Sub-Wavelength Plasmonic Crystals: Dispersion Relations and Effective Properties

TL;DR

This paper develops a convergent power series expansion for the dispersion relation of subwavelength plasmonic crystals, enabling precise computation of their effective properties and fields with explicit error bounds.

Contribution

It introduces a rigorous, explicit series expansion for the dispersion relation of plasmonic crystals, relating coefficients to cell problem solutions and geometry.

Findings

Explicit series coefficients and convergence radius are derived.

Accurate dispersion relations can be obtained with few series terms.

Error estimates confirm the effectiveness of the approximation.

Abstract

We obtain a convergent power series expansion for the first branch of the dispersion relation for subwavelength plasmonic crystals consisting of plasmonic rods with frequency-dependent dielectric permittivity embedded in a host medium with unit permittivity. The expansion parameter is $\eta=kd=2\pi d/\lambda$, where $k$ is the norm of a fixed wavevector, $d$ is the period of the crystal and $\lambda$ is the wavelength, and the plasma frequency scales inversely to $d$, making the dielectric permittivity in the rods large and negative. The expressions for the series coefficients (a.k.a., dynamic correctors) and the radius of convergence in $\eta$ are explicitly related to the solutions of higher-order cell problems and the geometry of the rods. Within the radius of convergence, we are able to compute the dispersion relation and the fields and define dynamic effective properties in a mathematically rigorous manner. Explicit error estimates show that a good approximation to the true dispersion relation is obtained using only a few terms of the expansion. The convergence proof requires the use of properties of the Catalan numbers to show that the series coefficients are exponentially bounded in the $H^1$ Sobolev norm.