How Lasers Work — Question 4

Question 4

A four-level cycle gains from a rapidly emptied lower laser state

In a supplied unsaturated four-level model a pump feeds the upper laser level at rate R. Its lifetime is $\tau_2$; every decay feeds the lower laser level, whose lifetime is $\tau_1$. Use $\dot N_2=R-N_2/\tau_2$ and $\dot N_1=N_2/\tau_2-N_1/\tau_1$.

1. Find steady populations and inversion for equal degeneracies.

2. For $R=2.0\times10^{18}\,\mathrm{s^{-1}}$, $\tau_2=1.0\,\mathrm{ms}$ and $\tau_1=10\,\mathrm{ns}$, calculate the inversion and its fractional reduction relative to an empty lower level.

Question 4 - Solution

A four-level cycle gains from a rapidly emptied lower laser state

(a) Method and calculation

$$\dot N_2=0\Rightarrow\boxed{N_2=R\tau_2};\quad\dot N_1=0\Rightarrow\boxed{N_1=R\tau_1}.$$$$\boxed{\Delta N=R(\tau_2-\tau_1)};\quad\tau_2>\tau_1\text{ gives inversion}.$$

(b) Interpretation and further calculation

$$\boxed{N_2=2.0\times10^{15},\quad N_1=2.0\times10^{10},\quad\Delta N=1.99998\times10^{15}}.$$$$\boxed{(N_2-\Delta N)/N_2=\tau_1/\tau_2=10^{-5}}.$$

Physical check

The model treats R as a prescribed feed rate and neglects pump depletion and stimulated emission. The four-level advantage is a nearly empty lower laser level, not a removal of the cavity-loss threshold.