Dispersive and effective properties of two-dimensional periodic media

TL;DR

This paper analyzes electromagnetic wave propagation in a two-dimensional periodic composite, deriving effective property tensors and their frequency corrections using asymptotic expansions for low frequencies.

Contribution

It provides a rigorous asymptotic expansion for the effective properties of 2D periodic media, including explicit frequency correction terms.

Findings

Eigenfunctions and eigenvalues are analytic functions of quasimomentum.

Derived asymptotic expansion for effective property tensor.

Explicit frequency correction term for effective properties.

Abstract

We consider transverse propagation of electromagnetic waves through a two-dimensional composite material containing a periodic rectangular array of circular cylinders. Propagation of waves is described by the Helmholtz equation with the continuity conditions the tangential components of the electric and magnetic fields on the boundaries of the cylinders. We assume that the dimensionless wave frequency $\nu \ll 1$ that allows us to view the governing equation as a perturbation of the Laplace equation. We show that the eigenfunctions and the eigenvalues are even analytic functions of the magnitude of the quasimomentum vector and provide a rigorously justified asymptotic expansion the tensor of effective properties. We also determine explicitly a frequency correction term to the tensor of effective properties.