L03 · Laser foundations

Population Inversion Workbench

Build a population route with two, three or four levels. Store excitation in the upper laser level, empty the lower level, and watch a bottleneck or 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.

Rates change live without discarding the population. Changing topology or selecting a preset starts a new ground-state ensemble. Gray controls do not belong to the selected topology.

Closed, equal-degeneracy rate network with bidirectional incoherent optical pumping. All atoms begin in the ground state. Fractions are deterministic; 80 rounded display markers are not atom trajectories. The density and cross section used for weak-probe gain are stated in the lesson formulas below. No probe depletion, cavity loss, thermal model or lasing threshold is included.

Start with the essentials

Focus question: Can a four-level medium lose inversion even while its pump remains on?
One-sentence intuition: A long-lived upper state can store excitation, and fast lower-state emptying can reduce absorption. Both must compete with leakage and the reverse optical pump transition.

Core mathematical model

Every transfer conserves emitters

\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 has equal forward and reverse rates. Relaxation and leakage return population to explicit destinations; atoms never disappear.

Continuous two-level limit

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

This bound applies to equal-degeneracy incoherent continuous pumping with positive relaxation, starting from the ground state. It does not forbid transient inversion by a coherent pulse.

Inversion and a calibrated weak-probe gain

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

The signed coefficient is positive for gain and negative for absorption. Upper-state radiative decay feeds the lower laser level; nonradiative leakage feeds the ground state.

Explicit teaching parameters

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

These are assumed parameters, not a fit to ruby or a particular commercial gain medium. The three-level topology resembles a ground-state lower laser level; the four-level topology adds a separate lower state.

Common difficulties

Transparency is not oscillation threshold

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

Better mental modelA resonator must feed a mode back and overcome its complete loss. This experiment has no resonator, and its gain readout alone cannot establish oscillation.

A four-level label is not a guarantee

Typical misconceptionFour-level systems always invert at arbitrarily low pump.

Better mental modelThat ideal limit assumes fast lower-state emptying and suitable relaxation. Finite rates and leakage can alter or destroy 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 approaches equal populations; it cannot sustain positive inversion under the model assumptions.
02
Store excitation
Choose Three levels and advance. Reduce the pump-to-upper relaxation rate or increase nonradiative leakage while retaining the population state.
What to observe: The storage level competes with pump return and upper-state loss. Merely occupying a high pump level does not guarantee inversion on the selected laser transition.
03
Repair a bottleneck
Choose Lower-state bottleneck and advance 10 ms. Change only lower-to-ground emptying to 20 per millisecond and advance again.
What to observe: A populated lower laser level can make a four-level network absorbing. Clearing it restores a favorable population difference without increasing pump rate.
04
Switch the pump off
Create inversion, turn Pump on off, and step through the decay. Compare the time trace with the updated steady target.
What to observe: Stored population relaxes after pumping stops. The dashed steady reference describes the current settings, not the prior history. Total population remains conserved.