Cloaking via anomalous localized resonance for doubly complementary media in the quasistatic regime

This paper is devoted to the study of cloaking via anomalous localized resonance (CALR) in the two- and three-dimensional quasistatic regimes. CALR associated with negative index materials was discovered by Milton and Nicorovici [21] for constant plasmonic structures in the two- dimensional quasistatic regime. Two key features of this phenomenon are the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others, and the connection between the localized resonance and the blow up of the power of the fields as the loss of the material goes to 0. An important class of negative index materials for which the localized resonance might appear is the class of reflecting complementary media introduced in [24]. It was shown in [29] that the complementarity property is not enough to ensure a connection between the blow up of the power and the localized resonance. In this paper, we study CALR for a subclass of complementary media called doubly complementary media. This class is rich enough to allow us to cloak an arbitrary source concentrating on an arbitrary smooth bounded manifold of codimension 1 placed in an arbitrary medium via anomalous localized resonance; the cloak is independent of the source. The following three properties are established for doubly complementary media: P1. CALR appears if and only if the power blows up; P2. The power blows up if the source is located "near" the plasmonic structure; P3. The power remains bounded if the source is far away from the plasmonic structure. Property P2, the blow up of the power, is in fact established for reflecting complementary media. The proofs are based on several new observations and ideas. One of the difficulties is to handle the localized resonance. To this end, we extend the reflecting and removing localized singularity techniques introduced in [24-26], and implement the separation of variables for Cauchy problems for a general shell. The results in this paper are inspired by and imply recent ones of Ammari et al. [3] and Kohn et al. [16] in two dimensions and extend theirs to general non-radial core-shell structures in both two and three dimensions.

[1]  S. Anantha Ramakrishna,et al.  Focusing light using negative refraction , 2003 .

[2]  V. Veselago.  The Electrodynamics of Substances with Simultaneously Negative Values of ∊ and μ , 1968 .

[3]  Viktor A. Podolskiy,et al.  A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance , 2005, Proceedings of the Royal Society A.

[4]  이현대,et al.  Spectral Theory of a Neumann-Poincare-Type Operator and Analysis of Cloaking Due to Anomalous Localized Resonance , 2013 .

[5]  Hoai-Minh Nguyen,et al.  Superlensing using complementary media , 2013, 1311.5700.

[6]  J. Pendry,et al.  Negative refraction makes a perfect lens , 2000, Physical review letters.

[7]  Hoai-Minh Nguyen,et al.  Cloaking via change of variables for the Helmholtz equation in the whole space , 2010 .

[8]  S. Maier.  Plasmonics: Fundamentals and Applications , 2007 .

[9]  Daniel Grieser,et al.  The plasmonic eigenvalue problem , 2012, 1208.3120.

[10]  H. Brezis,et al.  Ginzburg-Landau Vortices , 1994 .

[11]  Hoai-Minh Nguyen,et al.  Superlensing using complementary media and reflecting complementary media for electromagnetic waves , 2015, Advances in Nonlinear Analysis.

[12]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[13]  Hoai-Minh Nguyen,et al.  Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients , 2015, 1507.01730.

[14]  Hoai-Minh Nguyen.  Cloaking via anomalous localized resonance. A connection between the localized resonance and the blow up of the power for doubly complementary media , 2015 .

[15]  Milton,et al.  Optical and dielectric properties of partially resonant composites. , 1994, Physical review. B, Condensed matter.

[16]  Martin Costabel,et al.  A direct boundary integral equation method for transmission problems , 1985 .

[17]  S Enoch,et al.  Finite wavelength cloaking by plasmonic resonance , 2008, 0805.1295.

[18]  Hoai-Minh Nguyen.  Cloaking via anomalous localized resonance for doubly complementary media in the finite frequency regime , 2015, Journal d'Analyse Mathématique.

[19]  Guy Bouchitté,et al.  Cloaking of Small Objects by Anomalous Localized Resonance , 2010 .

[20]  S. Müller,et al.  Estimates for Green's matrices of elliptic systems byLp theory , 1995 .

[21]  Ross C. McPhedran,et al.  Solutions in folded geometries, and associated cloaking due to anomalous resonance , 2008, 0804.3903.

[22]  Hoai-Minh Nguyen,et al.  Cloaking using complementary media in the quasistatic regime , 2013, 1310.5483.

[23]  Received May,et al.  Cloaking via anomalous localized resonance for doubly complementary media in the quasistatic regime , 2015 .

[24]  M. H. Protter,et al.  UNIQUE CONTINUATION FOR ELLIPTIC EQUATIONS( , 1960 .

[25]  G. Milton,et al.  On the cloaking effects associated with anomalous localized resonance , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[26]  Lucas Chesnel,et al.  T-COERCIVITY FOR SCALAR INTERFACE PROBLEMS BETWEEN DIELECTRICS AND METAMATERIALS , 2011 .

[27]  Habib Ammari,et al.  Anomalous localized resonance using a folded geometry in three dimensions , 2013, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[28]  Andrew E. Thaler,et al.  Sensitivity of anomalous localized resonance phenomena with respect to dissipation , 2014, 1406.7044.

[29]  Luca Rondi,et al.  The stability for the Cauchy problem for elliptic equations , 2009, 0907.2882.

[30]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[31]  R. Shelby,et al.  Experimental Verification of a Negative Index of Refraction , 2001, Science.

[32]  Oscar P. Bruno,et al.  Superlens-cloaking of small dielectric bodies in the quasistatic regime , 2007 .

[33]  Hoai-Minh Nguyen.  Asymptotic behavior of solutions to the Helmholtz equations with sign changing coefficients , 2012, 1204.1518.

[34]  Robert V. Kohn,et al.  A Variational Perspective on Cloaking by Anomalous Localized Resonance , 2012, Communications in Mathematical Physics.

[35]  Daniel Grieser,et al.  Perturbation theory for plasmonic eigenvalues , 2009, 1103.5212.

[36]  Michael Grüter,et al.  The Green function for uniformly elliptic equations , 1982 .

[37]  Huanyang Chen,et al.  Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell. , 2008, Physical review letters.

[38]  U. Leonhardt,et al.  General relativity in electrical engineering , 2006, SPIE Optics + Optoelectronics.

[39]  Kyoungsun Kim,et al.  Cloaking Due to Anomalous Localized Resonance in Plasmonic Structures of Confocal Ellipses , 2013, SIAM J. Appl. Math..

[40]  N. Engheta,et al.  Achieving transparency with plasmonic and metamaterial coatings. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[41]  Wolfgang Desch,et al.  Progress in nonlinear differential equations and their applications, Vol. 80 , 2011 .

[42]  Habib Ammari,et al.  Spectral Theory of a Neumann–Poincaré-Type Operator and Analysis of Cloaking Due to Anomalous Localized Resonance , 2011, 1212.5066.

[43]  L. Nguyen,et al.  Cloaking using complementary media for the Helmholtz equation and a three spheres inequality for second order elliptic equations , 2015, 1504.07378.