On spectral properties of Neuman–Poincaré operator and plasmonic resonances in 3D elastostatics

Abstract

We consider plasmon resonances and cloaking for the elastostatic system in R 3 via the spectral theory of Neumann-Poincaré operator. We first derive the full spectral properties of the Neumann-Poincaré operator for the 3D elastostatic system in the spherical geometry. The spectral result is of significant interest for its own sake, and serves as a highly nontrivial extension of the corresponding 2D study in [8]. The derivation of the spectral result in 3D involves much more complicated and subtle calculations and arguments than that for the 2D case. Then we consider a 3D plasmonic structure in elastostatics which takes a general core-shell-matrix form with the metamaterial located in the shell. Using the obtained spectral result, we provide an accurate characterisation of the anomalous localised resonance and cloaking associated to such a plasmonic structure.

Spectral Properties of Neumann-Poincaré Operator and Anomalous Localized Resonance in Elasticity Beyond Quasi-Static Limit

“…We refer to the papers [3,19,28,30,33] and references therein for more discussions. Similar resonance phenomena were observed and investigated in elasticity [8,9,16,17,21,22,24], which are referred to as polariton resonances in the literature. In this paper, we are mainly concerned with the polariton resonances and their application for CALR for the elastic system within the finite frequency regime beyond the quasi-static approximation.…”

“…Remark 3.1. By taking ω → +0 in the spectral results in Theorem 3.2 and applying the asymptotic properties of the spherical Bessel and Hankel functions, j n (t) and h n (t), for t 1 in (2.9), one can obtain after straightforward though tedious calcuations the spectral system of the N-P operator K 0 S R in the static case, which coincides with that established in [16].…”

On novel elastic structures inducing polariton resonances with finite frequencies and cloaking due to anomalous localized resonances

“…There have been intensive and extensive studies in the literature for optics and we refer to [1, 7-9, 13, 18-25] for the relevant results in electrostatics governed by the Laplace equation, [2,6,11] in acoustics governed by the Helmholtz equation and [3] in electromagnetism governed by the Maxwell system. The plasmonic resonance associated to the Lamé system has been recently studied in [5,10,16,17]. According to our earlier discussion, the first crucial ingredient is the appropriate choice of the plasmon Lamé parameters ǫ 1 λ 0 and ǫ 2 µ 0 such that the following homogeneous Lamé system possesses nontrivial solutions,…”

Localization and geometrization in plasmon resonances and geometric structures of Neumann-Poincaré eigenfunctions

“…Plasmon materials are a type of metamaterials that are artificially engineered to allow the presence of negative material parameters. We refer to [3,4,8,21], [1,2,5,12,13,14,23,28,31,32,33,34,37,38,39,40,41,42,43] and [6,7,9,10,15,25,26,29,22,24] and the references therein for the relevant studies in acoustics, electromagnetism and elasticity, respectively.…”