= Solution
Define $\sigma(\omega)$ as the <emission cross section> for an upper-level atom, averaged over the participating sublevels, and let $W=\sigma I/(\hbar\omega)$. Detailed balance gives the upward stimulated rate per lower-level atom as $rW$, where $r=g_2/g_1$. With $D=N_2-rN_1$ the complete <laser rate equations> are
$$
\dot N_2=R_2-AN_2-WD,\qquad
\dot N_1=AN_2+WD-N_1/\tau_1.
$$
Adding them eliminates the internal transitions. Therefore steady state has $N_1=R_2\tau_1$ \b[even when the radiation is strong]. Substitution into the upper equation gives
$$
N_2=\frac{R_2(1+rW\tau_1)}{A+W},\qquad
D=\frac{R_2(1-rA\tau_1)}{A+W}
=\frac{D_0}{1+W/A}.
$$
One net stimulated transition changes the beam <energy> by $\hbar\omega$, so the coefficient of <optical gain> for <intensity> is $\alpha(\omega,I)=\sigma(\omega)D$. At line center, with $\sigma_0=\sigma(\omega_{21})$,
$$
\boxed{\alpha=\frac{\alpha_0}{1+I/I_s},\qquad
\alpha_0=\sigma_0R_2\left(A^{-1}-r\tau_1\right),\qquad
I_s=\frac{\hbar\omega_{21}A}{\sigma_0}.}
$$
Here $\alpha_0$ is the unsaturated \b[<intensity>] <optical gain> per length, which may be negative, and $I_s$ is the <intensity> at which its magnitude is halved in this model with <homogeneous broadening>. The cancellation of $\tau_1$ from $I_s$ follows from exclusive upper-to-lower decay and constant upper-state pumping. Keeping $N_1=A\tau_1N_2$ in the stimulated problem would discard the stimulated feeding term and give an incorrect saturation scale. Other decay branches or different pump constraints lead to different expressions.
For a traveling beam with uniform pump, no additional loss and local steady populations,
$$
\frac{dI}{dz}=\frac{\alpha_0 I}{1+I/I_s},\qquad
\boxed{\ln\frac{I(z)}{I(0)}+\frac{I(z)-I(0)}{I_s}=\alpha_0z.}
$$
If $\alpha_0>0$ and $I\gg I_s$, the increment becomes $dI/dz\simeq\alpha_0I_s=\hbar\omega_{21}R_2(1-rA\tau_1)$, approximately constant. <Intensity> then grows \b[linearly], rather than exponentially, with distance while that strong-saturation regime persists. Negative $\alpha_0$ gives <saturable absorption> and a decreasing beam.
\Image[/past-exam-of-the-quantum-atomic-and-molecular-physics-course-of-the-university-of-oxford/2016-open-pumped-amplifier-saturation.png]
{title=Saturated gain and the crossover from exponential to approximately linear amplification}
{description=Original steady-state calculation for the specified open pumped model. The propagation curves assume positive unsaturated gain and no added loss; a negative gain example is absorptive.}
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