Gain saturation transitions

The interpretation of the saturation intensity result, Eq. (8), contains a snbtlety. In the conservative two-state system nnder discnssion, a molecule removed from the upper state by laser-stimulated emission at the rate aJe(hc/Xu) per molecule must appear in the lower state. There it itmnediately is subjected to a pump rate (per molecule) of a l j hcjX returning it to the upper state. Thus for stimulated emission to produce a reduction of the small-signal upper-state population by half, it must be at a transition rate per molecule equal to the sum of the spontaneous decay rate plus the return rate, yielding Eq. (8). This makes the saturation intensity a linear function of the pump intensity at high pump rates, the e bleaches, the small-signal gain saturates at the total inversion value AT,CTe, and the output power increases with pump rate solely through the h term in Eq. (9). 

The different gain saturation of homogeneous and inhomogeneous transitions strongly affects the frequency spectrum of multimode lasers, as can be understood from the following arguments ... 

From equation (13.6) it would appear that the intensity of radiation bouncing to and fro inside the laser cavity would increase indefinitely. This is an unphysical situation and occurs because we have so far neglected the effect of stimulated emission on the populations of the laser levels. As soon as oscillation commences, stimulated transitions start to reduce the inversion density below the value that it had in the absence of oscillation. In a time of the order of the inverse of the cavity linewidth, Au, a steady state is established in which the gain at the oscillation frequency is reduced to a value equal to that required to replace the cavity losses. This process is known as gain saturation. 

We now proceed to consider gain saturation in detail for the case of a homogeneously-broadened transition. Although experimentally this is not the most common situation the analysis required is somewhat less involved than that necessary in the case of the inhomogeneously-broadened lines. 

This situation corresponds to the well-known saturation effect in the emission of most gas laser transitions, where, for the same reason, fewer upper-state molecules can contribute to the gain of the laser transition at the center of the doppler-broadened fluorescence line than nearby. When tuning the laser frequency across the doppler-line profile, the laser intensity therefore shows a dip at the centerfrequen-cy, called the Bennet hole or Lamb dip after W.R. Bennet who discovered and explained this phenomen, and W.E. Lamb 2) who predicted it in his general theory of a laser. 

Equations (5) and (6) indicate, however, that the intensities associated with j <- o transitions will first increase proportionally with I when I is small and ultimately increase with a smaller slope where I is large, j 1 transitions on the other hand will gain Intensity as I for low incident laser powers while at high laser powers they will become proportional to the laser power. Hence both types of signals will show a saturation-like behavior as a function of I. The ratio of any S, which includes all antistokes components, to any Sgj which forms the "main" RR progression (the progression which ordinarily exists when excited vibrational states are not populated) has the form... 

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