= Solution
For a nonzero <photon number density>[photon density], divide the <laser rate equations> to remove time:
$$
\frac{dn}{dN^*}=-\frac{f_c}{\beta}\left(1-\frac{N_{\rm th}^*}{N^*}\right).
$$
Integrating from the initial inversion and retaining a small seed density $n_i$ gives
$$
n-n_i=\frac{f_c}{\beta}\left[N_i^*-N^*
+N_{\rm th}^*\ln\frac{N^*}{N_i^*}\right].
$$
The source's pulse approximation neglects $n_i$ compared with the macroscopic pulse density, giving
$$
\boxed{n(t)=\frac{f_c}{\beta}\left[
N_{\rm th}^*\ln\frac{N^*(t)}{N_i^*}+N_i^*-N^*(t)\right].}
$$
A 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 $n(t)=0$, $N^*(t)=N_i^*$ 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.
Back to article page
