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

Abstract

We calculate analytically the effective index of refraction ${n}_{eff}$ of a periodic arrangement of nonmagnetic metallic cylinders in the low-frequency limit. At $\ensuremath{\omega}\ensuremath{\rightarrow}0$ the dielectric constant of the cylinders is singular, ${ϵ}_{m}(\ensuremath{\omega})\ensuremath{\approx}\ensuremath{-}({\ensuremath{\omega}}_{p}∕\ensuremath{\omega}{)}^{2}$, 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 $\ensuremath{\delta}\ensuremath{\sim}{10}^{\ensuremath{-}5}\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$. 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 ${ϵ}_{m}\ensuremath{\rightarrow}\ensuremath{\infty}$ and $\ensuremath{\omega}\ensuremath{\rightarrow}0$.