Resonance and double negative behavior in metamaterials
In this work, a generic class of metamaterials is introduced and is shown to exhibit frequency dependent double negative effective properties. We develop a rigorous method for calculating the frequency intervals where either double negative or double positive effective properties appear and show how these intervals imply the existence of propagating Bloch waves inside sub-wavelength structures. The branches of the dispersion relation associated with Bloch modes are shown to be explicitly determined by the Dirichlet spectrum of the high dielectric phase and the generalized electrostatic spectra of the complement. For numerical purposes, we consider a metamaterial 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. We make use of the method of Rayleigh to numerically calculate the generalized electrostatic resonances. We apply these resonances together with the Dirichlet resonances of the core material to calculate the branches of the dispersion relation to leading order. We compare the leading order dispersion relations to the dispersion relations obtained by direct numerical simulation. These calculations show that the leading order dispersion relations is a good predictor of the dispersive behavior of the metamaterial.
Metamaterials, dispersion relations, Bloch waves, simulations
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