= Saturable absorption in a degenerate two-level system
{title2=$I_{s,a}=\hbar\omega/[\sigma_{12}\tau(1+g_1/g_2)]$}
For a closed <two-level system> with total population $N$, excited-state lifetime $\tau$ and statistical sublevel populations, let $W_a=\sigma_{12}I/(\hbar\omega)$. The steady <rate equation> $0=W_a[N_1-(g_1/g_2)N_2]-N_2/\tau$ gives <absorption coefficient> $\kappa=N\sigma_{12}/(1+I/I_{s,a})$. For a uniform slab of thickness $L$ and transmission $T=I_{\rm out}/I_{\rm in}$, integrating $dI/dz=-\kappa I$ proves $\ln T+I_{\rm in}(T-1)/I_{s,a}=-N\sigma_{12}L$. Resolved sublevel <optical pumping> requires a more detailed model.
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