Gain saturation inhomogeneously

As before, the steady-state inversion density is obtained by requiring that (v) (v) =0, giving 

The saturation parameter, Ig(v,oj), in this case depends on both V and w and is given by 

As explained in section 13.3, the increase in the hole width at high intensities is the result of an effective reduction in the lifetimes of the upper and lower laser levels caused by stimulated emission and absorption. 

The actual number of longitudinal modes which oscillate simultaneously in an inhomogeneously-broadened gas laser is determined mainly by the Dopper width and the intermode frequency difference, equation (13.5), although the maximum inversion density and the distributed cavity loss will 

By some further manipulation this expression may be recast in the form of a folding integral of Lorentzian and Gaussian distributions whose widths are specified by the linewidth parameters + I ((o)/Ig(iOj j ) and A respectively. 

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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 ... 

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... 

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. 

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

The frequency dependence of the gain coefficient a(v) is related to the line profile g(y — vq) of the amplifying transition. Without saturation effects (i.e., jfor small intensities), a(v) directly reflects this line shape, for homogeneous as well as for inhomogeneous profiles. According to (2.83) we obtain with the Einstein coefficienct Bik... 

Figure 5.25 Saturation of gain profiles a for a homogeneous profile b for an inhomogeneous profile...

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). 

Finally, we discuss how much the output power of a laser at a given pump power Pp decreases when the losses y sre increased by a small amount Ay, The dependence I(Y Pp) of laser intensity on losses is different for homogeneous and for inhomogeneous gain profiles. The saturated gain is, according... 

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