Archive for Rational Mechanics and Analysis

, Volume 229, Issue 3, pp 1197–1221 | Cite as

Effective Maxwell’s Equations for Perfectly Conducting Split Ring Resonators

  • Robert Lipton
  • Ben Schweizer


We analyze the time harmonic Maxwell’s equations in a geometry containing perfectly conducting split rings. We derive the homogenization limit in which the typical size η of the rings tends to zero. The split rings act as resonators and the assembly can act, effectively, as a magnetically active material. The frequency dependent effective permeability of the medium can be large and/or negative.


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  1. 1.
    Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci., 21(9):823–864, 1998.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bensoussan, A., Lions, J.-L., Papanicolaou, G.C.: Homogenization in deterministic and stochastic problems. In Stochastic problems in dynamics (Sympos., Univ. Southampton, Southampton, 1976), pages 106–115. Pitman, London, 1977Google Scholar
  4. 4.
    Bouchitté, G., Bourel, C., Felbacq, D.: Homogenization of the 3D Maxwell system near resonances and artificial magnetism. C. R. Math. Acad. Sci. Paris, 347(9–10):571–576, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bouchitté, G., Felbacq, D.: Homogenization near resonances and artificial magnetism from dielectrics. C. R. Math. Acad. Sci. Paris, 339(5):377–382, 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bouchitté, G., Felbacq, D.: Homogenization of a wire photonic crystal: the case of small volume fraction. SIAM J. Appl. Math., 66(6):2061–2084, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bouchitté, G., Schweizer, B.: Homogenization of Maxwell's equations in a split ring geometry. Multiscale Model. Simul., 8(3):717–750, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bouchitté, G., Schweizer, B.: Plasmonic waves allow perfect transmission through sub-wavelength metallic gratings. Netw. Heterog. Media, 8(4):857–878, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, Y., Lipton, R.: Tunable double negative band structure from non-magnetic coated rods. New Journal of Physics, 12(8):083010, 2010.ADSCrossRefGoogle Scholar
  10. 10.
    Chen, Y., Lipton, R.: Double negative dispersion relations from coated plasmonic rods. Multiscale Modeling and Simulation, 209(3):835–868, 2013.MathSciNetzbMATHGoogle Scholar
  11. 11.
    Chen, Y., Lipton, R.: Multiscale methods for engineering double negative metamaterials. Photonics and Nanostructures Fundamentals and Applications, 11(4):442–454, 2013.ADSCrossRefGoogle Scholar
  12. 12.
    Chen, Y., Lipton, R.: Resonance and double negative behavior in metamaterials. Arch. Ration. Mech. Anal., 11(1):192–212, 2013.MathSciNetzbMATHGoogle Scholar
  13. 13.
    Costabel, M.: A coercive bilinear form for Maxwell's equations. J. Math. Anal. Appl., 157(2):527–541, 1991.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Felbacq, D., Bouchitté, G.: Homogenization of a set of parallel fibres. Waves Random Media, 7(2):245–256, 1997.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jikov, V., Kozlov, S., Oleinik, O.: Homogenization of Differential Operators and Integral Functionals. Springer, New York, 1994.CrossRefzbMATHGoogle Scholar
  16. 16.
    Kafesaki, M., Tsiapa, I., Katsarakis, N., Koschny, Th., Soukoulis, C.M., Economou, E.N.: Left-handed metamaterials: The fishnet structure and its variations. Phys. Rev. B 75:235114–1–235114–9, 2007Google Scholar
  17. 17.
    Kohn, R.V., Lu, J., Schweizer, B., Weinstein, M.I.: A variational perspective on cloaking by anomalous localized resonance. Comm. Math. Phys., 328(1):1–27, 2014.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kohn, R.V., Shipman, S.: Magnetism and homogenization of micro-resonators. Multiscale Model. Simul., 7(1):62–92, 2007.CrossRefzbMATHGoogle Scholar
  19. 19.
    Lamacz, A., Schweizer, B.: Effective Maxwell equations in a geometry with flat rings of arbitrary shape. SIAM J. Math. Anal., 45(3):1460–1494, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lamacz, A., Schweizer, B.: Effective acoustic properties of a meta-material consisting of small Helmholtz resonators. DCDS-S, 2016Google Scholar
  21. 21.
    Lamacz, A., Schweizer, B.: A negative index meta-material for Maxwell's equations. SIAM J. Math. Anal., 48(6):4155–4174, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lipton, R., Polizzi, A., Thakur, L.: Novel metamaterial surfaces from perfectly conducting subwavelength corrugations. SIAM J. Appl. Math., 77(4):1269–1291, 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Milton, G.W., Nicorovici, N.-A.P.: On the cloaking effects associated with anomalous localized resonance. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462:3027–3059, 2006.zbMATHGoogle Scholar
  24. 24.
    Movchan, A.B., Guenneau, S.: Split-ring resonators and localized modes. Phys. Rev. B, 70(12):125116, 2004.ADSCrossRefGoogle Scholar
  25. 25.
    Pendry, J.B.: Negative refraction makes a perfect lens. Phys. Rev. Lett., 85:3966, 2000.ADSCrossRefGoogle Scholar
  26. 26.
    Pendry, J.B., Schurig, D., Smith, D.R.: Controlling electromagnetic fields. Science, 312:5781, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Schweizer, B.: Resonance meets homogenization: construction of meta-materials with astonishing properties. Jahresber. Dtsch. Math. Ver., 119(1):31–51, 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Schweizer, B.: Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma. INdAM-Springer series, Trends on Applications of Mathematics to Mechanics, 2018.CrossRefGoogle Scholar
  29. 29.
    Schweizer, B., Urban, M.: Effective maxwells equations in general periodic microstructures. Appl. Anal., 0(0):1–21, 2017.Google Scholar
  30. 30.
    Sellier, A., Burokur, S. N., Kanté, B., de Lustrac, A.: Novel cut wires metamaterial exhibiting negative refractive index. IEEE, 2009Google Scholar
  31. 31.
    Shelby, R.A., Smith, D.R., Schultz, S.: Experimental verification of a negative index of refraction. Science, 292(5514):77–79, 2001.ADSCrossRefGoogle Scholar
  32. 32.
    Veselago, V.G.: The electrodynamics of substances with simultaneously negative values of ε and μ Sov. Phys. Uspekhi, 10:509–514, 1968.ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA
  2. 2.Fakultät für MathematikTU DortmundDortmundGermany

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