staple
With cedram.org
logo JEP
Table of contents for this volume | Previous article
Eric Bonnetier; Hoai-Minh Nguyen
Superlensing using hyperbolic metamaterials: the scalar case
(Propriété de superlensing de dispositifs constitués de méta-matériaux hyperboliques : le cas scalaire)
Journal de l'École polytechnique — Mathématiques, 4 (2017), p. 973-1003, doi: 10.5802/jep.61
Article PDF | TeX source
Class. Math.: 35B30, 35B40, 35J05, 35J70, 35M10, 35L53, 78A25
Keywords: Negative index materials, hyperbolic meta-materials, superlensing, degenerate elliptic equations

Résumé - Abstract

This paper is devoted to superlensing using hyperbolic metamaterials: the possibility to image an arbitrary object using hyperbolic metamaterials without imposing any conditions on the size of the object and the wave length. To this end, two types of schemes are suggested and their analysis are given. The superlensing devices proposed are independent of the object. It is worth noting that the study of hyperbolic metamaterials is challenging due to the change of type of the modeling equations, elliptic in some regions, hyperbolic in some others.

Bibliography

[1] H. Ammari, G. Ciraolo, H. Kang, H. Lee & G. W. Milton, “Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance”, Arch. Rational Mech. Anal. 208 (2013) no. 2, p. 667-692 Article |  MR 3035988
[2] G. Bouchitté & B. Schweizer, “Cloaking of small objects by anomalous localized resonance”, Quart. J. Mech. Appl. Math. 63 (2010) no. 4, p. 437-463 Article
[3] D. G. Bourgin & R. Duffin, “The Dirichlet problem for a vibrating string equation”, Bull. Amer. Math. Soc. 45 (1939), p. 851-859 Article
[4] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011
[5] A. Cherkaev, R. V. Kohn (ed.), Topics in the mathematical modelling of composite materials, Progress in Nonlinear Differential Equations and their Applications 31, Birkhäuser Boston, Inc., Boston, MA, 1997
[6] J. Droxler, J. Hesthaven & H-M. Nguyen, In preparation
[7] P. Grisvard, Elliptic problems in nonsmooth domains, Classics in Applied Mathematics 69, SIAM, Philadelphia, PA, 2011
[8] Z. Jacob, L. V. Alekseyev & E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit”, Optics Express 14 (2006), p. 8247-8256 Article
[9] F. John, “The Dirichlet problem for a hyperbolic equation”, Amer. J. Math. 63 (1941), p. 141-154 Article
[10] R. V. Kohn, J. Lu, B. Schweizer & M. I. Weinstein, “A variational perspective on cloaking by anomalous localized resonance”, Comm. Math. Phys. 328 (2014) no. 1, p. 1-27 Article
[11] Y. Lai, H. Chen, Z. Zhang & C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell”, Phys. Rev. Lett. 102 (2009), 093901 Article
[12] Z. Liu, H. Lee, C. Sun & Z. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects”, Science 315 (2007), p. 1686-1686 Article
[13] G. W. Milton & N.-A. Nicorovici, “On the cloaking effects associated with anomalous localized resonance”, Proc. Roy. Soc. London Ser. A 462 (2006) no. 2074, p. 3027-3059 Article
[14] H-M. Nguyen, “Asymptotic behavior of solutions to the Helmholtz equations with sign changing coefficients”, Trans. Amer. Math. Soc. 367 (2015) no. 9, p. 6581-6595 Article
[15] H-M. Nguyen, “Superlensing using complementary media”, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015) no. 2, p. 471-484 Article
[16] H-M. Nguyen, “Cloaking via anomalous localized resonance for doubly complementary media in the quasistatic regime”, J. Eur. Math. Soc. (JEMS) 17 (2015) no. 6, p. 1327-1365 Article
[17] H-M. Nguyen, “Localized and complete resonance in plasmonic structures”, ESAIM Math. Model. Numer. Anal. 49 (2015) no. 3, p. 741-754 Article
[18] H-M. Nguyen, “Cloaking using complementary media in the quasistatic regime”, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016) no. 6, p. 1509-1518 Article
[19] H-M. Nguyen, “Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients”, J. Math. Pures Appl. (9) 106 (2016) no. 2, p. 342-374 Article
[20] H-M. Nguyen, “Reflecting complementary and superlensing using complementary media for electromagnetic waves”, arXiv:1511.08050, 2015
[21] H-M. Nguyen, “Cloaking via anomalous localized resonance for doubly complementary media in the finite frequency regime”, J. Anal. Math. (to appear), arXiv:1511.08053
[22] H-M. Nguyen, “Cloaking an arbitrary object via anomalous localized resonance: the cloak is independent of the object: the acoustic case”, SIAM J. Math. Anal. (to appear), arXiv:1607.06492
[23] H-M. Nguyen & L. H. Nguyen, “Cloaking using complementary media for the Helmholtz equation and a three spheres inequality for second order elliptic equations”, Trans. Amer. Math. Soc. Ser. B 2 (2015), p. 93-112 Article
[24] N. A. Nicorovici, R. C. McPhedran & G. W. Milton, “Optical and dielectric properties of partially resonant composites”, Phys. Rev. B 49 (1994), p. 8479-8482 Article
[25] J. B. Pendry, “Negative refraction makes a perfect lens”, Phys. Rev. Lett. 85 (2000), p. 3966-3969 Article
[26] J. B. Pendry, “Perfect cylindrical lenses”, Optics Express 1 (2003), p. 755-760 Article
[27] A. Poddubny, I. Iorsh, P. Belov & Y. Kivshar, “Hyperbolic metamaterials”, Nature Photonics 7 (2013), p. 948-957 Article
[28] M. H. Protter, “Unique continuation for elliptic equations”, Trans. Amer. Math. Soc. 95 (1960), p. 81-91 Article
[29] S. A. Ramakrishna & J. B. Pendry, “Spherical perfect lens: solutions of Maxwell’s equations for spherical geometry”, Phys. Rev. B 69 (2004), 115115
[30] V. Veselago, “The electrodynamics of substances with simultaneously negative values of $\varepsilon $ and $\mu $”, Uspehi Fiz. Nauk 92 (1964), p. 517-526 Article