Surface localization of plasmons in three dimensions and convexity

TL;DR

This paper analyzes how plasmons, eigenfunctions of the Neumann--Poincaré operator, localize on smooth surfaces in three dimensions, revealing faster decay rates on convex domains and different behaviors on non-convex surfaces like the Clifford torus.

Contribution

It provides quantitative decay rates of plasmons on smooth domains, proves the absence of cloaking in convex cases, and explores spectral differences on non-convex surfaces through numerical analysis.

Findings

Plasmons decay at rate j^{-1/2} on general smooth domains.

Decay rate is faster than any polynomial on strictly convex domains.

Non-convex surfaces like the Clifford torus exhibit different spectral properties.

Abstract

The Neumann--Poincar\'e operator defined on a smooth surface has a sequence of eigenvalues converging to zero, and the single layer potentials of the corresponding eigenfunctions, called plasmons, decay to zero, i.e., are localized on the surface, as the index of the sequence $j$ tends to infinity. We investigate quantitatively the surface localization of the plasmons in three dimensions. The results are threefold. We first prove that on smooth bounded domains of general shape the sequence of plasmons converges to zero off the boundary surface almost surely at the rate of $j^{-1/2}$. We then prove that if the domain is strictly convex, then the convergence rate becomes $j^{-\infty}$, namely, it is faster than $j^{-N}$ for any integer $N$. As a consequence, we prove that cloaking by anomalous localized resonance does not occur on three-dimensional strictly convex smooth domains. We then look into the surface localization of the plasmons on the Clifford torus by numerical computations. The Clifford torus is taken as an example of non-convex surfaces. The computational results show that the torus exhibits the spectral property completely different from strictly convex domains. In particular, they suggest that there is a subsequence of plasmons on the torus which has much slower decay than other entries of the sequence.