 Solution

For a nonzero photon density, divide the laser rate equations to remove time:Integrating from the initial inversion and retaining a small seed density $n_i$ givesThe source's pulse approximation neglects $n_i$ compared with the macroscopic pulse density, givingA positive seed, or a spontaneous-emission source omitted from these equations, is nevertheless needed to start the pulse. Setting $n(0)=0$ exactly in the unforced equations gives the counterexample $n(t)=0$, $N^{\ast }(t)=N_i^{\ast }$ for all time, even above threshold. The displayed formula is therefore used in the negligible-seed limit, rather than as an assertion that a pulse starts from an identically zero field.