Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients

We analyze the reflection and transmission coefficients calculated from transfer matrix simulations on finite lengths of electromagnetic metamaterials, to determine the effective permittivity (\ensuremath{\varepsilon}) and permeability (\ensuremath{\mu}). We perform this analysis on structures composed of periodic arrangements of wires, split ring resonators (SRRs), and both wires and SRRs. We find the recovered frequency-dependent \ensuremath{\varepsilon} and \ensuremath{\mu} are entirely consistent with analytic expressions predicted by effective medium arguments. Of particular relevance are that a wire medium exhibits a frequency region in which the real part of \ensuremath{\varepsilon} is negative, and SRRs produce a frequency region in which the real part of \ensuremath{\mu} is negative. In the combination structure, at frequencies where both the recovered real parts of \ensuremath{\varepsilon} and \ensuremath{\mu} are simultaneously negative, the real part of the index of refraction is also found to be unambiguously negative.