High-frequency homogenization for layered hyperbolic metamaterials

Abstract

We propose an analytical approach for calculation of the homogenized dielectric functions ${\ensuremath{\epsilon}}_{\ensuremath{\parallel}}(\ensuremath{\omega})$ and ${\ensuremath{\epsilon}}_{\ensuremath{\perp}}(\ensuremath{\omega})$ of one-dimensional periodic metal-dielectric structure. The obtained formulas are valid at high frequencies near the points of topological transition from an elliptic to hyperbolic regime. The proposed method of high-frequency homogenization takes into account rapidly varying electromagnetic fields within the metallic component of a unit cell, in particular, the evanescent character of the plasmonic mode and oscillatory behavior of the waveguidelike modes. Our results show good correspondence to the exact solution of the Rytov's dispersion equation and significant deviation from the widely used quasistatic formulas obtained by spatial averaging along the direction of periodicity $z$ of $\ensuremath{\epsilon}(z)$ and $1/\ensuremath{\epsilon}(z)$. The quasistatic approach ignores $z$ dependence of the fields that leads to its limited applicability near the frequency of topological transition.