Optics 102 · Population inversion

Pumping & Population Inversion

Why can't pumping a two-level system produce population inversion, when three- and four-level schemes can? In a two-level system, pumping harder can at best make the two populations equal; there absorption and stimulated emission cancel, so inversion never arrives. Three- and four-level schemes pump through extra levels. A four-level laser keeps its lower laser level nearly empty, so even a modest number of excited atoms gives inversion.

Pumping two levels only bleaches them. A three-level gas needs over half its atoms pumped; a four-level gas gains at once. Watch all three under one lamp.

Two-level gas: inversion $\text{—}$

Three-level gas: inversion $\text{—}$

Four-level gas: inversion $\text{—}$

Three-level threshold pump rate (analytic) $\text{—}$

Probe gain at the end: 2 / 3 / 4 levels $\text{—}$

Analytic steady inversion: 2 / 3 / 4 levels $\text{—}$

Four-level gas: highest inversion and when $\text{—}$

Three-level gas: fraction left in the pump band $\text{—}$

Four-level gas: fraction in the lower laser level $\text{—}$

Pump times upper lifetime / pump-band and lower-level time ratios $\text{—}$

Probe optical depth with every atom in the lower laser level $\text{—}$

Population bookkeeping residual $\text{—}$

Recorded time after pump on $\text{—}$

Optics 102

Why can't pumping a two-level system produce population inversion, when three- and four-level schemes can?

Pump rate per atom

W_p

200\,\mathrm{s^{-1}}

Upper laser level lifetime

\tau_{21}

3\,\mathrm{ms}

Pump band decay time

\tau_{32}

0.1\,\mathrm{\mu s}

Lower laser level emptying time, four levels

\tau_{10}

0.1\,\mathrm{\mu s}

Probe optical depth with every atom in the lower laser level

\alpha_0 L

2

Every preset records from the moment the pump turns on. The record is computed once; the replay never integrates.

Each cell holds atoms whose levels are not degenerate and follow linear rate equations; every atom is in the ground level when the pump turns on. The pump rate per atom acts on the pump transition in both directions, absorption and stimulated emission at the same rate. In the two-level cell the pump light therefore drives the laser transition itself. In the three- and four-level cells it lifts atoms into a pump band, which decays without radiating into the upper laser level. The upper laser level decays at the upper lifetime: into the ground level in the three-level cell, and into a separate lower laser level in the four-level cell, which then empties into the ground level. The probe is weak. It reads the population difference without changing it, and its gain in decibels is 4.34 times the probe optical depth times the inversion per atom. Each record covers five of the longest lifetimes and is computed with one exact matrix exponential per sample step; the analytic steady states shown beside it come from the same equations. Degeneracy factors, the spectral overlap of pump light and pump band, excited-state absorption, pump depletion along the cell, amplified spontaneous emission, cavity feedback and temperature are not included. Cell lengths, atom positions and the lamp are schematic. The drawn atoms show the recorded level fractions, not the histories of individual atoms.

Start with the essentials

Focus question: Why does the same pump leave one gas absorbing, one barely clear and one amplifying?
One-sentence intuition: Pump light on the laser transition itself drives atoms down as often as up, so two levels never pass one half. A third level lets the pump fill the upper level by another route, but the lower laser level is still the ground level, so more than half of all atoms must be moved. A fourth level empties the lower laser level, so any pump gives gain as long as that level drains faster than the upper level decays.

Core mathematical model

Two levels: the pump only bleaches

\frac{\Delta N}{N}=-\frac{1}{1+2W_p\tau_{21}}

Delta N over N is the upper minus the lower laser-level fraction, W_p the pump rate per atom and tau_21 the upper lifetime. The inversion approaches zero from below and never turns positive.

Three levels: gain only past a threshold

\begin{gathered}\frac{\Delta N}{N}=\frac{x(1-\beta)-1}{1+x(1+2\beta)}\\ x=W_p\tau_{21},\quad \beta=\frac{\tau_{32}}{\tau_{21}}\end{gathered}

Beta compares the pump-band decay time with the upper lifetime. With a fast pump band this becomes (x minus 1) over (x plus 1).

The three-level threshold

W_{\rm th}=\frac{1}{\tau_{21}(1-\beta)}

Below this pump rate the three-level gas still absorbs. A slow pump band raises the threshold, and when it drains slower than the upper level, no pump inverts.

Four levels: gain from the start

\begin{gathered}\frac{\Delta N}{N}=\frac{x(1-\gamma)}{1+x(1+2\beta+\gamma)}\\ \gamma=\frac{\tau_{10}}{\tau_{21}}\end{gathered}

Gamma compares the time that empties the lower laser level with the upper lifetime. Any pump inverts the gas while gamma is below one.

How long the three-level gas takes to turn clear

t_0=\frac{\tau_{21}}{1+x}\,\ln\frac{2x}{x-1}

With a fast pump band the upper fraction rises toward x over 1 plus x, and the inversion crosses zero at this time. Close to threshold the wait grows without limit.

What the probe sees

G_{\rm dB}=\frac{10}{\ln 10}\,\alpha_0 L\,\frac{\Delta N}{N}

Alpha_0 L is the probe optical depth with every atom in the lower laser level. Positive inversion is gain, negative inversion is loss.

Common difficulties

A brighter pump does not invert two levels

Typical misconceptionWith enough pump light, a two-level gas will eventually invert.

Better mental modelLight at the transition drives absorption and stimulated emission at the same rate. A stronger pump only brings the populations closer to equal, so the gas bleaches toward transparency.

A long upper lifetime is not enough for three levels

Typical misconceptionA three-level gas inverts as soon as the upper level lives long enough to collect atoms.

Better mental modelIts lower laser level is the ground level, which starts full. More than half of all atoms must be pumped out before the gas stops absorbing, so it needs a threshold pump rate.

Four levels can fail too

Typical misconceptionA four-level gas always inverts.

Better mental modelAtoms arriving in the lower laser level must leave faster than the upper level decays. If they leave more slowly, they pile up there and the gain dies after a short start.

Run the experiment

01
Pump all three gases weakly
Keep the default preset, Weak pump: only four levels gain, and watch one replay.
What to observe: The three-level and four-level cells fill their upper levels equally, but only the four-level cell amplifies. Its lower laser level is empty, while the three-level gas keeps most atoms in its absorbing ground level.
02
Push the three-level gas past threshold
Load Three levels pass threshold, or drag the pump dial right.
What to observe: The three-level inversion turns positive once the pump rate passes the threshold readout. The four-level cell gains even more.
03
Pump the two-level gas hard
Load Strong pump: two levels only bleach.
What to observe: The two-level cell turns almost clear, but its inversion stays below zero. The other two cells now hold most atoms in the upper level.
04
Slow down the lower laser level
Load Slow lower level: gain, then loss.
What to observe: The four-level cell gains for less than four milliseconds. Then blue atoms pile up in its lower laser level and the gain turns into loss.

Pumping & Population Inversion

Pump three gases with one lamp and see which one amplifies

Core mathematical model

Two levels: the pump only bleaches

Three levels: gain only past a threshold

The three-level threshold

Four levels: gain from the start

How long the three-level gas takes to turn clear

What the probe sees

Common difficulties

A brighter pump does not invert two levels

A long upper lifetime is not enough for three levels

Four levels can fail too

Run the experiment

Pump all three gases weakly

Push the three-level gas past threshold

Pump the two-level gas hard

Slow down the lower laser level

Related experiments and articles

Create an inversion

Controlled light and precision measurement