A negative index meta-material for Maxwell's equations

We derive the homogenization limit for time harmonic Maxwell's equations in a periodic geometry with periodicity length $\eta>0$. The considered meta-material has a singular sub-structure: the permittivity coefficient in the inclusions scales like $\eta^{-2}$ and a part of the substructure (corresponding to wires in the related experiments) occupies only a volume fraction of order $\eta^2$; the fact that the wires are connected across the periodicity cells leads to contributions in the effective system. In the limit $\eta\to 0$, we obtain a standard Maxwell system with a frequency dependent effective permeability $\mu^{\mathrm{eff}}(\omega)$ and a frequency independent effective permittivity $\varepsilon^{\mathrm{eff}}$. 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 $\mu^{\mathrm{eff}}\neq 1$ 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.