L07 · Synthesis and measurement

Build & Debug Your First Laser

When a laser you built won't lase, how do you find the fault? Check the excited population, the return path through the mirrors and the losses on each round trip. Extra pump cannot replace missing feedback. A narrow aperture can raise the threshold beyond the available pump. Once the laser oscillates, compare useful output, absorbed-pump efficiency, mode purity and heat.

Assemble a pump, a three- or four-level medium, curved mirrors, an aperture and a detector. Find hidden faults from the 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 medium whose lower laser level is the ground state.Pump connectedToggling this changes the hardware; the laser keeps its current state.Gain medium installedToggling this changes the hardware; the laser keeps its current state.Rear mirror installedToggling this changes the hardware; the laser keeps its current state.Output coupler installedToggling this changes the hardware; the laser keeps its current state.Aperture installedToggling this changes the hardware; the laser keeps its current state.Detector connectedToggling this changes the hardware; the laser keeps its current 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}

The startup response is recorded automatically over twenty microseconds. Advanced controls contain hidden faults, hardware assembly and design evaluation. A score of 75 with a settled output passes the teaching challenge.

This is an illustrative classical rate model with three modes: ten billion emitters, emission at 1064 nm, an absorbed pump at 808 nm, an upper-level lifetime of 2 μs and a peak cross section of 0.000000000000000008 square metres. Concave mirrors have positive radii. Paraxial Gaussian geometry sets the mode area and the clipping at the aperture, which the beam passes twice per round trip. Marginal and unstable geometries have no supported mode in this model; real unstable-resonator lasers are outside its scope. The thin gain medium and aperture share the cavity midpoint in the model; their separation in the drawing is schematic. Axial length, transverse size and mirror curvature use separate display scales. The Gaussian envelope is capped at the viewport display limit. The three chosen longitudinal modes share one homogeneous population. Spontaneous emission seeds the modes as a continuous average source. Pump depletion, spatial hole burning, thermal lensing and optical phase are left out. Heat is estimated by its minimum, the quantum defect, and the score reflects teaching goals.

Start with the essentials

Focus question: Can you find a hidden fault from the measured populations, modal gain and output, and then fix it?
One-sentence intuition: Check stored excitation, the return path and round-trip losses separately to find what prevents growth before increasing the pump.

Core mathematical model

Three- versus four-level inversion

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

In a four-level system the lower laser level empties instantly; in a three-level system the lower level is the ground state. So the same upper-level fraction can amplify in one scheme and absorb in the other.

Three modes compete for one population

\begin{aligned}\dot n_u&=R(1-n_u)-\frac{n_u}{\tau_u}\\&\quad-\frac{d}{N_a}\sum_jG_jS_j\end{aligned}

Stimulated transitions drain the same population that sets the gain of every mode. Photon growth includes stimulated gain, logarithmic cavity loss and a small, continuous spontaneous source.

Growth of each mode and the output

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

The output coupler removes photons continuously; multiplying its escape rate by the stored photon energy gives power. Only three chosen longitudinal modes are included. Changing the detuning can shift the gain toward a neighboring chosen mode. Distant modes and spatial hole burning are not modeled.

Stable cavity geometry and aperture loss

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

The Gaussian beam that reproduces itself after one round trip of the ray matrix sets the mode radius at the medium. The beam passes the central aperture twice per round trip. Degenerate geometries on the stability boundary are excluded from this model.

Pump energy and minimum heat

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

Pump power here means the power absorbed in the effective one-way pump model. An absorbed pump photon has more energy than an emitted laser photon. The minimum heat comes only from that energy difference, the quantum defect: it leaves out nonradiative processes and does not calculate temperature or thermal lensing.

Common difficulties

Unstable cavities are outside this model

Typical misconceptionAn unstable cavity can never lase.

Better mental modelThis Lab handles only stable cavities with paraxial Gaussian modes. Real unstable-resonator lasers need a different diffraction model. The degenerate confocal boundary can support Gaussian modes, but this implementation excludes that boundary from its general matrix solution.

The score is for teaching

Typical misconceptionA high score proves the laser could be manufactured.

Better mental modelThe material constants are illustrative, only three modes are kept and heat is a minimum estimate, so engineering limits are missing. The score judges the design within this model only.

Run the experiment

01
Build it and let it settle
Choose Working four-level design and inspect the two startup curves. The twenty-microsecond response is already recorded; open All numerical results to check the last-microsecond stability flag.
What to observe: The population builds up first, then stimulated emission clamps the inversion. The output can overshoot before it settles.
02
Remove an essential part
In Advanced controls, remove the rear mirror, then put it back. Each edit records the next twenty microseconds from the existing population. Separately, disconnect the detector.
What to observe: Without feedback the oscillator cannot run. Disconnecting the detector only takes away the measurement; the population dynamics carry on unchanged.
03
Diagnose, then repair
Load a hidden fault in Advanced controls and pick a diagnosis from the recorded response. Repair it with the controls, without loading a preset, then evaluate.
What to observe: The fault may be too little pump, no inversion, a spectral mismatch, clipping, the output coupling, the geometry or missing feedback.
04
Trade power against quality
Compare the Too little output coupling and Too much output coupling presets, then Broad gain competition. Try to improve the working design with less pump.
What to observe: Too little extraction can waste useful output; too much raises the threshold. Broad gain lets more modes compete. The score rewards several goals at once.

Build & Debug Your First Laser

A working laser is a balanced system

Core mathematical model

Three- versus four-level inversion

Three modes compete for one population

Growth of each mode and the output

Stable cavity geometry and aperture loss

Pump energy and minimum heat

Common difficulties

Unstable cavities are outside this model

The score is for teaching

Run the experiment

Build it and let it settle

Remove an essential part

Diagnose, then repair

Trade power against quality

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