Population Inversion, Gain, and the Laser

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, 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:

Solving for the field that keeps the populations stationary,

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

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.¹

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

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

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.

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 $\int g(\nu )\mathbf{d}\nu =1$, determined by the line-broadening mechanisms. The stimulated-emission cross section collects the atomic factors:

with $n$ the refractive index of the host medium.² 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

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

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 $1\to 0$.

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

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

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, $G_{\text{rt}}=1$.

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.

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

spaced by the free spectral range

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.

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

where ${\gamma }_0$ is the small-signal gain, $I$ the intracavity intensity, and $I_{\text{sat}}$ the saturation intensity at which the gain halves.³ The steady state is the intensity at which the saturated gain has fallen exactly to threshold:

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.

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

with $\Delta {\nu }_c$ the passive-cavity linewidth and $P_{\text{out}}$ the output power.⁴ 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 and optical clocks.

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 that beat the Doppler width, and the frequency comb that turns a laser into an absolute ruler for optical frequencies.