 Three-level laser rate equation

Consider an original reduced model with a fixed number $N$ of active emitters. Let $u=N_2/N$ be the upper lasing-state fraction, $1-u=N_1/N$ the ground-state fraction, and $s=n/N$ the number of cavity photons per emitter. Assume the pump state empties rapidly, so its population can be neglected. Measure time by $x=t/\tau$, where $\tau$ is the upper state's spontaneous lifetime. Let $r=W\tau$ be the effective pump rate, $C>0$ the dimensionless stimulated-coupling coefficient, $K>0$ the photon-loss rate in these units, and $0\le \beta \le 1$ the fraction of spontaneous decays feeding this optical normal mode. The rate equations areEach term has a physical role. Pumping transfers emitters out of the ground state; spontaneous emission removes upper-state population. The net stimulated term includes emission from fraction $u$ and absorption by fraction $1-u$, giving their difference $2u-1$. A stimulated transition removes one stored excitation and creates one cavity photon, or does the reverse. The factor $\beta$ accounts for spontaneous emission into the selected mode without pretending that all spontaneous emission enters it.

For negligible cavity light, the upper-state equation is linear. A constant pump applied to $u(0)=0$ givesThus this model first permits population inversion at $r>1$. Positive optical gain is still insufficient for an oscillator: the gain must exceed the complete laser threshold, including photon loss. The small-signal photon growth coefficient changes sign whenIf $C\le K$, even complete inversion cannot exceed the assumed loss in this model. For chosen $C=100$ and $K=20$, the threshold population is $u_{\mathrm{th}}=0.6$ and the pump threshold is $r_{\mathrm{th}}=1.5$. At $r=2$, the unsaturated population first reaches $0.6$ at $x=(\log 10)/3\simeq 0.7675$. These numbers are stipulated examples, not ruby material parameters.