Laser rate equation

A laser rate equation approximates coupled evolution of population inversion and intracavity photon number, including pumping, spontaneous emission, stimulated emission and optical cavity losses. A simple model is $\dot{S}=[g(N)-1/{\tau }_c]S$, $\dot{N}=R_p-N/{\tau }_u-g(N)S$, where compatible normalizations absorb mode-volume factors. Large excess optical gain drives rapid field growth until inversion depletion reduces it below threshold.

If a fraction $f$ of an otherwise uniform optical cavity contains gain medium, define upper-state density per gain volume but photon density per whole cavity volume. Stimulated upper-state depletion is $v_g\sigma Nn$, while cavity photon production is $f{v}_g\sigma Nn$. Thus $\dot{N}=S-v_g\sigma Nn-N/{\tau }_2$ and $\dot{n}=(f{v}_g\sigma N-1/{\tau }_c)n$. Above threshold $N=N_{\mathrm{th}}$ and $n=f{\tau }_c{N}_{\mathrm{th}}(r-1)/{\tau }_2$, where $r=S{\tau }_2/N_{\mathrm{th}}$.

Small perturbations of a class-B laser exchange excitation between the upper-state population and cavity photon number. Linearization about a four-level lasing state gives $\ddot{w}+(r/{\tau }_2)\dot{w}+(r-1)w/({\tau }_c{\tau }_2)=0$. Its damped oscillation frequency is $(2\pi {)}^{-1}\sqrt{(r-1)/({\tau }_c{\tau }_2)-r^2/(4{\tau }_2^2)}$. Long population lifetime compared with photon lifetime permits underdamped relaxation.

Large startup excursions of a laser can accumulate population above threshold and then release it through a short stimulated-emission pulse. Inversion depletion and pump recovery can produce a damped sequence of spikes. This is a nonlinear transient, distinguished from small-signal laser relaxation oscillations and from externally imposed Q-switching.