A Negative Index Meta-Material for Maxwell's Equations

Abstract

We derive the homogenization limit for time harmonic Maxwell's equations in a periodic geometry with periodicity length η > 0. The considered meta-material has a singular sub-structure: the permittivity coefficient in the inclusions scales like η −2 and a part of the substructure (corresponding to wires in the related experiments) occupies only a volume fraction of order η 2 ; the fact that the wires are connected across the periodicity cells leads to contributions in the effective system. In the limit η → 0, we obtain a standard Maxwell system with a frequency dependent effective permeability µ eff (ω) and a frequency independent effective permittivity ε eff . Our formulas for these coefficients show that both coefficients can have a negative real part, the meta-material can act like a negative index material. The magnetic activity µ eff = 1 is obtained through dielectric resonances as in previous publications. The wires are thin enough to be magnetically invisible, but, due to their connectedness property, they contribute to the effective permittivity. This contribution can be negative due to a negative permittivity in the wires.

Effective Maxwell’s Equations for Perfectly Conducting Split Ring Resonators

“…Very recently, also the combination of resonator elements with wire elements was successfully analyzed. The setting is very close to the experimental set-up of the early negative index constructions; the mathematical analysis reveals that it is possible to obtain an effective system with two coefficients ε eff and µ eff , which have both a negative real part [21]. With that contribution, we have a mathematical confirmation and an effective description of a negative index meta-material.…”

“…This definition was also used, e.g., in [21]. Our new definition is in terms of line integrals and has the advantage that more complex obstacles Σ can be treated, e.g.…”

Effective Behaviour of Critical-Contrast PDEs: Micro-resonances, Frequency Conversion, and Time Dispersive Properties. I

“…Here in the second equality we use the fact that Γ soft 0 S soft z = I, and in the third equality we use Proposition 2.5, see also (46). Invoking (14) completes the proof in relation to the bottom-left element of (49).…”

Mathematical analysis of transmission properties of electromagnetic meta-materials

“…Regarding known results on non-compactly contained inclusions we mention the thin wires in [9] and [27], and the dimensionally reduced analysis of the metal plates Σ 3 in [11].…”