How Lasers Work — Question 7

Question 7

An emission cross section converts threshold gain into an inversion density

An equal-degeneracy transition has gain $g=\sigma\Delta n$, where $\sigma=3.00\times10^{-23}\,\mathrm{m^2}$ and $\Delta n$ is inversion per volume. Required gain is $g_{\mathrm{th}}=.300\,\mathrm{m^{-1}}$. The active volume is $V_a=2.00\times10^{-8}\,\mathrm{m^3}$, with a nearly empty lower level and upper lifetime $\tau=200\,\mu\mathrm s$.

1. Find threshold inversion density and number of excited atoms.

2. Find the absorbed pump power needed to replace spontaneous decay at threshold if each useful excitation requires one pump photon of energy $2.50\,\mathrm{eV}$. Use $1\,\mathrm{eV}=1.602\times10^{-19}\,\mathrm J$ and unit pump quantum efficiency.

Question 7 - Solution

An emission cross section converts threshold gain into an inversion density

(a) Method and calculation

$$\boxed{\Delta n_{\mathrm{th}}=g_{\mathrm{th}}/\sigma=1.00\times10^{22}\,\mathrm{m^{-3}}}.$$$$\boxed{N_{\mathrm{th}}=\Delta n_{\mathrm{th}}V_a=2.00\times10^{14}}.$$

(b) Interpretation and further calculation

$$R_{\mathrm{th}}=N_{\mathrm{th}}/\tau=1.00\times10^{18}\,\mathrm{s^{-1}}.$$$$\boxed{P_{\mathrm{abs,th}}=R_{\mathrm{th}}(2.50)(1.602\times10^{-19})=.4005\,\mathrm W}.$$

Physical check

This is an ideal absorbed threshold power. Pump delivery losses, imperfect excitation efficiency, spatial overlap, and additional decay paths would increase an actual incident threshold.