Past exam of the laser science and quantum information processing course of the University of Oxford / 2019 / 1 / b

The intended quantity is $\boxed{N_{\mathrm{th}}^{\ast }}$, the threshold population inversion density of the cavity after switching to low loss. At this value stimulated gain exactly balances the photon-loss rate:Above threshold a nonzero seed grows; below threshold it decays. The unstarred $N_{\mathrm{th}}$ in the subpart is shorthand for the starred threshold quantity in the displayed equations, rather than a different total atomic population. The photon lifetime ${\tau }_c$ fixes the photon-density loss rate $n/{\tau }_c$.

Write $N=N^{\ast }$, $N_i=N_i^{\ast }$ and $N_f=N_f^{\ast }$ during this derivation. Eliminating the stimulated term between the two laser rate equations gives the photon balanceIntegrate over the pulse, taking negligible initial and final stored photon densities:The photons lost from the cavity carry total energy $\hslash {\omega }_L{V}_c\int n/{\tau }_{c}dt$. Since $f_c{V}_c=V_g$,This identifies the supplied energy formula under the ideal extraction convention that all photon loss is useful output. If only a fraction $\zeta ={\kappa }_{\mathrm{out}}/{\kappa }_{\mathrm{total}}$ of the loss exits the output coupler, the measured output energy is $\zeta E$. The rate equations by themselves do not distinguish useful output from parasitic loss.

The Q-switched energy extraction efficiency in the ideal loss convention of ii isThe first factor reflects the definition of inversion: one stimulated transition reduces the upper population and increases the lower population, changing $N^{\ast }=N_2-(g_2/g_1)N_1$ by $1+g_2/g_1=\beta$. Thus one emitted photon consumes $\beta$ units of inversion. The second factor accounts for the inversion left after the pulse; finite pumping above threshold does not generally extract all the initially stored inversion. This $\eta$ is not the wall-plug efficiency and does not include the energy cost of pumping or the factor $\zeta$ for imperfect output extraction.