Homogenization of Maxwell's Equations in a Split Ring Geometry

Abstract

We analyze the time harmonic Maxwell equations in a complex three-dimensional geometry. The scatterer $\Omega\subset\mathbb{R}^3$ contains a periodic pattern of small wire structures of high conductivity, and the single element has the shape of a split ring. We rigorously derive effective equations for the scatterer and provide formulas for the effective permittivity and permeability. The latter turns out to be frequency dependent and has a negative real part for appropriate parameter values. This magnetic activity is the key feature of a left-handed metamaterial.