# Gain Saturation

A gas which has been pumped to achieve inversion ($N_2 > N_1$) which is used to amplify light via stimulated emission is an optical amplifier.  The amount of amplification that can be extracted from an optical amplifier is limited, which is not surprising.  The physical mechanism that limits the amplification is called gain saturation, and we’ll explore it here.

The physical mechanism underlying gain saturation can be explained pretty easily.  The stimulated emission rate is proportional to the density of excited atoms $N_2$ as well as the intensity of the light $I_{\nu}$.  As always, the only way to obtain a large $N_2$ is by pumping the gas.  As the intensity of the light increases, the stimulated emission rate increases as well.   Each stimulated emission event adds a photon to the optical beam, but also removes an excited atom, lowering $N_2$.  If the rate of stimulated emission exceeds the rate at which atoms are pumped into the excited state, $N_2$ will decrease, thus decreasing the stimulated emission rate and therefore the amplification.  So, you can see that an amplifier can be self-limiting – if it amplifies the optical intensity too much, it will speed up the stimulated emission rate sufficiently to lower $N_2$ and reduce the optical amplification.

All of this implies that the gain per length of the amplifier $\gamma(\nu)$ may depend upon the intensity of the light $I_{\nu}$.  We can show this explicitly by solving the rate equations we’ve derived.  Let’s take the rate equation for $N_2$, and add in a pumping rate $R$ that represents some pumping mechanism (discharge, optical to another level, or whatever):

$\displaystyle \frac{dN_2}{dt} = R - A_{21} N_2 - \frac{\sigma (\nu) I_{\nu}}{h \nu} \left[ N_2 - \frac{g_2}{g_1} N_1 \right]$

Let’s assume that the pump rate $R$ and the intensity $I_{\nu}$ are very large, so that we can approximately neglect the spontaneous emission rate.  Solving this equation in steady state under these approximations gives us

$\displaystyle R = \frac{\sigma (\nu) I_{\nu}}{h \nu} \left[ N_2 - \frac{g_2}{g_1} N_1 \right]$

This is just saying that in the limit of large pumping and large intensity, in steady state the pump rate + absorption rate must exactly balance the stimulated emission rate.  Now recall that the gain per length $\gamma(\nu) = \sigma(\nu) [ N_2 - (g_2 / g_1) N_1]$.  Substituting in, we can see that in this particular limit,

$\displaystyle \gamma(\nu) = \frac{R h \nu}{I_{\nu}}$

So for large intensities and pump rates, the gain per length varies inversely with the intensity, due to the physical effect we described at the top of this section.  This is gain saturation.