25 Jul 2025
Laser Physics
Einstein coefficients, laser rate equations, population inversion, three- and four-level lasers, Ruby and He-Ne systems, pulses, and Q-switching.
Einstein coefficients
For two levels $E_2-E_1=h\nu$ in radiation of spectral energy density $\rho(\nu)$, absorption, stimulated emission, and spontaneous emission occur at rates
\[R_{12}=N_1B_{12}\rho,\qquad R_{21}^{st}=N_2B_{21}\rho,\qquad R_{21}^{sp}=N_2A_{21}.\]Thermal equilibrium requires
\[N_1B_{12}\rho=N_2B_{21}\rho+N_2A_{21}.\]With $N_2/N_1=(g_2/g_1)e^{-h\nu/k_BT}$,
\[\rho=\frac{A_{21}/B_{21}}{(g_2B_{12}/g_1B_{21})e^{h\nu/k_BT}-1}.\]Comparison with Planck’s law gives
\[g_1B_{12}=g_2B_{21},\qquad \frac{A_{21}}{B_{21}}=\frac{8\pi h\nu^3}{c^3}.\]Stimulated emission duplicates the incident photon’s frequency, phase, polarization, and direction. This coherence distinguishes laser amplification from ordinary spontaneous light.
Gain, inversion, and threshold
For equal degeneracies, the net stimulated rate is proportional to $N_2-N_1$. A medium amplifies only when
\[\Delta N=N_2-N_1>0,\]the population-inversion condition. Optical pumping transfers energy into the active medium. A metastable upper laser level is useful because its long lifetime permits population to accumulate.
If $I(z)$ is resonant intensity,
\[\frac{dI}{dz}=gI\quad\Rightarrow\quad I(z)=I(0)e^{gz}.\]For a resonator of length $L$, mirror reflectivities $R_1,R_2$, and distributed loss $\alpha$, one round trip multiplies intensity by $R_1R_2e^{2(g-\alpha)L}$. Threshold is therefore
\[g_{th}=\alpha+\frac1{2L}\ln\frac1{R_1R_2}.\]Two- and three-level rate equations
In a true two-level system the same resonant radiation drives absorption and stimulated emission at rate $W=B\rho$. With $A=1/\tau$ and $N=N_1+N_2$,
\[\frac{dN_2}{dt}=WN_1-WN_2-AN_2 =WN-(2W+A)N_2.\]For $N_2(0)=0$, separation or the integrating-factor method gives
\[N_2(t)=\frac{WN}{2W+A}\left[1-e^{-(2W+A)t}\right].\]Consequently
\[\frac{N_2(\infty)}N=\frac{W}{2W+A}<\frac12,\]and even infinitely strong resonant pumping only equalizes the populations. A separate pump route is needed for inversion.
For a three-level laser, pumping $1\to3$ is followed by rapid nonradiative decay $3\to2$, while $2\to1$ is the laser transition. A minimal set of rate equations is
\[\begin{aligned} \dot N_3&=W_pN_1-\frac{N_3}{\tau_{32}},\\ \dot N_2&=\frac{N_3}{\tau_{32}}-\frac{N_2}{\tau_{21}} -W_l(N_2-N_1),\\ \dot N_1&=\frac{N_2}{\tau_{21}}+W_l(N_2-N_1)-W_pN_1. \end{aligned}\]Adding them gives $d(N_1+N_2+N_3)/dt=0$. If $\tau_{32}\ll\tau_{21}$ and the laser field is initially absent, $N_3\simeq W_p\tau_{32}N_1$ and $N_2\simeq W_p\tau_{21}N_1$. Thus inversion requires $W_p\tau_{21}>1$, so more than half the active atoms must leave the ground state. A four-level laser terminates on a level 1 that rapidly empties to the ground level 0; since $N_1\simeq0$, inversion is easier.
Ruby and helium-neon lasers
Ruby is $\mathrm{Al_2O_3}$ doped with $\mathrm{Cr^{3+}}$. Broad optical pumping bands feed a metastable level by rapid nonradiative decay; the red $694.3\ \mathrm{nm}$ transition returns to the ground manifold. It is a three-level laser and therefore normally operates in pulses.
In a He-Ne discharge, electron collisions excite helium metastable states. Near-resonant collisions transfer energy to neon, producing inversion between neon levels. The common $632.8\ \mathrm{nm}$ line is a four-level transition; rapid depletion of the lower laser level enables continuous operation.
Pulsed operation and Q-switching
The resonator quality factor is
\[Q=\omega\frac{\text{stored energy}}{\text{power loss}}.\]In Q-switching the cavity is initially held at low $Q$, suppressing oscillation while pumping stores energy in the inversion. Switching rapidly to high $Q$ reduces loss below gain, and the stored inversion is released as a short, intense pulse. The pulse energy is supplied by the previously stored excitation; Q-switching compresses that energy in time rather than creating additional energy.
Discussion