= Fibre-laser gain-volume rate equations
{title2=$N_{\mathrm{th}}=(fv_g\sigma\tau_c)^{-1}$}
If a fraction $f$ of an otherwise uniform <optical cavity> contains gain medium, define upper-state <density> per gain volume but <photon> <density> per whole cavity volume. Stimulated upper-state depletion is $v_g\sigma Nn$, while cavity <photon> production is $fv_g\sigma Nn$. Thus $\dot N=S-v_g\sigma Nn-N/\tau_2$ and $\dot n=(fv_g\sigma N-1/\tau_c)n$. Above threshold $N=N_{\mathrm{th}}$ and $n=f\tau_cN_{\mathrm{th}}(r-1)/\tau_2$, where $r=S\tau_2/N_{\mathrm{th}}$.
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