Determination of effective material parameters for a metamaterial based on analysis of local fields

There is considerable interest in the design of metamaterials for specific unusual electromagnetic properties. We want materials with particular values of epsilon and mu that are less than zero. We desired a calculation procedure that could take a geometric structure and evaluate the effective material parameters, both for design validation and in a search for new shapes. There have been other descriptions of similar efforts in the literature; our method is an improvement over the existing methods by reducing the errors due to phase variation across a structure while not requiring a solver that calculates the charge and current directly. We derive general formulas/algorithms for computing effective electric permittivity ε(ω) and magnetic permeability μ(ω) for polarizable media from first-principles classical E and M. We apply them to filamentary, conducting structures that are candidates for making double-negative metamaterials. Specifically we show we can analyze specific structures and design a material where ε(ω) and μ(ω) are simultaneously negative. We use the split-ring resonators and capacitively-end-loaded linear electric dipoles as examples and show that they form Lorentz materials.