# zbMATH — the first resource for mathematics

Resonance and double negative behavior in metamaterials. (English) Zbl 1301.35165
A transmission eigenvalue problem in a unit square with periodic boundary conditions is considered for an operator of the form $$-\nabla\cdot\gamma\nabla$$, where the scalar multiplication operator $$\gamma$$ is allowed to change the sign in a sub-region $$P$$ of the unit square $$Y$$, is considered. Outside of $$P$$ the coefficient is piecewise constant with pieces $$R$$ (large constant) and $$H$$ (constant 1). The study is motivated by issues associated with so-called electromagnetic metamaterials, where material properties encoded in $$\gamma$$, which for classical materials would be strictly positive definite, are generalized to allow for sign changing $$\gamma$$. The problem is approached by a (initially formal) power series ansatz for a solution and the eigenparameter with respect to an introduced oscillation parameter $$\eta$$ producing a cascade of equations for the coefficient functions. The resulting equations involve a continuous linear operator $$T_{z}=\left(\nabla\cdot\nabla\right)^{-1}\nabla\cdot\gamma_{z}\nabla$$, where $$\nabla$$ acts on periodic functions with weak $$L^{2}$$-derivatives restricted to the subset $$Y\setminus\overline{R}=H\cup P$$ of the unit square, which are perpendicular to $$1$$, and $$-\nabla \cdot$$ denoting the corresponding adjoint $$\nabla^{*}$$. Here the coefficient $$\gamma_{z}$$ is specified by $$\gamma_{z} := z$$ for $$x\in P$$ and $$1$$ otherwise. The spectral properties of the operator $$T_{z}$$, which for real $$z$$ is selfadjoint with pure point spectrum, are studied in detail and a spectral representation is given. The analysis of $$T_{z}$$ is linked to eigenvalue problems associated with the special case $$z=-1$$. Based on an invertibility result for $$T_{z}$$ solvability of the cascading system of equations can be established recursively. Finally, convergence of the power series ansatz is established.

##### MSC:
 35Q60 PDEs in connection with optics and electromagnetic theory 35J70 Degenerate elliptic equations 78A48 Composite media; random media in optics and electromagnetic theory
##### Keywords:
double negative index material; metamaterials; Bloch waves
Full Text:
##### References:
 [1] Alu, A., Engheta, N: Dynamical theory of artificial optical magnetism produced by rings of plasmonic nanoparticles. Phys. Rev. B78, 085112 (2008) [2] Adams, D.: Sobolev Spaces. Elsevier, Waltham, 2003 · Zbl 1098.46001 [3] Ammari, H.; Kang, H.; Lee, H., Asymptotic analysis of high-contrast phononic crystals and a criterion for the band-gap opening, Arch. Ration. Mech. Anal., 193, 679-714, (2009) · Zbl 1170.74023 [4] Ammari, H., Kang, H., Soussi, S., Zribi, H.: Layer potential techniques in spectral analysis. Part II: sensitivity analysis of spectral properties of high contrast band-gap materials. SIAM Multiscale Model. Simul. 5, 646-663 (2006) · Zbl 1115.81072 [5] Bergman, D.J., The dielectric constant of a simple cubic array of identical spheres, J. Phys. C, 12, 4947-4960, (1979) [6] Bohren, C.F., Huffman, D.H.: Absorption and Scattering of Light by Small Particles. Wiley, Weinheim, 2004 [7] Bouchitté, G., Bourel, C.: Homogenization of finite metallic fibers and 3D-effective permittivity tensor. Proc. SPIE7029, 702914 (2008). doi:10.1117/12.794935 [8] Bouchitté, G., Schweizer, B.: Homogenization of Maxwell’s equations in a split ring geometry. SIAM Multiscale Model. Simul. 