nguyen, H-M. Cloaking an arbitrary object via anomalous localized resonance: the cloak is independent of the object: the acoustic case, SIAM J. Math. Anal., Volume 49 (2017), pp. 3208-3232

[30] Nguyen, H-M.; Nguyen, L. Generalized impedance boundary conditions for scattering by strongly absorbing obstacles for the full wave equation: the scalar case, Math. Models Methods Appl. Sci., Volume 25 (2015), pp. 1927-1960

[31] Nguyen, H-M.; Vogelius, M.S. Approximate cloaking for the full wave equation via change of variables: the Drude–Lorentz model, J. Math. Pures Appl., Volume 106 (2016), pp. 797-836

[32] Nicorovici, N.A.; McPhedran, R.C.; Milton, G.W. Optical and dielectric properties of partially resonant composites, Phys. Rev. B, Volume 49 (1994), pp. 8479-8482

[33] Nussenzveig, H.M. Causality and Dispersion Relations, Academic Press, New York, 1972

[34] Ola, P. Remarks on a transmission problem, J. Math. Anal. Appl., Volume 196 (1995), pp. 639-658

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

[36] Pendry, J.B. Perfect cylindrical lenses, Opt. Express, Volume 1 (2003), pp. 755-760

[37] Poddubny, A.; Iorsh, I.; Belov, P.; Kivshar, Y. Hyperbolic metamaterials, Nat. Photonics, Volume 7 (2013), pp. 948-957

[38] Ramakrishna, S.A.; Pendry, J.B. Spherical perfect lens: solutions of Maxwell's equations for spherical geometry, Phys. Rev. B, Volume 69 (2004)

[39] Shelby, R.A.; Smith, D.R.; Schultz, S. Experimental verification of a negative index of refraction, Science, Volume 292 (2001), pp. 77-79

[40] Sihvola, A.H. Electromagnetic modeling of bi-isotropic media, Prog. Electromagn. Res., Volume 9 (1994), pp. 45-86

[41] Tip, A. Linear absorptive dielectrics, Phys. Rev. A, Volume 57 (1998), p. 4818

[42] Veselago, V.G. The electrodynamics of substances with simultaneously negative values of ε and μ, Usp. Fiz. Nauk, Volume 92 (1964), pp. 517-526

[43] Vinoles, V. Problèmes d'interface en présence de métamatériaux: modélisation, analyse et simulations, Paris-Saclay University, 2016 (PhD thesis)

Cited by Sources: