How Lasers Work — Question 1

Question 1

Net stimulated gain requires inversion per magnetic sublevel

Two levels have populations $N_1,N_2$, degeneracies $g_1,g_2$, and Einstein coefficients satisfying $g_1B_{12}=g_2B_{21}$. At spectral energy density u, stimulated emission and absorption rates are $B_{21}uN_2$ and $B_{12}uN_1$.

1. Derive the condition for positive net stimulated emission.

2. For $g_1=2$, $g_2=6$, and fixed $N=N_1+N_2$, find the upper-population fraction at transparency and decide whether $N_2=.70N$ gives gain.

Question 1 - Solution

Net stimulated gain requires inversion per magnetic sublevel

(a) Method and calculation

$$R_{\mathrm{net}}=B_{21}u[N_2-(g_2/g_1)N_1].$$$$R_{\mathrm{net}}>0\Rightarrow\boxed{N_2/g_2>N_1/g_1}.$$

(b) Interpretation and further calculation

$$N_2=3N_1,\ N=N_1+N_2\Rightarrow\boxed{N_2/N=3/4\text{ at transparency}}.$$$$N_2=.70N:\quad N_2-3N_1=.70N-.90N=-.20N\Rightarrow\boxed{\text{net absorption}}.$$

Physical check

A larger total upper-level population is insufficient when that level has more sublevels. The inversion condition compares population per sublevel.