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

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.

Homogenization of plasmonic crystals: seeking the epsilon-near-zero effect

“…This description implies precise conditions for the ENZ effect. Our work extends previous homogenization results for time-harmonic Maxwell's equations [22,39,2,3,12]. In particular, we develop the following aspects of the homogenization of plasmonic crystals:…”

“…The derivation presented in this paper is based on a formal asymptotic analysis in the spirit of [5]. Note that in [22] a rigorous approach invoking two-scale convergence is applied to plasmonic crystals without a line charge density along edges. In the framework of homogenization theory for time-harmonic Maxwell's equations, we should also mention a number of other related, rigorous or formal, results [31,39,12,2,3].…”

“…The derivation presented in this paper is based on a formal asymptotic analysis in the spirit of [28]. Note that in [10] a rigorous approach invoking two-scale convergence is applied to plasmonic crystals without a line charge density along edges. [12][13][14][15]29].…”

“…This effect calls for designing novel plasmonic crystals made of stacked metallic or semi-metallic 2D material structures arranged periodically with subwavelength spacing, and embedded in a dielectric host [5,6,10,11]. The growing need to describe, engineer and tune the optical properties of such plasmonic crystals motivates the present paper.…”

Adaptive finite element simulations of waveguide configurations involving parallel 2D material sheets

“…For example, [15] finds a frequency for which reflection in a double-layer graphene system, with both equal and different surface conductivities, is zero, leading to exponentially amplified transmitted modes. [25,26] shows that for plasmonic crystals, which consist of stacked metallic layers arranged periodically with subwavelength distance, embedded in a dielectric medium, the TM polarized waves experience an effective dielectric function that combines a bulk energy of the microstructure of the ambient dielectric medium and surface average of the surface conductivity of each sheet. Homogenization of layered structures and extension to a general hypersurface are also discussed.…”