Relation between the spatial growth rate of a traveling wave and the temporal growth rate of a standing wave

The relation between the spatial growth rate of a traveling wave and the temporal growth rate of the corresponding standing wave is examined. It is shown that, when radiation propagates in a gain medium with a sufficiently narrow gain line and a high amplification coefficient in the line center, the frequency dependences of the spatial and temporal growth rates of the field amplitude can differ significantly. In particular, at a fixed population inversion, the unbounded narrowing of the gain line, which results in an unbounded increase in the spatial growth rate and the narrowing of its frequency profile, is accompanied by neither an unbounded increase in the maximum value of the temporal growth rate nor an unbounded narrowing of the frequency profile of this growth rate.