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Resonance Meets Homogenization

Construction of Meta-Materials with Astonishing Properties

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Abstract

Meta-materials are assemblies of small components. Even though the single component consists of ordinary materials, the meta-material may behave effectively in a way that is not known from ordinary materials. In this text, we discuss some meta-materials that exhibit unusual properties in the propagation of sound or light. The phenomena are based on resonance effects in the small components. The small (sub-wavelength) components can be resonant to the wave-length of an external field if they incorporate singular features such as a high contrast or a singular geometry. Homogenization theory allows to derive effective equations for the macroscopic description of the meta-material and to verify its unusual properties. We discuss three examples: Sound-absorbing materials, optical materials with a negative index of refraction, perfect transmission through grated metals.

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References

  1. Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Allaire, G.: Dispersive limits in the homogenization of the wave equation. Ann. Fac. Sci. Toulouse Math. (6) 12(4), 415–431 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Allaire, G., Briane, M.: Multiscale convergence and reiterated homogenisation. Proc. R. Soc. Edinb. A 126(2), 297–342 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Allaire, G., Conca, C.: Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl. (9) 77(2), 153–208 (1998). doi:10.1016/S0021-7824(98)80068-8

    Article  MathSciNet  MATH  Google Scholar 

  5. Allaire, G., Piatnitski, A.: Homogenization of nonlinear reaction-diffusion equation with a large reaction term. Ann. Univ. Ferrara, Sez. 7: Sci. Mat. 56(1), 141–161 (2010). doi:10.1007/s11565-010-0095-z

    Article  MathSciNet  MATH  Google Scholar 

  6. Beale, J.T.: Scattering frequencies of reasonators. Commun. Pure Appl. Math. 26, 549–563 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bellieud, M., Bouchitté, G.: Homogenization of elliptic problems in a fiber reinforced structure. Nonlocal effects. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 26(3), 407–436 (1998)

    MathSciNet  MATH  Google Scholar 

  8. Bouchitté, G., Bellieud, M.: Homogenization of a soft elastic material reinforced by fibers. Asymptot. Anal. 32(2), 153–183 (2002)

    MathSciNet  MATH  Google Scholar 

  9. 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). doi:10.1016/j.crma.2009.02.027

    Article  MathSciNet  MATH  Google Scholar 

  10. Bouchitté, G., Felbacq, D.: Low frequency scattering by a set of parallel metallic rods. In: Mathematical and Numerical Aspects of Wave Propagation, Santiago de Compostela, 2000, S. 226–230. SIAM, Philadelphia, PA (2000)

    Google Scholar 

  11. Bouchitté, G., Felbacq, D.: Homogenization near resonances and artificial magnetism from dielectrics. C. R. Math. Acad. Sci. Paris 339(5), 377–382 (2004). doi:10.1016/j.crma.2004.06.018

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bouchitté, G., Schweizer, B.: Homogenization of Maxwell’s equations in a split ring geometry. Multiscale Model. Simul. 8(3), 717–750 (2010). doi:10.1137/09074557X

    Article  MathSciNet  MATH  Google Scholar 

  14. Bouchitté, G., Schweizer, B.: Plasmonic waves allow perfect transmission through sub-wavelength metallic gratings. Netw. Heterog. Media 8(4), 857–878 (2013). doi:10.3934/nhm.2013.8.857

    Article  MathSciNet  MATH  Google Scholar 

  15. Brown, R., Hislop, P.D., Martinez, A.: Eigenvalues and resonances for domains with tubes: Neumann boundary conditions. J. Differ. Equ. 115(2), 458–476 (1995). doi:10.1006/jdeq.1995.1023

    Article  MathSciNet  MATH  Google Scholar 

  16. Cao, Q., Lalanne, P.: Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits. Phys. Rev. Lett. 88, 057403 (2002). doi:10.1103/PhysRevLett.88.057403

    Article  Google Scholar 

  17. Chen, Y., Lipton, R.: Resonance and double negative behavior in metamaterials. Arch. Ration. Mech. Anal. 209(3), 835–868 (2013). doi:10.1007/s00205-013-0634-8

    Article  MathSciNet  MATH  Google Scholar 

  18. Cherednichenko, K.D., Smyshlyaev, V.P., Zhikov, V.V.: Non-local homogenized limits for composite media with highly anisotropic periodic fibres. Proc. R. Soc. Edinb. A 136(1), 87–114 (2006). doi:10.1017/S0308210500004455

    Article  MathSciNet  MATH  Google Scholar 

  19. Cherednichenko, K.D., Smyshlyaev, V.P., Zhikov, V.V.: Non-local homogenized limits for composite media with highly anisotropic periodic fibres. Proc. R. Soc. Edinb. A 136(1), 87–114 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cioranescu, D., Damlamian, A., Griso, G.: Periodic unfolding and homogenization. C. R. Math. Acad. Sci. Paris 335(1), 99–104 (2002). doi:10.1016/S1631-073X(02)02429-9

    Article  MathSciNet  MATH  Google Scholar 

  21. Dohnal, T., Lamacz, A., Schweizer, B.: Bloch-wave homogenization on large time scales and dispersive effective wave equations. Multiscale Model. Simul. 12(2), 488–513 (2014). doi:10.1137/130935033