8(3), 717-750 (2010) · Zbl 1228.35028 [9] Bouchitté, G.; Felbacq, D., Negative refraction in periodic and random photonic crystals, New J. Phys., 7, 159, (2005) [10] Bouchitté, G.; Felbacq, D., Homogenization near resonances and artificial magnetism from dielectrics, C. R. Acad. Sci. Paris I, 399, 377-382, (2004) · Zbl 1055.35016 [11] Chen, Y.; Lipton, R., Tunable double negative band structure from non-magnetic coated rods, New J. Phys., 12, 083010, (2010) [12] Cherdantsev, M., Spectral convergence for high-contrast elliptic periodic problems with a defect via homogenization., Mathematika, 55, 29-57, (2009) · Zbl 1196.35036 [13] Chern, R.L.; Felbacq, D., Artificial magnetism and anticrossing interaction in photonic crystals and split-ring structures, Phys. Rev. B, 79, 075118, (2009) [14] Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Springer, New York, 1998 · Zbl 0893.35138 [15] Dolling, G.; Enrich, C.; Wegener, M.; Soukoulis, C.M.; Linden, S., Low-loss negative-index metamaterial at telecommunication wavelengths, Opt. Lett., 31, 1800-1802, (2006) [16] Felbacq, D.; Bouchitte, G., Homogenization of wire mesh photonic crystals embedded in a medium with a negative permeability, Phys. Rev. Lett., 94, 183902, (2005) [17] Felbacq, D.; Guizal, B.; Bouchitte, G.; Bourel, G., Resonant homogenization of a dielectric metamaterial, Microw. Opt. Technol. Lett., 51, 2695-2701, (2009) [18] Farhat, M.; Guenneau Enoch, S.; Movchan, A.B., Negative refraction, surface modes, and superlensing effect via homogenization near resonances for a finite array of split-ring resonators, Phys. Rev. E, 80, 046309, (2009) [19] Figotin A.; Kuchment, P., Band-gap structure of the spectrum of periodic dielectric and acoustic media. I. scalar model, SIAM J. Appl. Math., 56, 68-88, (1996) · Zbl 0852.35014 [20] Figotin, A.; Kuchment, P., Band-gap structure of the spectrum of periodic dielectric and acoustic media. II. 2D photonic crystals, SIAM J. Appl. Math., 56, 1561-1620, (1996) · Zbl 0868.35009 [21] Figotin, A.; Kuchment, P., Spectral properties of classical waves in high contrast periodic media, SIAM J. Appl. Math., 58, 683-702, (1998) · Zbl 0916.35011 [22] Folland, G.: Introduction to Partial Differential Equations. Princeton University Press, Princeton, 1995 · Zbl 0841.35001 [23] Fortes, S.P., Lipton, R.P., Shipman, S.P.: Sub-wavelength plasmonic crystals: dispersion relations and effective properties. Proc. R. Soc. Lond. Ser. A Mat. (2009). doi:10.1098/rspa.2009.0542 · Zbl 1190.82045 [24] Fortes, S.P.; Lipton, R.P.; Shipman, S.P., Convergent power series for fields in positive or negative high-contrast periodic media, Commun. Partial Differ. Equ., 36, 1016-1043, (2011) · Zbl 1231.35245 [25] Hempel, R.; Lienau, K., Spectral properties of periodic media in the large coupling limit, Commun. Partial Differ. Equ., 25, 1445-1470, (2000) · Zbl 0958.35089 [26] Huangfu, J., Ran, L., Chen, H., Zhang, X., Chen, K., Grzegorczyk, T.M., Kong, J.A.: Experimental confirmation of negative refractive index of a metamaterial composed of Ω-like metallic patterns. Appl. Phys. Lett. 84, 1537 (2004) · Zbl 0916.35011 [27] Huang, K.C.; Povinelli, M.L.; Joannopoulos, J.D., Negative effective permeability in polaritonic photonic crystals, Appl. Phys.Lett., 85, 543, (2004) [28] Jelinek, L.; Marques, R., Artificial magnetism and left-handed media from dielectric rings and rods. J. phys, Condens. Matter., 22, 025902, (2010) [29] Kamotski, V.; Matthies, M.; Smyshlyaev, V., Exponential homogenization of linear second order elliptic PDEs with periodic coefficients, SIAM J. Math. Anal., 38, 1565-1587, (2007) · Zbl 1213.35062 [30] Kohn, R.; Shipman, S., Magnetism and homogenization of micro-resonators, SIAM Multiscale Model. Simul., 7, 62-92, (2008) · Zbl 1159.78350 [31] Lepetit, T.; Akmansoy, E., Magnetism in high-contrast dielectric photonic crystals, Microw. Opt. Technol. Lett., 50, 909-911, (2008) [32] McPhedran, R.C.; Milton, G.W., Bounds