---
title: Population Inversion, Gain, and the Laser
module: Lasers and Spectroscopy
moduleNumber: 8
lessonNumber: 1
order: 801
summary: >
  A laser is an optical amplifier placed inside a resonant cavity. Amplification
  requires that stimulated emission outrun absorption, which requires more atoms
  in the upper level than the lower one — a population inversion that the Einstein
  relations forbid in thermal equilibrium and that no two-level pump can produce.
  Three- and four-level schemes reach it by routing atoms through auxiliary states.
  The gain coefficient sets how strongly a weak beam grows, the cavity fixes the
  threshold and selects a comb of longitudinal modes, and gain saturation clamps
  the steady-state inversion at its threshold value.
topics: [Lasers and Spectroscopy]
sources:
  - book: Demtröder
    ref: "Ch. 8 — Lasers; §8.1 Basic Physics, §8.2 Optical Resonators, §8.3 Laser Modes"
  - book: Foot
    ref: "Ch. 1 §1.7 (stimulated emission), Ch. 7 — The Interaction of Atoms with Radiation"
  - book: Loudon
    ref: "The Quantum Theory of Light, 3rd ed., Ch. 1 — Planck's radiation law and stimulated emission"
  - book: Tipler & Llewellyn
    ref: "Ch. 9 — the laser"
draft: false
---

The word names the mechanism: light amplification by stimulated emission of
radiation. Three ideas have to hold together for it to work. Stimulated emission
lets one photon provoke a second identical photon from an excited atom, so a beam
can grow as it propagates. Growth requires that stimulated emission beat
absorption, which requires more atoms in the upper level of a transition than in
the lower — a condition that never holds in thermal equilibrium and that a
two-level system cannot be pumped into. And a single pass through a laboratory-scale
gain medium amplifies by a fraction of a percent, so the medium is placed between
mirrors and the light is forced to traverse it thousands of times.

This lesson derives each piece. The Einstein relations, established for
[radiative transitions](/atomic-physics/radiative-transitions-and-line-shapes/dipole-approximation-einstein-coefficients),
fix the ratio of stimulated to spontaneous rates and show why equilibrium forbids
gain. Rate equations for three- and four-level pumping schemes show how an
inversion is built and set the threshold pump rate. The gain coefficient converts
an inversion into an amplification per unit length, the cavity converts a
distributed gain into an oscillation condition, and gain saturation fixes the
steady state.

## The three radiative processes and detailed balance

Two bound levels of an atom, a lower level $1$ of energy $E_1$ and degeneracy
$g_1$ and an upper level $2$ of energy $E_2$ and degeneracy $g_2$, exchange energy
with a radiation field at the Bohr frequency $\nu_0 = (E_2 - E_1)/h$ through three
processes. Let $\rho(\nu)$ be the spectral energy density of the field and $N_1$,
$N_2$ the number densities of atoms in each level.

- **Absorption.** An atom in $1$ absorbs a photon and moves to $2$ at rate
  $B_{12}\,\rho(\nu_0)$ per lower-level atom.
- **Spontaneous emission.** An atom in $2$ decays to $1$ with no field present, at
  rate $A_{21}$ per upper-level atom, emitting a photon of random phase and
  direction.
- **Stimulated emission.** An atom in $2$, struck by a photon of frequency $\nu_0$,
  is induced to emit a second photon identical in frequency, phase, polarization,
  and direction, at rate $B_{21}\,\rho(\nu_0)$ per upper-level atom.

