Mathematical analysis of plasmonic nanoparticles: the scalar case

TL;DR

This paper provides a rigorous mathematical framework for understanding plasmonic resonances in metallic nanoparticles, analyzing their spectral properties, scattering enhancements, and potential for super-resolution imaging.

Contribution

It introduces a precise mathematical definition of plasmonic resonance, analyzes how size and shape affect resonance, and links these to scattering and super-resolution capabilities.

Findings

Resonance shifts with nanoparticle size and shape.

Derived bounds on electromagnetic field enhancement.

Demonstrated potential for super-resolution using plasmonic nanoparticles.

Abstract

Localized surface plasmons are charge density oscillations confined to metallic nanoparticles. Excitation of localized surface plasmons by an electromagnetic field at an incident wavelength where resonance occurs results in a strong light scattering and an enhancement of the local electromagnetic fields. This paper is devoted to the mathematical modeling of plasmonic nanoparticles. Its aim is threefold: (i) to mathematically define the notion of plasmonic resonance and to analyze the shift and broadening of the plasmon resonance with changes in size and shape of the nanoparticles; (ii) to study the scattering and absorption enhancements by plasmon resonant nanoparticles and express them in terms of the polarization tensor of the nanoparticle. Optimal bounds on the enhancement factors are also derived; (iii) to show, by analyzing the imaginary part of the Green function, that one can achieve super-resolution and super-focusing using plasmonic nanoparticles. For simplicity, the Helmholtz equation is used to model electromagnetic wave propagation.