Superpositions of Lorentzians as the class of causal functions

TL;DR

This paper demonstrates that all causal functions satisfying the Kramers-Kronig relations can be approximated by superpositions of Lorentzian functions, enabling new approaches to designing metamaterials with desired electromagnetic responses.

Contribution

It proves that Lorentzian superpositions can approximate any causal response function, expanding their use in metamaterial engineering and analysis.

Findings

Any causal function obeying Kramers-Kronig can be approximated by Lorentzian superpositions.

Lorentzian resonances can serve as fundamental building blocks for engineered responses.

Explicit examples include a negative susceptibility medium and a perfect lens with broad bandwidth.

Abstract

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 text-book 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/gain, and an optimal realization of a perfect lens over a bandwidth. Error bounds are derived for the approximation.