The coefficients $A_{21}$, $B_{12}$, $B_{21}$ are properties of the atom alone.
Their relations follow from requiring that a gas of these atoms come to equilibrium
with blackbody radiation. In steady state the upward and downward fluxes balance:

$$
N_1 B_{12}\,\rho(\nu_0) = N_2 A_{21} + N_2 B_{21}\,\rho(\nu_0).
$$

Solving for the field that keeps the populations stationary,

$$
\rho(\nu_0) = \frac{A_{21}/B_{21}}{\dfrac{N_1 B_{12}}{N_2 B_{21}} - 1}.
$$

In thermal equilibrium the populations follow the Boltzmann ratio
$N_2/N_1 = (g_2/g_1)\,e^{-h\nu_0/k_{\mathrm B}T}$, and the field must reduce to the
Planck spectrum

$$
\rho(\nu) = \frac{8\pi h \nu^3}{c^3}\,\frac{1}{e^{h\nu/k_{\mathrm B}T} - 1}.
$$

Matching the two expressions term by term forces two identities that then hold at
any temperature, and therefore hold for the isolated atom regardless of the
field.[^loudon-planck]

> **Theorem (Einstein relations).** The stimulated coefficients are tied to each
> other by the degeneracy ratio, and the spontaneous coefficient is fixed by the
> stimulated one:
> $$
> g_1 B_{12} = g_2 B_{21}, \qquad
> \frac{A_{21}}{B_{21}} = \frac{8\pi h \nu_0^3}{c^3}.
> $$

The first relation says absorption and stimulated emission are the same process
run in opposite directions, equal in strength once degeneracy is accounted for.
The second says spontaneous emission grows as $\nu^3$: it is negligible at radio
frequencies, where masers and electronic oscillators run on stimulated emission
alone, and dominant in the ultraviolet, which is why short-wavelength lasers are
hard to build.

## Why equilibrium forbids gain

A beam of intensity $I(\nu)$ traversing the medium along $z$ gains energy from
stimulated emission and loses it to absorption. Per unit length the two rates are
proportional to $N_2 B_{21}$ and $N_1 B_{12}$, so the net change is

$$
\frac{\d I}{\d z} \propto \left(N_2 B_{21} - N_1 B_{12}\right) I
= B_{21}\left(N_2 - \frac{g_2}{g_1}N_1\right) I,
$$

using $B_{12} = (g_2/g_1)B_{21}$. The beam grows only if the bracket is positive:

$$
N_2 - \frac{g_2}{g_1}N_1 > 0
\quad\Longleftrightarrow\quad
\frac{N_2}{g_2} > \frac{N_1}{g_1}.
$$

The **population inversion** condition demands that the per-state population of the
upper level exceed that of the lower level. In thermal equilibrium
$N_2/N_1 = (g_2/g_1)e^{-h\nu_0/k_{\mathrm B}T} < g_2/g_1$ for every finite positive
$T$, so $N_2/g_2 < N_1/g_1$ always: an equilibrium medium absorbs. Formally an
inversion corresponds to a Boltzmann factor with $T < 0$, a signature that the
level populations are not a thermal distribution at all.

$$
% caption: Level populations in thermal equilibrium (left) fall monotonically with
% energy; an inversion (right) overpopulates the upper laser level, the arrangement
% a Boltzmann distribution at no positive temperature can produce.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % --- thermal ---
  \draw[->, black] (0,0) -- (0,4) node[above] {energy};
  \draw[->, black] (0,0) -- (3.4,0) node[right] {population};
  \foreach \y/\w in {0.5/2.8, 1.6/1.4, 2.6/0.55, 3.4/0.22} {
    \fill[acc!14] (0,\y-0.12) rectangle (\w,\y+0.12);
    \draw[acc] (0,\y-0.12) rectangle (\w,\y+0.12);
  }
  \node[anchor=south] at (1.7,-0.9) {thermal (equilibrium)};
  % --- inverted ---
  \begin{scope}[xshift=6cm]
    \draw[->, black] (0,0) -- (0,4) node[above] {energy};
    \draw[->, black] (0,0) -- (3.4,0) node[right] {population};
    \fill[acc!14] (0,0.5-0.12) rectangle (1.1,0.5+0.12);
    \draw[acc] (0,0.5-0.12) rectangle (1.1,0.5+0.12);
    \fill[acc!14] (0,1.6-0.12) rectangle (0.5,1.6+0.12);
    \draw[acc] (0,1.6-0.12) rectangle (0.5,1.6+0.12);
    \fill[acc!25] (0,2.6-0.12) rectangle (2.9,2.6+0.12);
    \draw[acc, very thick] (0,2.6-0.12) rectangle (2.9,2.6+0.12);
    \node[anchor=west, acc] at (2.95,2.6) {upper laser};
    \node[anchor=south] at (1.7,-0.9) {inverted};
  \end{scope}
\end{tikzpicture}
$$

