The plasmonic eigenvalue problem

TL;DR

This paper studies the mathematical properties of plasmons, special harmonic functions related to electromagnetic waves on metallic particles, proving their eigenvalues form a converging sequence and deriving related formulas.

Contribution

It introduces a new formulation of the plasmonic eigenvalue problem, proves eigenvalues converge to one, and derives a shape derivative formula for Dirichlet-Neumann operators.

Findings

Eigenvalues form a converging sequence to one.

Regularity and variational characterization of plasmons established.

Derived a shape derivative formula for Dirichlet-Neumann operators.

Abstract

A plasmon of a bounded domain $\Omega\subset\mathbb{R}^n$ is a non-trivial bounded harmonic function on $\mathbb{R}^n\setminus\partial\Omega$ which is continuous at $\partial\Omega$ and whose exterior and interior normal derivatives at $\partial\Omega$ have a constant ratio. We call this ratio a plasmonic eigenvalue of $\Omega$. Plasmons arise in the description of electromagnetic waves hitting a metallic particle $\Omega$. We investigate these eigenvalues and prove that they form a sequence of numbers converging to one. Also, we prove regularity of plasmons, derive a variational characterization, and prove a second order perturbation formula. The problem can be reformulated in terms of Dirichlet-Neumann operators, and as a side result we derive a formula for the shape derivative of these operators.