We prove that all functions obeying the Kramers-Kronig relations can be approximated as superpositions of Lorentzian functions, to any precision. As a result, the typical textbook analysis of dielectric dispersion response functions in terms of Lorentzians may be viewed as encompassing the whole class of causal functions. A further consequence is that Lorentzian resonances may be viewed as possible building blocks for engineering any desired metamaterial response, for example, by use of split-ring resonators of different parameters. Two example functions, far from typical Lorentzian resonance behavior, are expressed in terms of Lorentzian superpositions: a steep dispersion medium that achieves large negative susceptibility with arbitrarily low loss or gain and an optimal realization of a perfect lens over a bandwidth. Error bounds are derived for the approximation.

• Received 22 June 2013

DOI:https://doi.org/10.1103/PhysRevA.88.033834