---
title: Lasers, Masers, and Stimulated Emission
module: Molecular Spectra
moduleNumber: 2
lessonNumber: 4
order: 204
summary: >
  Einstein's three radiative processes — absorption, spontaneous emission, and
  stimulated emission — and the coefficients that relate them. Stimulated
  emission produces coherent photons, and inverting the level populations turns
  it into net amplification. We build the ruby three-level laser and the
  helium-neon four-level laser, and show why the fourth level makes inversion easy.
topics: [Molecular Spectra]
draft: false
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 9 — Molecular Structure and Spectra; §9-5 Scattering, Absorption, and Stimulated Emission, §9-6 Lasers and Masers"
---

When radiation meets a two-level system with energies $E_1 < E_2$ separated by
$hf_{12} = E_2 - E_1$, 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 $hf_{12}$ 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 $hf_{12}$ 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 $B_{21}\,u(f)$.

$$
% caption: The three radiative processes for a two-level system: absorption lifts
% an atom to E_2, spontaneous emission drops it back with a random photon, and
% stimulated emission produces a second photon coherent with the incident one.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % --- absorption ---
  \begin{scope}
    \draw[black] (0,0) -- (1.6,0) node[right] {$E_1$};
    \draw[black] (0,1.6) -- (1.6,1.6) node[right] {$E_2$};
    \draw[acc, ->, thick] (0.8,0.05) -- (0.8,1.55);
    \draw[acc, ->, thick] (-1.0,0.8) -- (-0.15,0.8);
    \node[anchor=south] at (0.8,1.75) {absorption};
  \end{scope}
  % --- spontaneous ---
  \begin{scope}[xshift=4.6cm]
    \draw[black] (0,0) -- (1.6,0) node[right] {$E_1$};
    \draw[black] (0,1.6) -- (1.6,1.6) node[right] {$E_2$};
    \draw[acc, ->, thick] (0.8,1.55) -- (0.8,0.05);
    \draw[acc, ->, thick] (1.9,0.8) -- (2.75,1.15);
    \node[anchor=south] at (0.8,1.75) {spontaneous};
  \end{scope}
  % --- stimulated ---
  \begin{scope}[xshift=9.2cm]
    \draw[black] (0,0) -- (1.6,0) node[right] {$E_1$};
    \draw[black] (0,1.6) -- (1.6,1.6) node[right] {$E_2$};
    \draw[acc, ->, thick] (0.8,1.55) -- (0.8,0.05);
    \draw[acc, ->, thick] (-1.05,0.9) -- (-0.2,0.9);
    \draw[acc, ->, thick] (1.9,0.9) -- (2.75,0.9);
    \draw[acc, ->, thick] (1.9,0.6) -- (2.75,0.6);
    \node[anchor=south] at (0.8,1.75) {stimulated};
  \end{scope}
\end{tikzpicture}
$$

> **Definition (Coherent radiation).** Photons are **coherent** when they share
> the same frequency, direction, and phase. Stimulated emission produces a photon
> coherent with the one that triggered it; a cascade of such emissions is the
> coherent beam of a laser.

## 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](/statistical-mechanics/foundations/classical-statistics-and-equipartition)
gives $N_2/N_1 = e^{-hf/kT}$, and the total upward and downward transition rates
balance:

$$
N_1 B_{12}\,u(f) = N_2\bigl(A_{21} + B_{21}\,u(f)\bigr).
$$

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](/statistical-mechanics/bose-systems/the-photon-gas-and-plancks-radiation-law) derivation
supplies, forces

$$
B_{12} = B_{21}, \qquad \frac{A_{21}}{B_{21}} = \frac{8\pi h f^3}{c^3}.
$$

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

$$
\frac{A_{21}}{B_{21}\,u(f)} = e^{hf/kT} - 1.
$$

$$
% caption: In thermal equilibrium the ratio of spontaneous to stimulated emission
% is e^{hf/kT} − 1: at optical frequencies (hf ≫ kT) spontaneous emission
% overwhelms stimulated; at microwave frequencies (hf ≪ kT) stimulated wins.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.0,0) node[below] {$\frac{hf}{kT}$};
  \draw[->, black] (0,0) -- (0,4.2) node[left] {$\frac{A_{21}}{B_{21}u}$};
  \draw[acc, very thick, domain=0.05:2.05, samples=90, variable=\x]
    plot ({\x*3},{exp(\x)-1});
  \draw[black, dashed] (0,1.0) -- (7.0,1.0);
  \node[black, anchor=west, font=\scriptsize] at (0.1,1.2) {ratio $= 1$};
  \node[acc, anchor=west, font=\scriptsize] at (0.15,0.35) {microwave: stimulated favored};
  \node[acc, anchor=east, font=\scriptsize] at (6.9,3.6) {visible: spontaneous dominates};
\end{tikzpicture}
$$

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

$$
N_2 > N_1,
$$

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.

> **Definition (Population inversion).** A nonequilibrium condition $N_2 > N_1$ in
> which an upper energy level holds more atoms than a lower one. It is the
> requirement for a medium to amplify rather than absorb radiation at the
> transition frequency, and it cannot be reached at any temperature in
> equilibrium.

