L07 · Synthesis and measurement

Build & Debug Your First Laser

Assemble a pump, three- or four-level medium, curved mirrors, aperture and detector. Diagnose hidden faults from population, modal gain and output, then balance efficiency, mode purity and heat.

Model time $\text{—}$

Upper population fraction $\text{—}$

Available inversion $\text{—}$

Unsaturated net round-trip gain $\text{—}$

Geometry stability product $\text{—}$

Mode radius at medium $\text{—}$

Ideal pump threshold $\text{—}$

Round-trip aperture power loss $\text{—}$

Measured useful output $\text{—}$

Dominant longitudinal mode share $\text{—}$

Absorbed-pump efficiency $\text{—}$

Minimum quantum-defect heat $\text{—}$

Last microsecond stability $\text{—}$

Design score out of 100 $\text{—}$

L07 · Build challenge

Make it lase, then make it better

Four-level mediumTurn off to use a three-level ground-state lower level.Pump connectedChange the physical assembly and retain the evolving state.Gain medium installedChange the physical assembly and retain the evolving state.Rear mirror installedChange the physical assembly and retain the evolving state.Output coupler installedChange the physical assembly and retain the evolving state.Aperture installedChange the physical assembly and retain the evolving state.Detector connectedChange the physical assembly and retain the evolving state.Effective pump rate

R

1.2\,\mathrm{\mu s^{-1}}

Mirror separation

L

45\,\mathrm{cm}

Rear mirror curvature radius

\mathcal R_b

90\,\mathrm{cm}

Output mirror curvature radius

\mathcal R_o

90\,\mathrm{cm}

Output coupler transmission

T

12\,\mathrm{\%}

Internal round-trip power loss

\ell

1.5\,\mathrm{\%}

Gain half-width

\gamma

0.15\,\mathrm{GHz}

Selected central mode offset

\Delta\nu

0\,\mathrm{GHz}

Central aperture radius

a

0.55\,\mathrm{mm}

Advance twice to inspect settling. Load a hidden fault, identify it, repair the settings and evaluate. A score of 75 with settled output passes the teaching challenge.

Illustrative three-mode classical rate model: ten billion emitters, 1064 nm emission, 808 nm absorbed pump, 2 μs upper lifetime and a peak cross section of 0.000000000000000008 square metres. Concave mirror radii are positive; paraxial Gaussian geometry determines mode area and twice-per-trip aperture clipping. Marginal and unstable geometries have no supported mode in this model; real unstable-resonator lasers are outside its scope. Three selected longitudinal modes share one homogeneous reservoir. Spontaneous seeding is a continuous mean source. Pump depletion, spatial hole burning, thermal lensing and optical phase are omitted. Heat is the minimum quantum-defect proxy; scores are teaching objectives.

Start with the essentials

Focus question: Can you diagnose a hidden fault from measured populations, modal gain and output, then repair it?
One-sentence intuition: Check the reservoir, feedback path and modal loss separately. Stronger pump cannot repair missing mirrors, unsupported geometry or a severely clipped mode.

Core mathematical model

Three- versus four-level inversion

d_4=n_u,\qquad d_3=2n_u-1

The four-level lower state empties instantly; the three-level lower state is the ground population. The same upper fraction can therefore amplify in one scheme and absorb in the other.

One reservoir, three competing modes

\dot n_u=R(1-n_u)-\frac{n_u}{\tau_u}-\frac{d}{N_a}\sum_jG_jS_j

Stimulated transitions change the reservoir that sets every modal gain. Photon growth includes stimulated gain, logarithmic cavity loss and a small continuous spontaneous source.

Modal growth and output

\dot S_j=(G_jd-\kappa)S_j+\frac{\beta N_an_u}{3\tau_u},\quad P_o=h\nu\kappa_o\sum_jS_j

Only three selected longitudinal modes are included. Changing detuning can move gain toward a neighboring selected mode. Unmodeled distant modes and spatial hole burning are excluded.

Stable geometry and aperture

g_i=1-\frac{L}{\mathcal R_i},\quad 0<g_bg_o<1,\quad Q_a=\left[1-e^{-2a^2/w_m^2}\right]^2

The Gaussian fixed point of the round-trip ray matrix determines the medium-plane radius. The central aperture is crossed twice per trip. Degenerate boundary geometries are excluded from this model.

Pump energy and minimum heat

P_p=h\nu_pN_aR(1-n_u),\quad P_{h,\min}=P_p\left(1-\frac{\nu}{\nu_p}\right)

Pump power is absorbed power in the effective one-way pump model. This minimum quantum-defect heat excludes nonradiative processes and does not calculate temperature or lensing.

Common difficulties

A geometry boundary is a model boundary

Typical misconceptionAn unstable cavity can never lase.

Better mental modelThis Lab supports the paraxial stable Gaussian family only. Real unstable-resonator lasers need a different transverse field and diffraction model.

The score is a teaching rubric

Typical misconceptionA high score certifies a manufacturable laser.

Better mental modelThe illustrative material constants, three-mode truncation and minimum heat proxy omit engineering limits. The score evaluates this stated model only.

Run the experiment

01
Assemble and settle
Choose Working four-level design. Advance twice and compare output, dominant-mode share and the last-microsecond stability flag.
What to observe: The reservoir first fills, then stimulated growth clamps the inversion. The output can overshoot before settling.
02
Remove a necessary link
Remove the rear mirror and advance again; reinstall it. Disconnect the detector separately.
What to observe: Missing feedback prevents the supported oscillator; disconnecting the detector changes the measurement availability rather than the reservoir dynamics.
03
Diagnose, then repair
Load a hidden fault, advance and choose a diagnosis. Repair the relevant controls without loading a preset, then advance and evaluate.
What to observe: The case may involve insufficient pump, no inversion, spectral mismatch, clipping, output coupling, geometry or missing feedback.
04
Trade power against quality
Compare the weak and strong output-coupling presets, then the broad-gain competition preset. Optimize the working design with less pump.
What to observe: Weak extraction can waste useful output; strong extraction raises threshold. Broad gain admits more competing modes. Scores reward several objectives.