L20 · Excitation and gain media

Laser Medium Zoo

How do different laser media reach population inversion? Ruby must deplete its absorbing ground state. Helium–neon uses collisions to transfer excitation from helium to neon. Four-level routes keep the lower laser state nearly empty; krypton fluoride has a repulsive lower state that dissociates. Useful gain depends on both the upper population and the population that can absorb the laser light.

Compare six representative pump routes. Watch population accumulate in the upper or absorbing state, and follow the resulting normalized output.

Acquired time / upper lifetime $\text{—}$

Selected medium $\text{—}$

Representative wavelength $\text{—}$

Population model $\text{—}$

Ground fraction $\text{—}$

Pump or donor fraction $\text{—}$

Recorded end upper fraction $\text{—}$

Recorded end absorbing fraction $\text{—}$

Total population fraction $\text{—}$

Recorded end inversion $\text{—}$

On-pump small-signal gain margin $\text{—}$

End-of-record gain / loss $\text{—}$

Normalized cavity photons $\text{—}$

Recorded end output rate $\text{—}$

Largest output in retained run $\text{—}$

End-of-record route feed $\text{—}$

Representative pump compatibility $\text{—}$

L20

How do different laser media reach population inversion?

Optical pump selectedWhen off, the representative discharge route is used.Pulsed pumpThree on units in every twelve; off gives continuous pumping.Cavity feedback connectedWhen off, light escapes the cavity quickly; the spontaneous background remains.Representative medium

m

2

Normalized pump feed

r

2

Lower-state clearance slowdown

b

1

Detuning / illustrative gain full width

\delta

0

Internal normalized loss

\kappa_i

0.2

Output escape rate

\kappa_o

0.3

Keep the pump feed the same when you compare where the population of each medium gets stuck. The normalized curves cannot rank real materials by power or efficiency.

All six media run in the same dimensionless population model; they are not one physical device. The names, representative wavelengths, pump routes and qualitative bandwidths come from published sources, but every rate, transfer yield, loss, cavity gain and output value is chosen for teaching. Time is measured in an upper-state lifetime chosen separately for each medium, so the same point on the time axis means a different number of seconds for different media. Population flows from a ground reservoir to a pump or donor reservoir and then to the upper laser level. In ruby the lower laser level is the absorbing ground state; the other media empty into a fast lower reservoir. For KrF that lower reservoir stands in for the flow of population into a repulsive continuum; it is not a bound level, and no chemistry is solved. The He–Ne and carbon dioxide donor reservoirs represent energy-transfer routes; they are not a literal, conserved count of atoms or molecules. The normalized total population of the one effective ensemble stays fixed. The pump switch picks the representative optical or discharge route. If that route does not match the medium, its feed is set to zero; this does not claim that every other way of pumping is impossible. Pulsed pumping is on for three time units out of every twelve. One Lorentzian factor for the selected mode applies to every medium, with detuning measured in one illustrative full width; it does not reproduce the real multi-line spectra or predict tuning ranges. Blocking the feedback adds a large escape rate, and the tiny spontaneous background, modeled as a single channel, remains. The lower-level clearance control affects only the four-reservoir media and does nothing to ruby. All readouts and plots come from the same integrated populations and cavity photons. The small-signal margin is a separate steady-state reference with no seed light; for pulsed pumping it uses the pump value while the pulse is on. Output is a normalized escape rate, not watts or efficiency, and it does not rank real materials. Chemical kinetics, gas flow, triplet states, bleaching, heating, real cross-sections and real cavity design are left out. Apparatus sizes and colors are schematic. Each selection integrates sixty upper-lifetime units. Replay reads retained samples; numerical readouts describe the end of the record. Pump-route mismatches give zero modeled feed. Material tint, infrared/UV colors and logarithmic field brightness are illustrative; green excitation glow follows the sampled upper fraction. Transport uses fixed brightness references of eight for pump rate, twelve for normalized cavity population and three for normalized output rate. The fixed-size collector circle has linear opacity in output up to the reference of three, then clips; its size is a drawing convention.

Compare the six mechanisms and sources

These are representative transitions and qualitative properties. The simulation uses separate, illustrative normalized rates.