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.

$$
% caption: Ruby is a three-level laser: pumping lifts ions to broad bands that
% decay into the metastable level E_2, and the laser transition E_2 → E_1 returns
% them to the ground state, which is also the lower laser level.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % ground
  \draw[black, thick] (0,0) -- (4.0,0);
  \node[anchor=west] at (4.05,0) {$E_1$ ground};
  % metastable E2
  \draw[acc, thick] (0,2.0) -- (4.0,2.0);
  \node[anchor=west, acc] at (4.05,2.0) {$E_2$ metastable};
  % pump band E3
  \draw[black, thick] (0,3.3) -- (4.0,3.3);
  \node[anchor=west, black] at (4.05,3.3) {$E_3$ pump band};
  % pump arrow
  \draw[acc, ->, thick] (1.0,0.05) -- (1.0,3.25);
  \node[anchor=east, font=\scriptsize] at (0.95,1.7) {pump};
  % nonradiative
  \draw[black, ->, thick] (2.0,3.25) -- (2.6,2.05);
  \node[anchor=west, font=\scriptsize, black] at (2.35,2.7) {fast decay};
  % laser transition
  \draw[acc, ->, very thick] (3.4,1.95) -- (3.4,0.05);
  \node[anchor=west, font=\scriptsize, acc] at (3.42,1.0) {694.3 nm};
\end{tikzpicture}
$$

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.

$$
% caption: The laser cavity: an inverted gain medium between a fully reflecting
% mirror and a partially reflecting output mirror, sustaining an axial standing
% wave that grows on each pass and partly escapes as the beam.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % mirrors
  \draw[black, very thick] (0,-0.9) -- (0,0.9);
  \node[anchor=south, font=\scriptsize] at (0,0.95) {full mirror};
  \draw[black, very thick] (6.0,-0.9) -- (6.0,0.9);
  \node[anchor=south, font=\scriptsize] at (6.0,0.95) {output mirror};
  % gain medium
  \draw[acc, thick, fill=acc!10] (0.5,-0.55) rectangle (5.5,0.55);
  \node[acc] at (3.0,0) {gain medium};
  % standing wave
  \draw[black, domain=0.5:5.5, samples=80, variable=\x]
    plot ({\x},{0.32*sin((\x-0.5)*180*4/5)});
  % output beam
  \draw[acc, ->, very thick] (6.1,0) -- (7.6,0);
  \node[acc, anchor=west, font=\scriptsize] at (7.0,0.25) {beam};
\end{tikzpicture}
$$

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.

$$
% caption: Helium-neon is a four-level laser: a discharge excites helium, which
% transfers energy to neon by collision; the laser transition ends on a normally
% empty level that then empties to the ground state, so the lower laser level
% stays nearly unpopulated.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % He side
  \draw[black, thick] (0,3.4) -- (1.8,3.4);
  \node[anchor=south, black, font=\scriptsize] at (0.9,3.42) {He metastable};
  \draw[black] (0,0) -- (1.8,0);
  \node[anchor=north, font=\scriptsize] at (0.9,0) {He ground};
  \draw[acc, ->, thick] (0.4,0.05) -- (0.4,3.35);
  \node[anchor=east, font=\scriptsize] at (0.35,1.7) {discharge};
  % transfer arrow
  \draw[black, ->, thick] (1.85,3.4) -- (3.15,3.5);
  \node[anchor=south, font=\scriptsize, black] at (2.5,3.5) {collision};
  % Ne side
  \draw[acc, thick] (3.2,3.5) -- (5.4,3.5);
  \node[anchor=west, acc, font=\scriptsize] at (5.45,3.5) {upper laser};
  \draw[acc, thick] (3.2,2.0) -- (5.4,2.0);
  \node[anchor=west, acc, font=\scriptsize] at (5.45,2.0) {lower laser (empty)};
  \draw[black] (3.2,0) -- (5.4,0);
  \node[anchor=west, font=\scriptsize] at (5.45,0) {Ne ground};
  % laser transition
  \draw[acc, ->, very thick] (4.3,3.45) -- (4.3,2.05);
  \node[anchor=west, font=\scriptsize, acc] at (4.32,2.75) {632.8 nm};
  % empty via spontaneous + wall
  \draw[black, ->, thick] (3.7,1.95) -- (3.7,0.05);
  \node[anchor=east, font=\scriptsize, black] at (3.65,1.0) {decay to ground};
\end{tikzpicture}
$$

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.

| Feature | Three-level (ruby) | Four-level (He-Ne) |
| --- | --- | --- |
| Lower laser level | ground state | empty excited state |
| Inversion condition | $N_2 > N_1/2$ | $N_2 > 0$ (nearly automatic) |
| Pump demand | over half the atoms | small |
| Operation | pulsed (must cool) | continuous |
| Output | $694.3\ \text{nm}$ | $632.8\ \text{nm}$ |

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](/condensed-matter/semiconductors/semiconductor-bands-and-junctions),
where the inverted populations are electrons and holes across a heavily forward-
biased junction.