Laser rate equations

Remarks

The steady states give the threshold. A steady state with $\phi > 0$ needs $B N = 1/\tau_c$, so $N = N_{\mathrm{th}} = 1/(B \tau_c)$. The first equation then gives $\phi = \tau_c (R_p - N_{\mathrm{th}}/\tau)$, which is positive exactly when $R_p > N_{\mathrm{th}}/\tau$, the threshold pump rate. Above threshold the inversion is clamped at $N_{\mathrm{th}}$ and the photon density grows linearly with the pump. Below threshold the only steady state is $\phi = 0$, $N = R_p \tau$.

Example: with $\tau_c = 1$, $\tau = 1$, $B = 1$ and $R_p = 3$, the threshold is $N_{\mathrm{th}} = 1$ and the steady state is $N = 1$, $\phi = 2$; one checks $3 - 1 - 2 = 0$ and $2 - 2 = 0$.

Non-example: the equations do not track the phase of the field or the polarization of the medium. So they cannot describe effects such as the Lorenz-type chaos of the Maxwell–Bloch equations. Many variants exist, for example with a term for spontaneous emission into the mode or for three-level media.