Past exam of the laser science and quantum information processing course of the University of Oxford / 2017 / 1 / c / Solution

Write $N_2=N_{\mathrm{th}}+u$ and $n=n_{\mathrm{ss}}+w$. Keeping terms linear in the perturbations givesEquivalently,The characteristic exponents areConsequently the above-threshold steady state has damped oscillations precisely whenTheir ordinary frequency isThe approximation neglects the damping correction when $r^2{\tau }_c\ll 4(r-1){\tau }_2$. The exponential envelope decays on time $2{\tau }_2/r$. The doped-volume fraction cancels out of this linear frequency once the pump is expressed through $r$. ^[ref]