Mathematical Analysis of Plasmonic Nanoparticles: The Scalar Case

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.Mathematics Subject Classification (MSC2000): 35R30, 35C20.

Mathematical and numerical framework for metasurfaces using thin layers of periodically distributed plasmonic nanoparticles

“…Recently, there have been several interesting mathematical works on plasmonic resonances for nanoparticles [2][3][4][5][6][7][8][9]. On the other hand, scattering of waves by periodic structures plays a central role in optics [10].…”

Estimation of a class of quasi‐resonances generated by multiple small particles with high surface impedances

“…However, it should be empty for the model describing magnetic nanoparticles. Indeed, the eigenfunctions of the corresponding quasi‐resonances, ie, the plasmonic ones, have average zero on the surfaces of the particles as they are solutions of the Neumann‐Poincaré operator, see Ammari et al for instance.…”

Superdimensional Metamaterial Resonators From Sub-Riemannian Geometry

“…For the numerics, SRO resonators are modeled to occupy a rectangle R of width 2 and height 1 (R = {−1 ≤ x ≤ 1, −0.5 ≤ y ≤ 0.5}). 1 We are looking for the solution to the Helmholtz-type equation with a point source excitation and the Dirichlet BC u = 0 on the boundary. The equation considered is of the form…”

“…The maximum degeneracy occurs at x = 0, and the anisotropy does not vary greatly in the strip |x| ≤ a; outside of this strip, the approximate SRO medium is close to the ideal medium (1). Consider the eigenvalues and eigenfunctions of (9) with the Dirichlet BC, u = 0.…”

“…The possibility of realizing such devices then depends on the ability to fabricate suitable metamaterial (MM) cells for the wave type and wavelengths of interest and the feasibility of assembling these into the array required by the design. The mathematical theory of the subwavelength resonators, such as plasmonic nanoparticles, Helmholtz resonators, and Minnaert bubbles, that could be used as the building blocks of metamaterials, has been recently investigated in [1,2,3,4].…”