L03 · Laser foundations

Population Inversion Workbench

Why can stronger pumping still fail to produce gain? Because gain needs population inversion, and pumping harder does not guarantee it. With only two levels, the pump light drives atoms down as readily as up, so the upper level can never outnumber the lower one. Three- and four-level schemes get around this, but inversion still fails if the lower laser level empties too slowly or atoms leak out of the upper level without emitting light.

Build a pumping scheme with two, three or four levels. Store excitation in the upper laser level, empty the lower level, and watch a bottleneck or a nonradiative leak destroy the inversion.

Model time $\text{—}$

Upper laser fraction $\text{—}$

Lower laser fraction $\text{—}$

Signed inversion $\text{—}$

Small-signal gain coefficient $\text{—}$

Current-setting steady inversion $\text{—}$

Conserved population sum $\text{—}$

Design the population route

A route to inversion

Pump onSwitch off during a run to observe stored excitation decay.Number of levels

N_{\mathrm{levels}}

3

Pump rate in each direction

W_p

2\,\mathrm{ms^{-1}}

Radiative upper-state lifetime

\tau_u

2\,\mathrm{ms}

Pump-to-upper relaxation

k_{pu}

20\,\mathrm{ms^{-1}}

Lower-to-ground emptying

k_{lg}

20\,\mathrm{ms^{-1}}

Upper-to-ground nonradiative leak

q

0.05\,\mathrm{ms^{-1}}

Try the strongest two-level pump. Then try four levels at low pump. Finally create a lower-state bottleneck and repair it by changing only the emptying rate.

Changing a rate takes effect immediately and keeps the current populations. Changing the number of levels or choosing a preset starts over with all atoms in the ground state. Grayed-out controls do not apply to the selected scheme.

The model is a closed rate network of levels with equal degeneracy, and the incoherent optical pump drives atoms both up and down. All atoms start in the ground state. The population fractions are deterministic; the 80 markers are a rounded display and do not follow individual atoms. The density and cross section used for the weak-probe gain are given in the lesson formulas below. Probe depletion, cavity loss, thermal effects and the lasing threshold are not included.

Start with the essentials

Focus question: Can a four-level medium lose its inversion while the pump is still on?
One-sentence intuition: A long-lived upper level can store excitation, and a lower level that empties quickly absorbs less. Both effects have to compete with leakage and with the pump driving atoms back down.

Core mathematical model

Rate equations that conserve atoms

\dot n_i=\sum_{j\ne i}(k_{ji}n_j-k_{ij}n_i),\qquad\sum_i n_i=1

All rates are per millisecond. Optical pumping moves atoms up and down at equal rates. Relaxation and leakage send atoms to definite levels, so no atom is ever lost.

Continuous two-level limit

n_{u,\infty}=\frac{W_p}{2W_p+\Gamma}<\frac12,\qquad\Gamma=\tau_u^{-1}+q

This bound holds for continuous incoherent pumping of levels with equal degeneracy and a positive relaxation rate, starting from the ground state. It does not rule out a brief inversion made by a coherent pulse.

Inversion and gain with assumed parameters

\Delta n=n_u-n_l,\qquad g_0=\sigma_0N\Delta n

The coefficient is positive for gain and negative for absorption. Radiative decay from the upper level feeds the lower laser level, and nonradiative leakage goes to the ground state.

Teaching values for density and cross section

N=10^{24}\,\mathrm{m}^{-3},\quad\sigma_0=2\times10^{-24}\,\mathrm{m}^{2}

These values are assumed; they are not fitted to ruby or to any particular commercial gain medium. In the three-level scheme, the lower laser level is the ground state; the four-level scheme adds a separate lower level.

Common difficulties

Transparency is not the lasing threshold

Typical misconceptionAny positive gain means the medium is already a laser.

Better mental modelA laser needs a resonator that feeds light back into a mode, and the gain must beat all the losses of that mode. This experiment has no resonator, so its gain readout alone cannot show oscillation.

Four levels do not guarantee inversion

Typical misconceptionA four-level system inverts at any pump rate, however low.

Better mental modelThat ideal limit assumes the lower level empties fast and relaxation sends atoms where they need to go. Finite rates and leakage can change or remove the advantage.

Run the experiment

01
Try to beat the two-level limit
Choose Two-level limit and advance 10 ms. Increase lifetime or remove leakage, then compare the upper and lower fractions.
What to observe: A stronger continuous pump brings the populations closer to equal, but under the model assumptions it never sustains a positive inversion.
02
Store excitation
Choose Three levels and advance. Lower the pump-to-upper relaxation rate or raise the nonradiative leak; the populations carry over when a rate changes.
What to observe: Storage in the upper level competes with atoms returning through the pump and with losses from the upper level. Putting atoms into a high pump level does not by itself invert the chosen laser transition.
03
Clear a bottleneck
Choose Lower-state bottleneck and advance 10 ms. Change only the lower-to-ground emptying rate to 20 per millisecond, then advance again.
What to observe: When atoms pile up in the lower laser level, even a four-level medium can absorb. Emptying that level restores a useful population difference without raising the pump rate.
04
Switch the pump off
Build an inversion, uncheck Pump on, and step through the decay. Compare the time trace with the updated steady-state value.
What to observe: The stored population relaxes once pumping stops. The dashed steady-state line belongs to the current settings, not to the earlier history. The total population stays the same.

Population Inversion Workbench

Engineer a population route

Core mathematical model

Rate equations that conserve atoms

Continuous two-level limit

Inversion and gain with assumed parameters

Teaching values for density and cross section

Common difficulties

Transparency is not the lasing threshold

Four levels do not guarantee inversion

Run the experiment

Try to beat the two-level limit

Store excitation

Clear a bottleneck

Switch the pump off

Related experiments and articles

Create an inversion