    Article  MathSciNet  MATH  Google Scholar 

  22. Felbacq, D., Bouchitté, G.: Homogenization of a set of parallel fibres. Waves Random Media 7(2), 245–256 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Felbacq, D., Bouchitté, G.: Left handed media and homogenization of photonic crystals. Opt. Lett. 30, 1189–1191 (2005)

    Article  Google Scholar 

  24. Felbacq, D., Bouchitté, G.: Negative refraction in periodic and random photonic crystals. New J. Phys. 7, 159 (2005)

    Article  Google Scholar 

  25. Fernández, C., Menzala, G.P.: Resonances of an elastic resonator. Appl. Anal. 76(1–2), 41–49 (2000). doi:10.1080/00036810008840864

    Article  MathSciNet  MATH  Google Scholar 

  26. Gadyl’shin, R.R.: On analogues of the Helmholtz resonator in averaging theory. Mat. Sb. 193(11), 43–70 (2002). doi:10.1070/SM2002v193n11ABEH000691

    Article  MathSciNet  MATH  Google Scholar 

  27. Gadyl’shin, R.R.: On domains with a narrow isthmus in the critical case. Proc. Steklov Inst. Math. 1, S68–S74 (2003). Asymptotic Expansions Approximation Theory Topology, suppl. 1

    MathSciNet  MATH  Google Scholar 

  28. Holloway, C.L., Kuester, E.F., Baker-Jarvis, J., Kabos, P.: A double negative (dng) composite medium composed of magnetodielectric spherical particles embedded in a matrix. IEEE Trans. Antennas Propag. 51(10), 2596–2603 (2003). doi:10.1109/TAP.2003.817563

    Article  Google Scholar 

  29. Jikov, V.V., Kozlov, S.M., Oleĭnik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994). Translated from the Russian by G.A. Yosifian [G.A. Iosifyan]

    Book  Google Scholar 

  30. Joly, P., Tordeux, S.: Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots. M2AN Math. Model. Numer. Anal. 40(1), 63–97 (2006). doi:10.1051/m2an:2006008

    Article  MathSciNet  MATH  Google Scholar 

  31. Joly, P., Tordeux, S.: Matching of asymptotic expansions for waves propagation in media with thin slots. II. The error estimates. M2AN Math. Model. Numer. Anal. 42(2), 193–221 (2008). doi:10.1051/m2an:2008004

    Article  MathSciNet  MATH  Google Scholar 

  32. Kohn, R., Shipman, S.: Magnetism and homogenization of micro-resonators. Multiscale Model. Simul. 7(1), 62–92 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. 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). doi:10.1137/120874321

    Article  MathSciNet  MATH  Google Scholar 

  34. Lamacz, A., Schweizer, B.: A negative index meta-material for Maxwell’s equations. Preprint 2015-06 of the TU Dortmund (2015, submitted). http://hdl.handle.net/2003/34176

  35. Lamacz, A., Schweizer, B.: Effective acoustic properties of a meta-material consisting of small Helmholtz resonators. Tech. rep., TU Dortmund (2016)

  36. Lipton, R., Schweizer, B.: Negative magnetic response in a material with perfectly conducting split rings (2016, in preparation)

  37. Milton, G.W.: Realizability of metamaterials with prescribed electric permittivity and magnetic permeability tensors. New J. Phys. 12(3), 033035 (2010)

    Article  Google Scholar 

  38. Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20(3), 608–623 (1989). doi:10.1137/0520043

    Article  MathSciNet  MATH  Google Scholar 

  39. O’Brien, S., Pendry, J.: Magnetic activity at infrared frequencies in structured metallic photonic crystals. J. Phys. Condens. Matter 14, 6383–6394 (2002)

    Article  Google Scholar 

  40. Pendry, J.: Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 3966 (2000)

    Article  Google Scholar 

  41. Porto, J.A., Garcia-Vidal, F.J., Pendry, J.B.: Transmission resonances on metallic gratings with very narrow slits. Phys. Rev. Lett. 83, 2845–2848 (1999). doi:10.1103/PhysRevLett.83.2845

    Article  Google Scholar 

  42. Sánchez-Palencia, E.: Nonhomogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 127. Springer, Berlin (1980)

    MATH  Google Scholar 

  43. Schweizer, B.: The low-frequency spectrum of small Helmholtz resonators. Proc. R. Soc. A 471(2174), 20140339 (2015). doi:10.1098/rspa.2014.0339

    Article  MathSciNet  Google Scholar 

  44. Shelby, R.A., Smith, D.R., Schultz, S.: Experimental verification of a negative index of refraction. Science 292(5514), 77–79 (2001). doi:10.1126/science.1058847

    Article  Google Scholar 

  45. Smith, D., Pendry, J., Wiltshire, M.: Metamaterials and negative refractive index. Science 305, 788–792 (2004)

    Article  Google Scholar 

  46. Veselago, V.: The electrodynamics of substances with simultaneously negative values of \(\varepsilon \) and \(\mu \). Sov. Phys. Usp. 10, 509–514 (1968)

    Article  Google Scholar 

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  1. Fakultät für Mathematik, TU Dortmund, Vogelpothsweg 87, 44227, Dortmund, Germany

    Ben Schweizer

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Schweizer, B. Resonance Meets Homogenization. Jahresber. Dtsch. Math. Ver. 119, 31–51 (2017). https://doi.org/10.1365/s13291-016-0153-2

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