Optics 101 · Gain or absorption

Absorption, Spontaneous & Stimulated Emission

What decides whether atoms absorb resonant light or amplify it? It depends on which level holds more atoms. Atoms in the lower level absorb resonant light; atoms in the upper level are stimulated by it to emit coherent copies of the photons. If the lower level is more crowded, absorption wins; if the upper level is, the light is amplified. Spontaneous emission happens at random either way, with or without the light.

A probe beam crosses a gas cell just after a pulse has excited most of its atoms. Watch the beam grow, then fade as the atoms decay, and count each absorbed, stimulated and spontaneous photon.

Probe gain just after preparation $\text{—}$

Gain turns into absorption at $\text{—}$

Phase delay of the output at the start $\text{—}$

Tagged atoms: stimulated / absorbed events $\text{—}$

Probe gain at the end of the record $\text{—}$

Tagged atoms: spontaneous events $\text{—}$

Ensemble expectation: stimulated / absorbed / spontaneous $\text{—}$

Crossing time for a weak probe (analytic) $\text{—}$

Upper-level fraction at the end, cell average $\text{—}$

Steady upper fraction at the entrance (analytic) $\text{—}$

Steady gain of an absorbing cell (analytic) $\text{—}$

Lorentzian line factor $\text{—}$

Line-centre optical depth with every atom in the lower level $\text{—}$

Atom bookkeeping residual $\text{—}$

Recorded time after preparation $\text{—}$

Optics 101

What decides whether atoms absorb resonant light or amplify it?

Upper-level fraction left by the preparation pulse

p_0

0.9

Probe intensity over saturation intensity

I/I_{\rm sat}

0.4

Level gap minus probe photon energy, in half linewidths

\delta

0

Cell length

L

5\,\mathrm{cm}

Line-centre absorption coefficient with every atom in the lower level

\alpha_0

0.4\,\mathrm{cm^{-1}}

Tagged-atom random seed

s

7

Every preset records five lifetimes from the preparation pulse. The record is computed once; the replay never adds photons or events.

Two-level atoms with equal degeneracy and one homogeneously broadened Lorentzian line, in the rate-equation limit: dephasing is much faster than spontaneous decay, so the atomic polarization follows the field and coherent Rabi oscillation is left out. A short preparation pulse sets the upper-level fraction at time zero. How that pulse prepares the atoms is outside the model; a resonant coherent pulse can do it, while continuous incoherent light on two levels cannot. Time is in spontaneous lifetimes. The probe enters with a fixed intensity and is followed through 64 slices of the cell. In each slice it grows or decays exponentially with the local population difference, and the atoms there see the slice-mean intensity, so the photons added to the beam equal stimulated minus absorbed events. Absorption and stimulated emission share one rate per atom, set by the local intensity and the Lorentzian line factor. Spontaneous emission goes into every direction and mode; its share in the probe mode is neglected. The phase delay comes from the same Lorentzian susceptibility and equals minus half the gap offset times the natural log of the transmission. The 120 tagged atoms are a random sample drawn with the selected seed. Each follows the ensemble rates of its slice and its events are counted one by one; the ensemble record does not depend on the seed. Doppler broadening, hyperfine structure, radiation trapping, optical pumping among sublevels, beam diffraction and the probe's travel time are not included. The wavelength, the drawn atoms and the photon markers are schematic. The drawn wave amplitude follows the square root of the transmission and is clipped above e times the input and below a tenth.

Physics tutorial

Watch a gas switch from amplifying light to absorbing it

BackgroundAn atom with two levels can do three things with light at its transition frequency: absorb a photon, be stimulated by one to emit a copy, or emit a photon on its own. Einstein showed in 1916 that all three are needed for matter and thermal radiation to reach equilibrium.

Why it mattersWhether a gas brightens a beam or dims it is a race between the first two processes. The record follows a probe beam through a cell whose atoms start mostly excited and then decay.

Start with the essentials

Focus question: When does a beam gain more photons from stimulated emission than it loses to absorption?
One-sentence intuition: Absorption and stimulated emission have the same rate per atom, so the beam grows exactly when more atoms are in the upper level than in the lower one. The probe itself drives the atomic dipole that radiates a stimulated photon, so that photon joins the probe in the same mode, with the same frequency, phase and polarization.