## The gain coefficient and the amplification cross section

Amplification per unit length is set by the inversion and by how strongly a single
atom couples to the beam at the beam's frequency. Real transitions are not
monochromatic: each carries a normalized lineshape $g(\nu)$ with $\int g(\nu)\d\nu = 1$,
determined by the [line-broadening](/atomic-physics/radiative-transitions-and-line-shapes/lifetimes-and-line-shapes)
mechanisms. The stimulated-emission cross section collects the atomic factors:

$$
\sigma(\nu) = \frac{c^2}{8\pi n^2 \nu^2}\,A_{21}\,g(\nu),
$$

with $n$ the refractive index of the host medium.[^demtroeder-gain] The cross
section has units of area; multiplying by the inversion density gives an inverse
length. Writing $\Delta N = N_2 - (g_2/g_1)N_1$ for the inversion density, the beam
obeys

$$
\frac{\d I}{\d z} = \gamma(\nu)\,I,
\qquad
\gamma(\nu) = \Delta N\,\sigma(\nu),
\qquad
I(z) = I(0)\,e^{\gamma(\nu) z}.
$$

> **Definition (Gain coefficient).** $\gamma(\nu) = \Delta N\,\sigma(\nu)$ is the
> fractional intensity growth per unit length of a weak beam at frequency $\nu$.
> It is positive (gain) when the medium is inverted and negative (absorption) in
> equilibrium, and it inherits the frequency dependence of the transition lineshape
> $g(\nu)$, peaking at line center.

The small-signal gain $\gamma_0(\nu)$ uses the unsaturated inversion; a strong beam
depletes the upper level and reduces $\Delta N$, the saturation effect treated
below. Typical laboratory gain media give $\gamma_0 L$ of order $10^{-2}$ per pass,
which is why a resonator is indispensable.

## Pumping schemes: why not two levels

Inverting a transition means pumping atoms into level $2$ faster than they leave and
keeping level $1$ empty. A **two-level** pump cannot do it. Suppose the pump drives
the same $1\leftrightarrow 2$ transition it is meant to invert. The pump adds atoms
to $2$ by absorption at rate $N_1 W$ and removes them by stimulated emission at rate
$N_2 W$, where $W = B_{21}\rho_{\text{pump}}$ (taking $g_1 = g_2$). The rate equation
for the inversion in steady state, including spontaneous decay $A_{21}$, is

$$
\frac{\d N_2}{\d t} = W(N_1 - N_2) - A_{21} N_2 = 0
\;\Longrightarrow\;
\frac{N_2}{N_1} = \frac{W}{W + A_{21}} < 1.
$$

However hard the pump drives, $N_2 \to N_1$ from below: the strongest possible
outcome is equal populations (transparency), never inversion. The pump that
populates the upper level empties it just as fast. Inversion requires auxiliary
levels so that the pumping transition and the lasing transition are different.

### The three-level scheme

The three-level laser, realized first in ruby, uses a broad pump band $3$, a
metastable upper laser level $2$, and the ground state as the lower laser level $1$.

- Pump light drives $1 \to 3$ over a broad absorption band, so an ordinary flashlamp
  can be used.
- Fast non-radiative relaxation $3 \to 2$ funnels atoms into the metastable level,
  where they accumulate because $A_{21}$ is small.
- The laser transition $2 \to 1$ returns atoms to the ground state.

