  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.
