Laser Rate-Equation Dynamics

A laser is a pumped gain medium in a resonator, governed by two coupled rate equations for the population inversion $n$ and the intracavity photon number $\phi$: $\dot n = r - n - n\phi$, $\dot\phi = n\phi - \phi/q_0 + s$. Net photon growth needs $n\gt 1/q_0$, so the threshold pump is $r_{th}=1/q_0$. Below it the cavity is dark. Above it the inversion clamps exactly at $n_{th}=1/q_0$ (gain clamping) no matter how hard you pump, and the output power rises linearly with pump from a sharp kink at threshold; on the way to steady state the photon number rings down in relaxation oscillations. Q-switching holds the cavity lossy (low $q_0$) so the gain charges far above threshold, then suddenly opens it: the stored inversion dumps as a giant pulse. Pick the regime, the pump and the cavity $q_0$ and watch the inversion bar, the $\phi(t)$ and $n(t)$ traces, and the operating point on the power-versus-pump curve. RK4, deterministic, gate-tested.