Because level $1$ is the ground state, it starts fully populated. Inversion
$N_2 > N_1$ requires pumping more than half of all atoms out of the ground state and
into level $2$. The threshold is high: a large fraction of the entire atomic
population must be lifted before the medium even reaches transparency.

### The four-level scheme

Adding one level removes that penalty. The four-level laser, realized in
neodymium-doped hosts, places the lower laser level $1$ above the ground state $0$
by more than $k_{\mathrm B}T$, so it is thermally empty, and drains it by fast
relaxation $1 \to 0$.

$$
% caption: Three-level (left) and four-level (right) pumping. In the four-level
% scheme the lower laser level sits above the ground state and empties quickly, so
% it stays nearly unpopulated and any pumping produces an inversion.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
  \definecolor{acc}{HTML}{4A6FA5}
  % ---- three-level ----
  \draw[very thick] (0,0) -- (2.6,0) node[right, black] {ground = lower};
  \draw[very thick] (0,2.4) -- (2.6,2.4) node[right, black] {upper laser};
  \draw[thick] (0,3.3) -- (2.6,3.3) node[right, black] {pump band};
  \draw[->, black!70, very thick] (0.5,0) -- (0.5,3.3);
  \node[black!70, anchor=east] at (0.5,1.7) {pump};
  \draw[->, black, thick] (1.4,3.3) -- (1.4,2.4);
  \node[black, anchor=west, font=\scriptsize] at (1.45,2.85) {fast};
  \draw[->, acc, very thick] (2.2,2.4) -- (2.2,0);
  \node[acc, anchor=west] at (2.25,1.2) {laser};
  \node[anchor=north] at (1.3,-0.55) {three-level};
  % ---- four-level ----
  \begin{scope}[xshift=7.4cm]
    \draw[very thick] (0,0) -- (2.6,0) node[right, black] {ground};
    \draw[thick] (0,0.9) -- (2.6,0.9) node[right, black] {lower laser};
    \draw[very thick] (0,2.6) -- (2.6,2.6) node[right, black] {upper laser};
    \draw[thick] (0,3.4) -- (2.6,3.4) node[right, black] {pump band};
    \draw[->, black!70, very thick] (0.5,0) -- (0.5,3.4);
    \node[black!70, anchor=east] at (0.5,1.7) {pump};
    \draw[->, black, thick] (1.4,3.4) -- (1.4,2.6);
    \node[black, anchor=west, font=\scriptsize] at (1.45,3.0) {fast};
    \draw[->, acc, very thick] (2.2,2.6) -- (2.2,0.9);
    \node[acc, anchor=west] at (2.25,1.75) {laser};
    \draw[->, black, thick] (1.4,0.9) -- (1.4,0);
    \node[black, anchor=west, font=\scriptsize] at (1.45,0.45) {fast};
    \node[anchor=north] at (1.3,-0.55) {four-level};
  \end{scope}
\end{tikzpicture}
$$

Model the four-level system with pump rate $R$ (atoms per volume per time lifted
into level $2$ via the pump band), upper-level lifetime $\tau_2 = 1/A_{21}$
against the laser transition, and a lower level so short-lived that $N_1 \approx 0$.
Below threshold, with no laser field, the upper level fills to

$$
\frac{\d N_2}{\d t} = R - \frac{N_2}{\tau_2} = 0
\;\Longrightarrow\;
N_2 = R\,\tau_2,
\qquad \Delta N = N_2 - N_1 \approx R\,\tau_2 .
$$

Since $N_1 \approx 0$, the inversion is positive for any pump rate $R > 0$: there is
no population barrier to overcome, only the resonator loss. This is why four-level
lasers reach threshold at pump powers one to two orders of magnitude below their
three-level counterparts, and it is the dominant design.

