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 laser gain medium at steady state becomes saturated and the lasing field experiences a flat gain equal to the cavity loss. A solution of the laser equations reveals that the lasing frequency experiences a dispersion that is linear over the allowed bandwidth. However, outside this band, the lasing stops, so that the dispersion is that of the unsaturated gain medium. The combined profile is therefore non-analytical, and 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 and its connection to causality. 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.