Gain saturation homogeneously broadened transitions

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. 

In the small-signal limit we see that equation (13.30) reduces to equation (11.7) as expected. Comparing equations (13.19) and (13.30) we find that the gain of a Doppler-broadened transition saturates more slowly than that of a homogeneously-broadened line. This is a consequence of the fact that in the inhomogeneous case only a small fraction of the total number of atoms are on speaking terms with radiation of angular frequency u. 

---