## The resonator: threshold and longitudinal modes

A gain medium of length $L_g$ amplifies a single pass by $e^{\gamma L_g}$. Placing it
between two mirrors of intensity reflectivities $R_1$ and $R_2$ separated by an
optical length $L$ makes the light re-traverse the medium indefinitely. Per round
trip the intensity is multiplied by

$$
G_{\text{rt}} = R_1 R_2\,e^{2\gamma L_g}\,e^{-2\alpha L},
$$

where $\alpha$ collects distributed losses (scattering, absorption in the host,
diffraction). Steady oscillation is a fixed point: the beam must reproduce itself
each round trip, $G_{\text{rt}} = 1$.

> **Definition (Threshold gain).** The gain at which round-trip amplification
> exactly balances round-trip loss,
> $$
> \gamma_{\text{th}} = \alpha + \frac{1}{2 L_g}\,\ln\!\frac{1}{R_1 R_2}.
> $$
> Below it a fluctuation dies out; above it, it grows. The corresponding threshold
> inversion is $\Delta N_{\text{th}} = \gamma_{\text{th}}/\sigma(\nu_0)$.

The logarithmic term is the mirror transmission loss written as an effective
distributed loss; a high-reflectivity output coupler ($R \gtrsim 0.99$) keeps
$\gamma_{\text{th}}$ small so a modest inversion suffices.

$$
% caption: A linear resonator: gain medium of length L_g between two mirrors of
% reflectivities R1 and R2. The beam is amplified on every pass and partially
% transmitted through the output coupler; oscillation is the round trip that
% reproduces the field.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
  \definecolor{acc}{HTML}{4A6FA5}
  % mirrors
  \draw[very thick] (0,-1) -- (0,1);
  \node[anchor=east] at (0,-1.25) {R1};
  \draw[very thick] (8,-0.8) -- (8,0.8);
  \node[anchor=west] at (8,-1.25) {R2 (output)};
  % gain medium
  \fill[acc!12] (2.4,-0.6) rectangle (5.6,0.6);
  \draw[acc] (2.4,-0.6) rectangle (5.6,0.6);
  \node[acc] at (4,0) {gain medium};
  \node[anchor=north] at (4,-0.7) {length $L_g$};
  % beam
  \draw[->, acc, thick] (0.15,0.25) -- (7.8,0.25);
  \draw[->, acc, thick] (7.8,-0.25) -- (0.2,-0.25);
  % output arrow
  \draw[->, acc, very thick] (8.15,0) -- (9.4,0);
  \node[acc, anchor=west] at (9.45,0) {output};
  % cavity length
  \draw[<->, black] (0,-1.6) -- (8,-1.6);
  \node[black, anchor=north] at (4,-1.6) {cavity length L};
\end{tikzpicture}
$$

Threshold is also a phase condition. Only fields that reproduce their phase after a
round trip survive, which quantizes the axial wavenumber: an integer number of
half-wavelengths must fit the cavity, $q\,(\lambda/2) = nL$. The allowed
**longitudinal modes** are the frequencies

$$
\nu_q = q\,\frac{c}{2nL}, \qquad q \in \mathbb{Z}^{+},
$$

spaced by the free spectral range

$$
\Delta\nu_{\text{FSR}} = \frac{c}{2nL}.
$$

For a $30\ \mathrm{cm}$ air-spaced cavity, $\Delta\nu_{\text{FSR}} = c/(2L)
\approx 500\ \mathrm{MHz}$. The gain profile $\gamma(\nu)$ has a width set by the
transition's Doppler and pressure broadening — often several $\mathrm{GHz}$ — so
many longitudinal modes can lie under the gain curve, and each one whose gain
exceeds threshold oscillates.

