How Lasers Work — Question 3

Question 3

A three-level laser must pump more than half its active atoms upward

In an ideal three-level laser the pump level relaxes instantly into the upper laser level 2; the laser transition ends in ground level 1. Below lasing threshold use $\dot N_2=W(N-N_2)-N_2/\tau$ and equal laser-level degeneracies.

1. Find the steady upper population and transparency pump rate.

2. A cavity requires inversion $N_2-N_1=.20N$. Find the required dimensionless pump $W\tau$.

Question 3 - Solution

A three-level laser must pump more than half its active atoms upward

(a) Method and calculation

$$0=WN-(W+1/\tau)N_2\Rightarrow\boxed{N_2/N=W\tau/(1+W\tau)}.$$$$N_2=N_1=N/2\Rightarrow\boxed{W_{\mathrm{tr}}\tau=1}.$$

(b) Interpretation and further calculation

$$(N_2-N_1)/N=.20\Rightarrow N_2/N=.60.$$$$.60=x/(1+x),\ x=W\tau\Rightarrow\boxed{W\tau=1.50}.$$

Physical check

Transparency balances stimulated emission and absorption but does not overcome cavity loss. The pump level is assumed negligibly populated because its relaxation is very fast.