Gain saturation inhomogeneously broadened

Note If the gain medium is inhomogeneously broadened, the saturated gain becomes... 

Desurvire E Sulhoff I.W., Zyskind J.L., Simpson J.R. Study of spectral dependence of gain saturation and effect of inhomogeneous broadening in erbium-doped aluminosilicate fibo" amplifiers. IEEE Photon. Technol. Lett. 1990 2 653-655... 

The Lamb dip is an important manifestation of the saturation of the gain which occurs in inhomogeneously-broadened transitions when the oscillation frequency coincides with the centre of the laser line. A similar phenomenon occurs if the laser frequency is tuned close to the centre of an absorption line of a sample of atoms or molecules interacting with the standing wave field of the laser. This saturated absorption has become an important new technique in atomic and molecular spectroscopy since it removes the limit on the attainable resolution which was formerly imposed by the Doppler broadening of absorption lines. 

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. 

We have discussed gain saturation in detail for a homogeneously-broadened line since simple explicit expressions can be obtained for the gain coefficient and the intensity of laser radiation. Unfortunately this is not possible in the case of inhomogeneous broadening which we examine in the following section. 

Then, using equation (8.40) for P(/k), the saturated gain coefficient in the limit of dominant inhomogeneous broadening becomes... 

In the case of a purely inhomogeneous gain profile, the different laser modes do not share the same molecules for their amplification, and no mode competition occurs if the frequency spacing of the modes is larger than the saturation-broadened line profiles of the oscillating modes. Therefore all laser modes within that part of the gain profile, which is above the threshold, can oscillate simultaneously. The laser output consists of all axial and transverse modes for which the total losses are less than the gain (Fig. 5.27a). 

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. 

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