$$
% caption: Longitudinal modes (vertical lines, spaced by the free spectral range)
% sit under the gain profile. Only modes whose gain exceeds the threshold set by
% cavity loss oscillate; here the three central modes lase.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-0.2,0) -- (9.4,0) node[right] {frequency};
  \draw[->, black] (-0.2,0) -- (-0.2,3.2) node[above] {gain};
  % gain profile (broad bump)
  \draw[acc, very thick] plot[smooth, domain=0.2:8.8, samples=60]
    ({\x}, {2.7*exp(-(\x-4.5)*(\x-4.5)/4.5)});
  \node[acc, anchor=south] at (4.5,2.55) {gain curve};
  % threshold line
  \draw[black, thick, dashed] (-0.2,1.35) -- (9.2,1.35);
  \node[black, anchor=west] at (8.4,1.55) {threshold};
  % longitudinal modes
  \foreach \x in {1.0,1.9,2.8,3.7,4.6,5.5,6.4,7.3,8.2} {
    \draw[black] (\x,0) -- (\x,{2.7*exp(-(\x-4.5)*(\x-4.5)/4.5)});
  }
  % mark the lasing modes with dots above threshold
  \foreach \x in {3.7,4.6,5.5} {
    \fill[acc] (\x,{2.7*exp(-(\x-4.5)*(\x-4.5)/4.5)}) circle (2.4pt);
  }
  \draw[<->, black] (4.6,-0.35) -- (5.5,-0.35);
  \node[black, anchor=north, font=\scriptsize] at (5.05,-0.35) {FSR};
\end{tikzpicture}
$$

## Gain saturation and the steady state

At threshold the round-trip gain equals one and the intracavity field starts to
grow from spontaneous emission. It cannot grow without bound: as the field
intensifies, stimulated emission empties the upper level faster than the pump
refills it, and the inversion $\Delta N$ — hence the gain — drops. For a
homogeneously broadened transition the saturated gain is

$$
\gamma(\nu) = \frac{\gamma_0(\nu)}{1 + I/I_{\text{sat}}},
$$

where $\gamma_0$ is the small-signal gain, $I$ the intracavity intensity, and
$I_{\text{sat}}$ the **saturation intensity** at which the gain halves.[^demtroeder-sat]
The steady state is the intensity at which the saturated gain has fallen exactly to
threshold:

$$
\frac{\gamma_0}{1 + I/I_{\text{sat}}} = \gamma_{\text{th}}
\;\Longrightarrow\;
I = I_{\text{sat}}\!\left(\frac{\gamma_0}{\gamma_{\text{th}}} - 1\right).
$$

Two consequences follow. The inversion is **clamped**: once oscillating, $\Delta N$
stays pinned at $\Delta N_{\text{th}}$ no matter how hard the pump is driven — extra
pumping raises the output intensity, not the inversion. And the steady-state output
power grows linearly with pump rate above threshold, with a sharp kink at the
threshold point.

$$
% caption: Below threshold the medium only fluoresces; above threshold the gain
% clamps at its threshold value and output power rises linearly with pump rate.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7,0) node[right] {pump rate};
  \draw[->, black] (0,0) -- (0,3.4) node[above] {output power};
  % pre-threshold flat, then linear
  \draw[acc, very thick] (0,0.12) -- (2.6,0.12);
  \draw[acc, very thick] (2.6,0.12) -- (6.4,3.0);
  \draw[black, dashed] (2.6,0) -- (2.6,0.12);
  \node[black, anchor=north] at (2.6,0) {threshold};
  \node[acc, anchor=south, rotate=32] at (4.6,1.6) {output slope};
\end{tikzpicture}
$$

## Coherence

Stimulated photons inherit the phase and direction of the photon that induced them,
so the laser field is coherent in a way thermal light is not. Two measures quantify
it.

