Spectral Theory of a Neumann–Poincaré-Type Operator and Analysis of Cloaking Due to Anomalous Localized Resonance

Abstract

If a body of dielectric material is coated by a plasmonic structure of negative dielectric constant with nonzero loss parameter, then cloaking by anomalous localized resonance (CALR) may occur as the loss parameter tends to zero. The aim of this paper is to investigate this phenomenon in two and three dimensions when the coated structure is radial, and the core, shell and matrix are isotropic materials. In two dimensions, we show that if the real part of the permittivity of the shell is −1 (under the assumption that the permittivity of the background is 1), then CALR takes place. If it is different from −1, then CALR does not occur. In three dimensions, we show that CALR does not occur. The analysis of this paper reveals that occurrence of CALR is determined by the eigenvalue distribution of the Neumann-Poincaré-type operator associated with the structure.

Cloaking via anomalous localized resonance. A connection between the localized resonance and the blow up of the power for doubly complementary media

“…The study of CALR has been mainly based on separation of variables see [1][2][3][4]6,7]. In addition to the separation of variables technique, there are two methods suggested by Ammari et al in [1] and Kohn et al in [5] to study the blow up of the power.…”

“…In addition to the separation of variables technique, there are two methods suggested by Ammari et al in [1] and Kohn et al in [5] to study the blow up of the power. They considered non-radial core-shell structures in which the shell has permittivity −1 + iδ and the core and the matrix have permittivity 1 in the two-dimensional quasistatic regime.…”

Negative index materials and their applications: Recent mathematics progress

“…Otherwise it is visible. Concerning the second feature of cloaking a source via ALR, the blow up of the power was studied for a more general setting in two dimensional quasistatic regime by Ammari et al in [5] and Kohn et al in [11]. More precisely, they considered nonradial core-shell structures in which the shell has permittivity −1−iδ and its complement has permitivity 1.…”

“…More precisely, they considered nonradial core-shell structures in which the shell has permittivity −1−iδ and its complement has permitivity 1. In [5], Ammari et al dealt with arbitrary shells and provided a characterization of sources for which the power blows up via the information of the spectral decomposition of the Neumann-Poincaré type operator. In [11], Kohn et al considered core-shell structures in which the outer boundary of the shell is round but the inner is not and established the blow up of the power for some class of sources using a variational approach.…”

“…When Ω j = B r j for j = 2, 3 and A = I in Ω 3 \ Ω 2 , more quantitative estimates on the blow up and the boundedness of the power are given in the following proposition in the spirit of [19, Theorems 1.2 and 1.3] (inspired by [5,11]). …”