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Comptes Rendus Mathématique
Volume 353, n° 1
pages 41-46 (janvier 2015)
Doi : 10.1016/j.crma.2014.10.014
Received : 25 August 2014 ;  accepted : 21 October 2014
Cloaking via anomalous localized resonance. A connection between the localized resonance and the blow up of the power for doubly complementary media
Invisibilité par résonance localisée anormale. Une liaison entre la résonance localisée et l'exposion de la puissance pour les milieux doublement complémentaires
 

Hoai-Minh Nguyen
 École polytechnique fédérale de Lausanne, SB MATHAA CAMA, Station 8, CH-1015 Lausanne, Switzerland 

Abstract

This paper is devoted to the study of cloaking via anomalous localized resonance (CALR) in two and three dimensions in the quasistatic regime. Two key figures of CALR are (i ) the localized resonance and (ii ) the connection between the localized resonance and the blow up of the power of the fields as the loss 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 and analyzed in [[8], [9], [10]]. It was shown in [[12]] that the complementary property of media is not enough to ensure such a connection. In this paper, we introduce a subclass of complementary media called the class of doubly complementary media. This class is rich enough to allow us to do cloaking via anomalous localized resonance for an arbitrary source concentrating on an arbitrary smooth bounded manifold of codimension 1 located in an arbitrary medium . The following three properties are established: 1) CALR appears if and only if the power blows up; 2) the power blows up if the source is “located” near the plasmonic structure; 3) 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 full text of this article is available in PDF format.
Résumé

Nous étudions l'invisibilité par résonance localisée anormale (CALR) en deux et trois dimensions en régime quasi-statique. Deux figures principales de CALR sont i ) la résonance localisée et ii ) la liaison entre la résonance localisée et l'exposion de la puissance quand la perte de la matériel tend vers 0. Une importante classe de matériels de l'indice négatif pour laquel la résonance localisée peut apparaître est la classe de milieux complémentaires introduite et analysée dans [[8], [9], [10]]. Il a été noté dans [[12]] que la propriété complémentaire ne suffit pas à assurer une telle liaison. Dans cette note, nous introduisons une sous-classe des milieux complémentaires s'appelée la classe des milieux doublement complémentaires. Cette classe est suffisament large pour accomplir l'invisibilité par résonance localisée anormale une source arbitraire concentrant sur une sous-variété arbitraire de codimension 1 placé dans un milieu arbitraire . Les trois propriétés suivantes sont établies : 1) CALR apparaît si et seulement si la puissance explose ; 2) la puissance explose si la source est « placée » près de la structure plasmonique ; 3) la puissance reste bornée si la source est loin de la structure plasmonique. Propriété P2), l'explosion de la puissance est en fait établie pour les milieux complémentaires .

The full text of this article is available in PDF format.
1    denotes   when  .
2  In ((6)) and ((7)), F and G denote some diffeomorphism extensions of F and G in a neighborhood of   and of  .
3  Recall that   is defined by ((3)).
4  In a recent discussion with Graeme Milton, we realize that the result of Kohn et al. on the boundedness of the power [[5]] can be derived from the radial setting using the Mobius transformation. We are grateful to him for the discussion.


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