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In this paper, we study theoretically a pump-probe model for theKramers-Kronig (KK) relations during laser operation. A laser gain medium atsteady state becomes saturated and the lasing field experiences a flat gainequal to the cavity loss. A solution of the laser equations reveals that thelasing frequency experiences a dispersion that is linear over the allowedbandwidth. However, outside this band, the lasing stops, so that the dispersionis that of the unsaturated gain medium. The combined profile is thereforenon-analytical, and cannot be explained in terms of the KK relations. In orderto interpret this situation, it is important to consider carefully the physicalbasis of the KK relations and its connection to causality. We conclude that theKK relation is expected to apply only to an independent probe applied to themedium, which is under excitation by the pump producing the gain as well as thelasing mode. The absorption/gain and dispersion profiles are then analytical,and satisfy the KK-relations. Specifically, these are variants of the so-calledMollow-Ezekiel spectra of probe absorption/gain and dispersion in the presenceof a pump, with the exception that in this case the medium is inverted.

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