Core mathematical model

Three processes, one shared rate

\frac{dn}{dt}=W(1-n)-W\,n-\frac{n}{\tau}

Here n is the upper-level fraction, W the stimulated rate per atom set by the local probe intensity and the line shape, and tau the spontaneous lifetime. The two levels have equal degeneracy, so absorption and stimulated emission share W.

The stimulated rate

W\tau=\frac{\ell(\delta)}{2}\,\frac{I}{I_{\rm sat}},\qquad \ell(\delta)=\frac{1}{1+\delta^{2}}

Delta is the level gap minus the photon energy, in half linewidths. Moving the gap off the photon energy lowers the rate along the Lorentzian line.

Gain or absorption along the cell

\frac{dI}{dz}=\alpha_0\,\ell(\delta)\,(2n-1)\,I

The population difference 2n minus 1 decides the sign. The beam grows wherever more than half the atoms are in the upper level.

The phase follows the gain

\Delta\phi=-\frac{\delta}{2}\,\ln T

For a Lorentzian line the same population difference sets the refractive index. Below resonance an absorbing gas delays the wave and an excited gas advances it; on resonance the added light is exactly in phase or out of phase.

When the gain disappears

t_{\times}=\tau\ln(2p_0)

With a weak probe every atom decays freely from the prepared fraction p0, and the gain vanishes when half the atoms are left up. A stronger probe drives atoms down faster, so the crossing comes earlier.

Continuous light never inverts two levels

n\to\frac{s/2}{1+s}<\frac{1}{2},\qquad s=\ell(\delta)\,\frac{I}{I_{\rm sat}}

The steady upper fraction under any constant intensity approaches one half from below.

How far a strong probe bleaches the cell

\ln T+\ell(\delta)\,\frac{I_{\rm in}}{I_{\rm sat}}\,(T-1)=-\alpha_0 L\,\ell(\delta)

Steady transmission of an absorbing cell. The record's end is compared with this curve in the collapsed diagnostic.

Common difficulties

A stimulated photon does not go its own way

Typical misconceptionA stimulated photon leaves in a random direction, like spontaneous light.

Better mental modelThe probe drives the atomic dipole, so the light it radiates adds to the same beam mode with the same frequency, phase and polarization. Only spontaneous emission spreads into every direction.

Strong light does not pass the halfway mark

Typical misconceptionBright enough light can push more than half the atoms up.

Better mental modelLight at the transition drives both directions at the same rate. Under constant light the upper fraction only approaches one half; inversion needs another route through extra levels.

Spontaneous light does not add to the beam on average

Typical misconceptionSpontaneous photons strengthen the probe beam.

Better mental modelThey go into every direction with random phase, so they add no coherent field to the beam. The tiny share that lands in the beam mode is noise, which this model leaves out.

Run the experiment

01
Watch the gas amplify, then absorb
Keep the default preset, Excited gas amplifies, then absorbs, and watch one replay.
What to observe: The output wave starts taller than the input. Purple atoms go dark as amber photons fly off, and after about half a lifetime the output is weaker than the input.
02
Start with every atom down
Load Ground-state gas absorbs.
What to observe: Absorption events outnumber stimulated ones and the gain stays below zero for the whole record.
03
Push the probe hard
Load Strong probe: clear, never gain.
What to observe: Stimulated events almost match absorptions. The upper fraction stalls just below one half and the cell becomes nearly transparent, but never amplifies.
04
Detune and compare the phase
Load the two off-resonance presets one after the other and compare the output wave with the faint empty-cell wave.
What to observe: The absorbing gas shifts the crests back; the excited gas shifts them forward. The phasor plot turns the opposite way.

Absorption, Spontaneous & Stimulated Emission

Watch a gas switch from amplifying light to absorbing it

Core mathematical model

Three processes, one shared rate

The stimulated rate

Gain or absorption along the cell

The phase follows the gain

When the gain disappears

Continuous light never inverts two levels

How far a strong probe bleaches the cell

Common difficulties

A stimulated photon does not go its own way

Strong light does not pass the halfway mark

Spontaneous light does not add to the beam on average

Run the experiment

Watch the gas amplify, then absorb

Start with every atom down

Push the probe hard

Detune and compare the phase

Related experiments and articles

Events and their average

Controlled light and precision measurement