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

Let $v_g\simeq c/n_r$ be the light's group velocity in the fibre, with unit mode overlap and a uniform transverse mode area. The fibre-laser gain-volume rate equations follow because the upper-level density is defined per doped volume, whereas the photon density is defined per full cavity volume. Stimulated emission therefore removes upper-level ions at rate $v_g{\sigma }_{21}{N}_{2}n$ per doped volume, but supplies cavity photons at the volume-averaged rate $f{v}_g{\sigma }_{21}{N}_{2}n$. ThusAt threshold the stimulated photon growth rate balances the photon loss rate, givingUsing this relation yields the stated equations, including the essential $1/f$ in the ion-depletion term:Above threshold, the nonzero stationary photon density requires $N_{2,\mathrm{ss}}=N_{\mathrm{th}}$. The remaining stationary equation then givesThe parameter $r$ is the pump rate in units of its lasing-threshold value. The above-threshold branch requires $r>1$; below threshold the ideal stationary solution is $n=0$, $N_2=S{\tau }_2$. Spontaneous emission into the lasing mode has been omitted in this deterministic model, so an exactly zero initial field would need a seed or noise to start, even when the zero-field branch is unstable.