 Solution

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.