and exact theories for transport properties of inhomogeneous media, Physics A, 26, 207-220, (1981) [33] Milton, G.W., Realizability of metamaterials with prescribed electric permittivity and magnetic permeability tensors, New J. Phys., 12, 033035, (2010) · Zbl 0970.37001 [34] Milton, G.W.: The Theory of Composites. Cambridge University Press, Cambridge, 2002 · Zbl 0993.74002 [35] O’Brien, S.; Pendry, J.B., Photonic band-gap effects and magnetic activity in dielectric composites, J. Phys. Condens. Matter., 14, 4035-4044, (2002) [36] PendryJ.; Holden, A.; Robbins, D.; Stewart, W., Magnetisim from conductors and enhanced nonlinear phenomena, IEEE Trans. Microw. Theory Tech., 47, 2075-2084, (1999) [37] Pendry, J.; Holden, A.; Robbins, D.; Stewart, W., Low frequency plasmons in thin-wire structures. J. phys, Condens. Matter., 10, 4785-4809, (1998) [38] Peng, L.; Ran, L.; Chen, H.; Zhang, H.; Kong, L.A.; Grzegorczyk, T.M., Experimental observation of left-handed behavior in an array of standard dielectric resonators, Phys. Rev. Lett., 98, 157403, (2007) [39] Shvets, G., Urzhumov, Y.: Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances. Phys. Rev. Lett. 93(24), 243902-1-4 (2004) [40] Shipman, S.: Power series for waves in micro-resonator arrays. Proceedings of the 13th International Conference on Mathematical Methods in Electrodynamic Theory, MMET 10, Kyiv, Ukraine, September 6-8, IEEE, 2010 [41] Veselago, V.G.: The electrodynamics of substances with simultaneously negative values of $$ε$$ and $$μ$$. Sov. Phys. Usp. 10, 509 (1968) [42] Vynck, K.; Felbacq, D.; Centeno, E.; Cabuz, A.I.; Cassagne, D.; Guizal, B., All-dielectric rod-type metamaterials at optical frequencies, Phys. Rev. Lett., 102, 133901, (2009) [43] Service, R.F.: Next Wave of metamaterials hopes to fuel the revolution. Science327(5962), 138-139 (2010) [44] Shalaev, V., Optical negative-index metamaterials, Nat. Photonics, 1, 41-48, (2007) [45] Shalaev, V.M.; Cai, W.; Chettiar, U.K.; Yuan, H.K.; Sarychev, A.K.; Drachev, V.P.; Kildishev, A.V., Negative index of refraction in optical metamaterials, Opt. Lett., 30, 3356-3358, (2005) [46] Smith, D., Padilla, W., Vier, D., Nemat-Nasser, S., Schultz, S.: Composite medium with simultaneously negative permeability and permittivity. Phys. Rev. Lett. 84(18) 4184-4187 (2000) [47] Wheeler, M.S.; Aitchison, J.S.; Mojahedi, M., Coated non-magnetic spheres with a negative index of refraction at infrared frequencies, Phys. Rev. B, 73, 045105, (2006) [48] Yannopapas, V.: Negative refractive index in the near-UV from Au-coated CuCl nanoparticle superlattices. Phys. Stat. Sol. (RRL)1(5), 208-210 (2007) [49] Yannopapas, V., Artificial magnetism and negative refractive index in three-dimensional metamaterials of spherical particles at near-infrared and visible frequencies, Appl. Phys. A, 87, 259-264, (2007) [50] Zhang, F., Potet, S., Carbonell, J., Lheurette, E., Vanbesien, O., Zhao, X., Lippens, D.: Negative-zero-positive refractive index in a prism-like omega-type metamaterial. IEEE Trans. Microw. Theory Tech. 56, 2566 (2008) [51] Zhang, S.; Fan, W.; Minhas, B.K.; Frauenglass, A.; Malloy, K.J.; Brueck, S.R.J., Midinfrared resonant magnetic nanostructures exhibiting a negative permeability, Phys. Rev. Lett., 94, 037402, (2005) [52] Zhikov, V.V.: Gaps in the spectrum of some elliptic operators in divergent form with periodic coefficients. Algebra i Analiz16, 34-58 (2004) [English translation St. Petersburg Math. J. 16(5), 773-790 (2005)] [53] Zhou, X.; Zhao, X.P., Resonant condition of unitary dendritic structure with overlapping negative permittivity and permeability. appl, Phys. Lett., 91, 181908, (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.