Constraints on the temporal dispersion of passive metamaterials

Metamaterial applications such as cloaking, perfect lenses, and artificial permeability are restricted by the frequency dependence of the permittivity, permeability, and index of refraction. Here, causality and passivity together with integral identities for Herglotz functions are used to construct 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 the minimum variations of the material parameters over a frequency interval. The results are illustrated with a numerical example for artificial permeability.