Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers

Abstract

We investigate a high-order homogenization (HOH) algorithm for periodic multi-layered stacks. The mathematical tool of choice is a transfer matrix method. Expressions for effective permeability, permittivity and magnetoelectric coupling are explored by frequency power expansions. On the physical side, this HOH uncovers a magnetoelectric coupling effect (odd-order approximation) and artificial magnetism (even-order approximation) in moderate contrast photonic crystals. Comparing the effective parameters' expressions of a stack with three layers against that of a stack with two layers, we note that the magnetoelectric coupling effect vanishes while the artificial magnetism can still be achieved in a centre-symmetric periodic structure. Furthermore, we numerically check the effective parameters through the dispersion law and transmission property of a stack with two dielectric layers against that of an effective bianisotropic medium: they are in good agreement throughout the low-frequency (acoustic) band until the first stop band, where the analyticity of the logarithm function of the transfer matrix () breaks down.

Compendium on Electromagnetic Analysis

“…The present conclusions have been rigorously proved and numerically checked in the onedimensional case in the reference [75]. In particular, it has been shown that the wavenumber k(ω) is an analytic function with respect to the frequency ω in the domain Im(ω) > 0 and that its imaginary part cannot vanish (passivity requirement).…”

Negative index materials: at the frontier of macroscopic electromagnetism

“…The classical quasistatic limit as been also overcome in the case of periodic metamaterials made of dielectric meta-atoms, by an approach relating the macroscopic fields to the microscopic fields averaged over the Floquet unit cell [30][31][32], which can be considered as an extension to periodic arrays of meta-atoms of the classical derivation of macroscopic Maxwell's equations [33]. Also, perturbative expansions with respect to the frequency have been proposed: when starting from the quasistatic limit [34], it has been shown that the first order in frequency reports magnetoelectric coupling while the second order in frequency reports effective magnetism (the higher orders bringing refined corrections to all these parameters), a mechanism similar to the expansion on the wave vector [3,35]; and when Figure 2. Effect of the folding of the dispersion law on group velocity.…”

“…The value of the effective permittivity of the multilayered structure at the quasistatic limit is given by [34]…”

Finite element analysis of electromagnetic waves in two-dimensional transformed bianisotropic media

“…where < f (x 1 ) >= d 0 f (x 1 ) dx 1 , with d the periodic cell size. Note also that these formulae can be deduced from the high-order homogenization approach in [21], by keeping only leading order terms.…”