How Lasers Work — Question 8

Question 8

Gain clamping makes output rise linearly above threshold in a rate model

A four-level single-mode model uses excited-atom number N and photon number S: $\dot S=(GN-\kappa)S$ and $\dot N=R-N/\tau-GNS$. Here G has units of inverse seconds per atom, $\kappa$ is photon-loss rate, and spontaneous seeding into the mode is neglected after startup.

1. Find steady N and the threshold pump rate for a nonzero photon population.

2. Derive steady S above threshold and evaluate it when $R=2R_{\mathrm{th}}$.

Question 8 - Solution

Gain clamping makes output rise linearly above threshold in a rate model

(a) Method and calculation

$$S>0,\ \dot S=0\Rightarrow\boxed{N_{\mathrm{th}}=\kappa/G}.$$$$R_{\mathrm{th}}=N_{\mathrm{th}}/\tau=\boxed{\kappa/(G\tau)}.$$

(b) Interpretation and further calculation

$$0=R-R_{\mathrm{th}}-\kappa S\Rightarrow\boxed{S=(R-R_{\mathrm{th}})/\kappa}.$$$$R=2R_{\mathrm{th}}\Rightarrow\boxed{S=R_{\mathrm{th}}/\kappa=1/(G\tau)}.$$

Physical check

The steady inversion is clamped at threshold while excess pumping supplies stimulated photons. The zero-photon deterministic solution needs a seed to begin lasing, which the stated above-startup model omits.