Localization and geometrization in plasmon resonances and geometric structures of Neumann-Poincar\'e eigenfunctions

This paper reports some novel and intriguing discoveries about the localization and geometrization phenomenon in plasmon resonances and the intrinsic geometric structures of Neumann-Poincar\'e 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 provide partial theoretical explanation and link with the geometric structures of the Neumann- Poincar\'e (NP) eigenfunctions. The spectrum of the Neumann-Poincar\'e operator has received significant attentions in the literature. We show for the first time some intrinsic geometric structures of the Neumann-Poincar\'e eigenfunctions near high-curvature points.