Homogenization of time-harmonic Maxwell's equations in nonhomogeneous plasmonic structures

TL;DR

This paper rigorously derives the effective dielectric permittivity for layered plasmonic structures with 2D metallic sheets, accounting for anisotropic and inhomogeneous surface conductivities, using homogenization techniques.

Contribution

It introduces a novel homogenization framework for Maxwell's equations in layered plasmonic structures with complex surface conductivities, deriving explicit effective parameters.

Findings

Effective permittivity expressed as bulk and surface averages.

Proved convergence of solutions to the homogenized limit.

Implications for modeling plasmonic crystals.

Abstract

We carry out the homogenization of time-harmonic Maxwell's equations in a periodic, layered structure made of two-dimensional (2D) metallic sheets immersed in a heterogeneous and in principle anisotropic dielectric medium. In this setting, the tangential magnetic field exhibits a jump across each sheet. Our goal is the rigorous derivation of the effective dielectric permittivity of the system from the solution of a local cell problem via suitable averages. Each sheet has a fine-scale, inhomogeneous and possibly anisotropic surface conductivity that scales linearly with the microstructure scale, $d$. Starting with the weak formulation of the requisite boundary value problem, we prove the convergence of its solution to a homogenization limit as $d$ approaches zero. The effective permittivity and cell problem express a bulk average from the host dielectric and a surface average germane to the 2D material (metallic layer). We discuss implications of this analysis in the modeling of plasmonic crystals.