= Saturated amplifier with exclusive upper-to-lower decay
{title2=$I_s=\hbar\omega A_{21}/\sigma_{21}$}
An open pumped <two-level system> amplifier with fixed upper-state source $R_2$, upper <spontaneous emission> only into the lower <energy level>, and lower-state removal rate $N_1/\tau_1$. Let $r=g_2/g_1$, $W=\sigma_{21}I/(\hbar\omega)$ and $D=N_2-rN_1$. Steady <rate equations> give $N_1=R_2\tau_1$ and $D=R_2(1-rA_{21}\tau_1)/(A_{21}+W)$. Consequently the <optical gain> is $\alpha=\alpha_0/(1+I/I_s)$ with $\alpha_0=\sigma_{21}R_2(A_{21}^{-1}-r\tau_1)$. Lower-state removal controls whether <population inversion> is possible, while its lifetime cancels from this model's <saturation intensity>. Different upper-state branching or a nonconstant pump generally changes that conclusion.
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