Ruby

$694.3\,\mathrm{nm}$

Optical bands → fast relaxation → chromium metastable state

Narrow electronic line

Often pulsed; ground-state absorption raises the pump burden

He–Ne

$632.8\,\mathrm{nm}$

Discharge → helium metastable atoms → collision transfer to neon

Narrow atomic gain line

Commonly continuous; gas mixture and lower-level removal matter

Nd:YAG

$1064\,\mathrm{nm}$

Optical absorption → relaxation → neodymium upper laser level

Narrow solid-state transition

Continuous or pulsed; the cavity and pump schedule decide

Carbon dioxide

$10600\,\mathrm{nm}$

Discharge → nitrogen vibration → resonant transfer to carbon dioxide

Many distinct vibrational–rotational lines

Continuous or pulsed; vibrational relaxation and gas cooling matter

KrF excimer

$248\,\mathrm{nm}$

Discharge chemistry → excited bound exciplex → repulsive lower continuum

Broad molecular band

Usually pulsed; formation, dissociation and gas chemistry limit cycling

Rhodamine 6G dye

$600\,\mathrm{nm}$

Optical pump → vibrational relaxation in excited singlet → emission band

Broad vibronic band; representative wavelength only

Continuous or pulsed; triplets, bleaching and flow matter

MIT: gain media and laser types, sections 7.2–7.3

Start with the essentials

Focus question: Which pump route does each medium use, and where does its population get stuck?
One-sentence intuition: How pumping builds an inversion depends on which state absorbs at the laser wavelength and how quickly that state empties.

Core mathematical model

The populations add up to one

n_g+n_d+n_u+n_\ell=1

The four fractions describe one effective, normalized ensemble. The donor fraction does not count real helium or nitrogen atoms.

Energy transfer through a donor

\begin{aligned}\dot n_d&=rn_g-k_dn_d\\F_u&=\eta_dk_dn_d\end{aligned}

A pump that matches the medium fills the donor reservoir. The part of the population that fails to transfer falls back to the ground reservoir.

The absorbing state changes the threshold

\Delta=\begin{cases}n_u-n_g&\text{ruby}\\n_u-n_\ell&\text{other proxies}\end{cases}

In ruby the upper level has to outnumber a well-filled ground state. When the lower level empties quickly, the upper level has a smaller population to beat.

Gain for one selected mode

\begin{aligned}G&=8\Delta/(1+4\delta^2)\\\dot s&=(G-\kappa)s+\beta n_u\end{aligned}

Detuning is measured in units of an illustrative full gain width. All media share this one line shape, so it cannot reproduce media that really have many lines.

Blocking feedback still leaves a background

\begin{aligned}\kappa&=\kappa_i+\kappa_o+\kappa_{\rm block}\\p_{\rm out}&=\kappa_os\end{aligned}

Blocking the feedback makes photons escape quickly, but the small spontaneous emission into the mode remains. Output is in normalized units.

Small-signal margin before the field depletes the inversion

M_0=8\Delta_0/(1+4\delta^2)-\kappa

The zero-field steady population supplies an onset reference. The growing field depletes it. For pulsed pumping this reference uses only the on-pump branch; the actual trajectory can approach a repeating cycle.

Common difficulties

Gain alone does not make a laser

Typical misconceptionIf the medium is pumped, it has to lase.

Better mental modelThe mode lases only if its gain makes up for all the losses and the population feeding it keeps being resupplied.

Know what the model leaves out

Typical misconceptionThese teaching curves predict how a commercial laser behaves.

Better mental modelThese normalized curves compare pumping mechanisms at chosen rates. A real power or efficiency comparison also needs measured material parameters, pump absorption and cavity losses.

Run the experiment

01
Compare the absorbing state
Compare Ground-state absorption and Fast lower-state clearance with the same Normalized pump feed. Each selection automatically acquires sixty lifetime units.
What to observe: Ruby's absorbing ground state is heavily populated, so it needs a larger upper population before it shows gain.
02
Cut the energy-transfer route
Choose Collision transfer, then switch on Optical pump selected.
What to observe: In this model He–Ne is pumped by a discharge, so the optical pump feeds nothing.
03
Slow the emptying of the lower level
Choose Fast lower-state clearance and raise Lower-state clearance slowdown. Its control is hidden for ruby, whose lower laser state is the ground state.
What to observe: Atoms pile up in the lower level, and both the inversion and the output drop.
04
Pumping is not the same as lasing
Compare continuous and pulsed pumping, then switch off Cavity feedback connected. The plots are reacquired; Pause replay stops only the transport animation.
What to observe: A filled upper level is not enough. Whether the laser oscillates still depends on the feedback and on gain beating loss.