Pump-probe model for the Kramers-Kronig relations in a laser

In this paper, we study theoretically a pump-probe model for the Kramers-Kronig (KK) relations during laser operation. A solution of laser equations reveals that the dispersion profile of a saturated laser gain medium is non-analytical at the boundary between the lasing and the non-lasing spectral region. Such a non-analyticity cannot be explained in terms of the KK relations. In order to interpret this situation, it is important to consider carefully the physical basis of the KK relations. We conclude that the KK relation is expected to apply only to an independent probe applied to the medium, which is under excitation by the pump producing the gain as well as the lasing mode. The absorption/gain and dispersion profiles are then analytical, and satisfy the KK relations. Specifically, these are variants of the so-called Mollow-Ezekiel spectra of probe absorption/gain and dispersion in the presence of a pump, with the exception that in this case the medium is inverted.