Constraints on the temporal dispersion of metamaterials

The frequency dependence of the permittivity, permeability, and index of refraction restrict metamaterial applications such as cloaking and perfect lenses. Here, the principles of causality and passivity together with identities for Herglotz functions are used to construct various sum rules. The sum rules relate the frequency dependence of the material parameters with their high- and low-frequency values. The corresponding physical bounds determine minimum variations of the material parameters over a frequency interval.