Lasers, Masers, and Stimulated Emission

When radiation meets a two-level system with energies $E_1<E_2$ separated by $h{f}_{12}=E_2-E_1$, three radiative processes can occur. Two were already familiar from atomic spectra; the third, identified by Einstein in 1917, is the basis of every laser.

• Absorption. A photon of energy $h{f}_{12}$ is absorbed and the system jumps $E_1\to E_2$. The rate per atom is $B_{12}u(f)$, proportional to the radiation energy density $u(f)$.
• Spontaneous emission. An atom in state $E_2$ decays to $E_1$ on its own, emitting a photon in a random direction and phase. The rate per atom is $A_{21}$, independent of $u(f)$. Its reciprocal ${\tau }_s=1/A_{21}$ is the mean lifetime of the excited state, typically $\sim 10^{-8}\text{s}$.
• Stimulated emission. An incident photon of energy $h{f}_{12}$ induces an atom in $E_2$ to emit a second photon in the same direction and phase as the incident one. The rate per atom is $B_{21}u(f)$.

The Einstein relations

The three coefficients are fixed relative to one another by demanding consistency with thermal equilibrium. Let $N_1,N_2$ be the level populations. In equilibrium the Boltzmann factor gives $N_2/N_1=e^{-hf/kT}$, and the total upward and downward transition rates balance:

Solving for $u(f)$ and requiring the result to match Planck's blackbody law, $u(f)=(8\pi h{f}^3/c^3)/(e^{hf/kT}-1)$, which the photon gas derivation supplies, forces

The absorption and stimulated-emission coefficients are equal, and the ratio of spontaneous to stimulated emission grows as $f^3$. Rewriting,

At $T=300\text{K}$, $kT\approx 0.026\text{eV}$. In the visible ($hf\approx 2\text{eV}$) the ratio is $e^{77}-1$, an astronomical number — spontaneous emission dominates, which is why stimulated emission is normally invisible. In the microwave ($hf\approx 10^{-4}\text{eV}$) the ratio is about $1/260$ and stimulated emission is favored. This frequency dependence is why the maser (microwave) preceded the laser (light).

Population inversion

To make stimulated emission outrun absorption, the upper level must be more populated than the lower. Since $B_{12}=B_{21}$, when $N_1>N_2$ absorption wins and a beam is attenuated; net amplification requires

a population inversion, which is impossible in thermal equilibrium (there $N_2<N_1$ always). It is achieved by three ingredients:

• A metastable upper level $E_2$, with a long lifetime, so atoms accumulate there instead of decaying immediately.
• Optical pumping: intense auxiliary radiation excites atoms to a level above $E_2$, from which they decay into the metastable $E_2$.
• Once inverted, a single spontaneously emitted photon triggers a cascade of coherent stimulated emissions.

The maser came first: Charles Townes and co-workers built one in 1953. The hydrogen maser, still used as an atomic-frequency standard, works on the transition between the hyperfine levels of the ground state — the same $21\text{cm}$ line that maps hydrogen clouds in interstellar space.

The ruby laser: three levels

Theodore Maiman built the first laser in 1960, using a ruby rod — ${\text{Al}}_2{\text{O}}_3$ with about 0.05% chromium — wrapped in a flashtube. The chromium ions ${\text{Cr}}^{3+}$ do the lasing. A flash pumps them into broad pump bands, from which they decay non-radiatively into a pair of metastable states $E_2$ at $1.79\text{eV}$ above the ground state. An intense enough flash puts more ions in $E_2$ than remain in the ground state, inverting the population. Spontaneous $E_2\to E_1$ photons at $1.79\text{eV}$ (wavelength $694.3\text{nm}$) then stimulate the rest.

The rod is a resonant optical cavity: both ends are silvered, one almost totally reflecting and the other about 99% reflecting so a fraction of the beam escapes. If the ends are parallel, standing waves form with $L=m\lambda /2$ for integer $m$, and photons traveling along the axis bounce back and forth, each pass stimulating more atoms until an intense coherent beam emerges.

Because it is a three-level laser, the lower laser level is the ground state, so more than half of all the atoms must be pumped up to invert the population. That demands enormous pump energy, most of it wasted as heat, so ruby lasers must be pulsed to cool between shots. The most intense pulses are produced by Q-switching: the cavity's resonance is spoiled while pumping builds a large inversion, then suddenly restored so the stored energy dumps in one giant pulse.

The helium-neon laser: four levels

The first continuous-wave laser (Javan, Bennett, and Herriott, 1961) was a gas tube of 15% helium and 85% neon. Helium is excited by an electric discharge to metastable states at $19.72\text{eV}$ and $20.61\text{eV}$; these energies almost exactly match excited neon states at $19.83\text{eV}$ and $20.66\text{eV}$, so collisions transfer the excitation to neon, the small energy mismatch ($\sim 0.05\text{eV}$) supplied by kinetic energy. Neon has a normally empty level $1.96\text{eV}$ below the $20.66\text{eV}$ state, so the inversion is automatic. The laser transition emits at $1.96\text{eV}$, wavelength $632.8\text{nm}$ — the familiar red line.

The advantage is structural. In a three-level laser the lower laser level is the ground state, so inversion needs $N_2>N_1/2$ — more than half the atoms excited. In a four-level laser the lower laser level is an excited state that empties quickly, so it stays nearly empty ($N_1\approx 0$) and any population in the upper level is already an inversion. Four-level lasers therefore run continuously with modest pumping.

The coherent, narrow, intense laser beam underlies holography, precision alignment and ranging (the Earth-Moon distance is measured to millimeters off Apollo corner reflectors), retinal surgery, and fusion research. The same stimulated emission also drives the semiconductor diode laser, where the inverted populations are electrons and holes across a heavily forward- biased junction.