 Laser gain clamping

In the steady limit with nonzero cavity light and negligible spontaneous seeding, the photon equation requires $C(2u-1)=K$. The upper-state fraction is therefore fixed near $u_{\mathrm{th}}$ while increased pumping principally increases the photon population. Substituting into the population equation givesFor $C=100$, $K=20$ and $r=3$, this predicts $u=0.6$ and $s=0.03$. The inversion does not rise indefinitely above threshold. A finite $\beta$ supplies seed photons and smooths the ideal onset; with $\beta =0$ and exactly $s(0)=0$, the deterministic equations remain on the zero-photon solution even when that solution is unstable. This distinguishes instability of an unseeded state from actual startup of a noisy optical field.