A fast Fourier transform implementation of the Kramers-Kronig relations: Application to anomalous and left handed propagation

In this this paper we quickly derive the Kramers-Kronig relations from simple causality considerations and propose a simple way to implement them using the Fast Fourier Transform. This work focuses on how to make these relations a usable tool even when their conditions of validity are not strictly respected. In this respect we emphasize on their application to the constant low level loss approximation at microwave frequencies. The method presented is demonstrated on various typical cases of fancy propagation: high velocity, negative phase velocity and evanescent waves.