Double Negative Dispersion Relations from Coated Plasmonic Rods

A metamaterial with frequency dependent double negative effective properties is constructed from a sub-wavelength periodic array of coated rods. Explicit power series are developed for the dispersion relation and associated Bloch wave solutions. The expansion parameter is the ratio of the length scale of the periodic lattice to the wavelength. Direct numerical simulations for finite size period cells show that the leading order term in the power series for the dispersion relation is a good predictor of the dispersive behavior of the metamaterial.