Homogenization of Maxwell’s equations with split rings

Abstract

We analyze the time harmonic Maxwell’s equations in a complex geometry. The scatterer Omega subset R^3 contains a periodic pattern of small wire structures of high conductivity, 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 meta-material.