= Solution
{c}
Define $r=g_2/g_1$ and let $\sigma=\sigma_{21}(\omega)$ be the stimulated-<emission cross section>, averaged over upper <magnetic sublevels>. The <Einstein coefficients> relation $g_1B_{12}=g_2B_{21}$ gives the <absorption cross section> $\sigma_{12}=r\sigma$. Assume equally populated <magnetic sublevels> within each <energy level>, a steady pump, and a monochromatic beam. The photon flux is $I/(\hbar\omega)$; define $W=\sigma I/(\hbar\omega)$ and the effective <population inversion> $D=N_2-rN_1$. Then the <rate equations> are
$$
\boxed{\dot N_2=R_2-A_{21}N_2-WD,\qquad
\dot N_1=A_{21}N_2+WD-\frac{N_1}{\tau_1}.}
$$
The $WD$ term includes both <stimulated emission> and <absorption>. In steady state, adding the two <rate equations> gives $N_1=R_2\tau_1$. The first equation then yields
$$
N_2=\frac{R_2+WrR_2\tau_1}{A_{21}+W},\qquad
D=\frac{R_2(1-rA_{21}\tau_1)}{A_{21}+W}.
$$
The directed beam gains <energy> at rate $\hbar\omega WD$ per volume, giving $dI/dz=\sigma DI$ and the <optical gain> coefficient $\alpha=\sigma D$. At line centre, write $\sigma_0=\sigma_{21}(\omega_{21})$. Therefore
$$
\boxed{\alpha=\frac{\alpha_0}{1+I/I_s},\qquad
\alpha_0=\sigma_0R_2\left(\frac1{A_{21}}-r\tau_1\right),\qquad
I_s=\frac{\hbar\omega_{21}A_{21}}{\sigma_0}.}
$$
Here $\alpha_0$ is the small-signal <optical gain> coefficient and $I_s$ is the <saturation intensity>, at which the local <optical gain> is half its small-signal value. Positive <optical gain> requires $rA_{21}\tau_1<1$: the lower <energy level> must empty sufficiently quickly.
The cancellation of $\tau_1$ from $I_s$ is specific to this model's steady fixed pump and the fact that \b[every upper <spontaneous emission> feeds the lower <energy level>]. Adding an extra lower-state accumulation factor to $I_s$ would contradict these <rate equations>. Alternative conventions using the <absorption cross section> must replace $\sigma_0$ by $\sigma_{12}/r$ throughout. <Homogeneous spectral broadening> permits the whole selected ensemble to share this local <saturation> response.
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