- **Temporal coherence** is set by the oscillating linewidth $\Delta\nu_{\text{osc}}$.
  The field stays phase-correlated for a coherence time $\tau_c \approx 1/\Delta\nu_{\text{osc}}$
  and over a coherence length $L_c = c\,\tau_c$. A single-longitudinal-mode laser
  with a kilohertz linewidth has a coherence length of tens of kilometers, against
  micrometers for a thermal source.
- **Spatial coherence** is set by the transverse mode structure. A resonator
  operating on its fundamental transverse mode emits a field with a single,
  well-defined wavefront across the beam, so the light can be focused to a
  diffraction-limited spot and produces high-contrast interference across the full
  aperture.

The ultimate linewidth of an oscillating laser is not the passive cavity width but
the far narrower Schawlow–Townes limit, set by the phase diffusion that spontaneous
emission adds to the coherent field. One spontaneously emitted photon per coherence
time randomizes the phase by a small increment; the accumulated phase walk broadens
the line by

$$
\Delta\nu_{\text{ST}} = \frac{\pi h \nu\,(\Delta\nu_c)^2}{P_{\text{out}}},
$$

with $\Delta\nu_c$ the passive-cavity linewidth and $P_{\text{out}}$ the output
power.[^demtroeder-st] The inverse-power scaling is why high-power single-mode
lasers achieve sub-hertz linewidths, the property that makes them the oscillators
behind
[optical frequency combs](/atomic-physics/lasers-and-spectroscopy/spectroscopy-techniques)
and [optical clocks](/atomic-physics/modern-atomic-physics/optical-clocks-precision).

## Assembling the laser

The pieces compose into a single operating picture. A pump lifts atoms into the
upper laser level of a three- or four-level scheme, building an inversion that no
two-level system could reach. The inversion gives the medium a gain coefficient
$\gamma_0(\nu) = \Delta N_0\,\sigma(\nu)$ peaked at line center. The medium sits in
a resonator whose loss sets a threshold gain $\gamma_{\text{th}}$; when the pump
raises $\gamma_0$ past $\gamma_{\text{th}}$, the field grows from spontaneous
emission until saturation clamps the gain back to $\gamma_{\text{th}}$. The cavity
selects a comb of longitudinal modes; those above threshold oscillate, each
inheriting the phase coherence of stimulated emission. The output is a beam narrow
in frequency, directional, and coherent — the amplifier, the inversion, and the
resonator working as one.

The next lesson uses these coherent, tunable sources as instruments: the
[spectroscopic techniques](/atomic-physics/lasers-and-spectroscopy/spectroscopy-techniques)
that beat the Doppler width, and the frequency comb that turns a laser into an
absolute ruler for optical frequencies.

[^loudon-planck]: Loudon, _The Quantum Theory of Light_, 3rd ed., Ch. 1 — Planck's law derived from detailed balance among the three Einstein processes; the requirement that the equilibrium field be the Planck spectrum forces $g_1 B_{12} = g_2 B_{21}$ and $A_{21}/B_{21} = 8\pi h\nu^3/c^3$. See also Foot, §1.7.
[^demtroeder-gain]: Demtröder, _Atoms, Molecules and Photons_, 2nd ed., §8.1 — the amplification cross section $\sigma(\nu) = (c^2/8\pi n^2\nu^2)A_{21}\,g(\nu)$ and the Beer–Lambert form $\d I/\d z = \Delta N\,\sigma\,I$ with $\Delta N = N_2 - (g_2/g_1)N_1$. <https://link.springer.com/book/10.1007/978-3-642-10298-1>
[^demtroeder-sat]: Demtröder, §8.1–8.2 — homogeneous gain saturation $\gamma = \gamma_0/(1 + I/I_{\text{sat}})$, the clamping of the steady-state inversion at $\Delta N_{\text{th}}$, and the linear output-versus-pump characteristic above threshold.
[^demtroeder-st]: Demtröder, §8.2 — the Schawlow–Townes linewidth as the spontaneous-emission floor on the oscillating linewidth, scaling as $(\Delta\nu_c)^2/P_{\text{out}}$; the passive-cavity width $\Delta\nu_c$ is set by the resonator finesse.