Low-frequency index of refraction for a two-dimensional metallodielectric photonic crystal

We calculate analytically the effective index of refraction $n_{eff}$ of a periodic arrangement of nonmagnetic metallic cylinders in the low-frequency limit. At $\omega \to 0$ the dielectric constant of the cylinders is singular, ${ϵ}_m\left(\omega \right)\approx -({\omega }_p∕\omega {)}^2$, allowing propagation in the plane of periodicity of a mode with the magnetic field parallel to the cylinders ($H$ polarization). The in-plane electric field induces eddy currents, which are localized in a narrow skin layer. We show that the magnetic moment of the eddy currents leads to diamagnetic response if the radius of the cylinders is larger than the skin depth $\delta \sim {10}^{-5}\mathrm{cm}$. Otherwise, the cylinders are transparent for the electromagnetic field and their magnetic moment can be neglected. Magnetization of the cylinders gives rise to distinct values of the quasistatic and static indices of refraction and explains a paradox with noncommuting limits ${ϵ}_m\to \infty$ and $\omega \to 0$.

Figure 1

(Color online) The effective index of refraction $n_{eff}=\sqrt{{\mu }_{eff}{\epsilon }_{eff}}$ for the $H$ mode for a photonic crystal of metallic cylinders arranged in a square lattice. The cylinders are of three different metals: aluminum (circles), copper (triangles), and bismuth (squares). We also plot the static effective index of refraction $\sqrt{{\epsilon }_{eff}}$, which almost coincides with the effective index of refraction of bismuth cylinders.

Figure 2

(Color online) The effective permeability vs filling fraction for (a) the whole range of filling fractions and (b) the transition region of small filling fractions.