Saturable absorption in a degenerate two-level system ($I_{s,a}=\hslash \omega /[{\sigma }_{12}\tau (1+g_1/g_2)]$)

For a closed two-level system with total population $N$, excited-state lifetime $\tau$ and statistical sublevel populations, let $W_a={\sigma }_{12}I/(\hslash \omega )$. 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 $T=I_{\mathrm{out}}/I_{\mathrm{in}}$, integrating $dI/dz=-\kappa I$ proves $\ln T+I_{\mathrm{in}}(T-1)/I_{s,a}=-N{\sigma }_{12}L$. Resolved sublevel optical pumping requires a more detailed model.