# Mathematical analysis of plasmonic resonance for 2-D photonic crystal

## TL;DR

This paper provides a mathematical framework for understanding plasmonic resonance in 2D photonic crystals with negative nanoparticles, deriving conditions for resonance and near-field energy enhancement.

## Contribution

It introduces a spectral analysis approach using layer potentials and the Neumann-Poincaré operator to analyze plasmonic resonance in 2D photonic crystals with Drude-model nanoparticles.

## Key findings

- Conditions for plasmonic resonance are established.
- Near-field energy blow-up rate is derived.
- Control parameters for maximizing energy are identified.

## Abstract

In this article, we study the plasmonic resonance of infinite photonic crystal mounted by the double negative nanoparticles in two dimensions. The corresponding physical model is described by the Helmholz equation with so called Bloch wave condition in a periodic domain. By using the quasi-periodic layer potential techniques and the spectral theorem of quasi-periodic Neumann-Poincar{\'e} operator, the quasi-static expansion of the near field in the presence of nanoparticles is derived. Furthermore, when the magnetic permeability of nanoparticles satisfies the Drude model, we give the conditions under which the plasmonic resonance occurs, and the rate of blow up of near field energy with respect to nanoparticle's bulk electron relaxation rate and filling factor are also obtained. It indicates that one can appropriately control the bulk electron relaxation rate or filling factor of nanoparticle in photonic crystal structure such that the near field energy attains its maximum, and enhancing the efficiency of energy utilization.