Fibre-laser gain-volume rate equations ($N_{\mathrm{th}}=(f{v}_g\sigma {\tau }_c{)}^{-1}$)

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 $f{v}_g\sigma Nn$. Thus $\dot{N}=S-v_g\sigma Nn-N/{\tau }_2$ and $\dot{n}=(f{v}_g\sigma N-1/{\tau }_c)n$. Above threshold $N=N_{\mathrm{th}}$ and $n=f{\tau }_c{N}_{\mathrm{th}}(r-1)/{\tau }_2$, where $r=S{\tau }_2/N_{\mathrm{th}}$.