Research Article

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

¹ Department of Mathematics, University of Helsinki Helsinki, Finland
² Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong SAR, PR China
³ Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong SAR, PR China
⁴ Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong SAR, PR China

^* Corresponding author: hongyu.liuip@gmail.com, hongyliu@cityu.edu.hk

Received: 7 May 2019
Accepted: 13 December 2019

Abstract

This paper reports some interesting discoveries about the localization and geometrization phenomenon in plasmon resonances and the intrinsic geometric structures of Neumann-Poincaré eigenfunctions. It is known that plasmon resonance generically occurs in the quasi-static regime where the size of the plasmonic inclusion is sufficiently small compared to the wavelength. In this paper, we show that the global smallness condition on the plasmonic inclusion can be replaced by a local high-curvature condition, and the plasmon resonance occurs locally near the high-curvature point of the plasmonic inclusion. We link this phenomenon with the geometric structures of the Neumann-Poincaré (NP) eigenfunctions. The spectrum of the Neumann-Poincaré operator has received significant attentions in the literature. We show that the Neumann-Poincaré eigenfunctions possess some intrinsic geometric structures near the high-curvature points. We mainly rely on numerics to present our findings. For a particular case when the domain is an ellipse, we can provide the analytic results based on the explicit solutions.