L06 · Gain and feedback

Laser Noise Startup Workbench

Where does the first photon in a laser cavity come from? In an unseeded cavity, spontaneous emission can put the first photon into the selected mode. If gain exceeds all cavity losses, stimulated emission amplifies that seed; depletion then limits its growth. An injected photon can provide a seed too.

Watch a recorded laser startup from random mode seeding. Move the output mirror, compare gain with loss, and seed an exact dark state with one photon.

Integrated record duration $\text{—}$

Elapsed round trips $\text{—}$

Recorded end upper fraction $\text{—}$

Loss-compensating upper fraction $\text{—}$

Recorded end mean mode photons $\text{—}$

Recorded end output power $\text{—}$

Current stimulated gain rate $\text{—}$

Total cavity loss rate $\text{—}$

Round-trip net logarithmic gain $\text{—}$

Ideal zero-noise pump threshold $\text{—}$

Cumulative spontaneous mode events $\text{—}$

L06 · Experiment

Where does the first photon in a laser cavity come from?

Logarithmic photon traceTurn off to inspect linear output power and relaxation oscillations.Pump onYou can switch it during a run; the stored excitation is kept.Spontaneous cavity seedingUse the exact dark state to test the role of the first photon.Effective pump rate

R

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

Upper radiative lifetime

\tau_u

2\,\mathrm{\mu s}

Vacuum-equivalent cavity length

L

50\,\mathrm{cm}

Peak stimulated cross section

\frac{\sigma_0}{10^{-18}\,\mathrm{m}^2}

2

Mode area scale

\frac{A}{10^{-8}\,\mathrm{m}^2}

1

Output mirror transmission

T

10\,\mathrm{\%}

Rear mirror reflectivity

R_b

99\,\mathrm{\%}

Internal round-trip power loss

\ell

2\,\mathrm{\%}

Gain half-width

\gamma

200\,\mathrm{GHz}

Mode offset from gain center

\Delta\nu

0\,\mathrm{GHz}

Random seed

s

7

Compare below and above threshold, then disable seeding and inject exactly one photon. Finally turn the pump off and compare upper-state and cavity decay times.

This is a hybrid single-mode rate model: spontaneous photons arrive as Poisson events and seed a classical mean photon population. The effective four-level pump works in one direction only, and the lower laser level empties instantly. The model fixes one billion active emitters, a 2 cm gain length, 1064 nm photons and a spontaneous mode fraction of 0.00000001. The cross section and lifetime are illustrative values, not fitted to a material. Loss is the logarithmic attenuation over one round trip. Transverse effects, competing modes, optical phase and quantum photon statistics are left out, and the transport markers are schematic. Each selection acquires ten microseconds. Pumping and seeding switches retain stored excitation and integrate ten more microseconds; numerical parameters prepare a fresh run. Readouts describe the record end, while replay samples the retained trajectory without adding events. The gain rod and amber event symbol are enlarged; the flash marks an accepted time, not a localized emitter. Mirror separation follows the vacuum-equivalent length with a schematic scale. Logarithmic brightness and moving markers do not count photons; uncollected rear and internal losses are not drawn separately. Only the last eighty spontaneous event times are shown; the counter includes all accepted events. Green glow follows the recorded upper fraction.

Start with the essentials

Focus question: If spontaneous emission into the mode is switched off, can a medium pumped above threshold stay exactly dark?
One-sentence intuition: Gain above loss lets a seed grow; if the model omits seeding, an exactly zero photon population stays zero.

Core mathematical model

Rate equations for stored excitation and cavity photons

\begin{aligned}\dot n&=R(1-n)\\&\quad-\frac{n}{\tau_u}-\frac{GnS_c}{N_a}\\\dot S_c&=(Gn-\kappa)S_c\\&\quad+\frac{\beta N_a n}{\tau_u}\end{aligned}

The first equation evolves the upper-state fraction; the second evolves the cavity photon population. There are one billion active emitters. Spontaneous mode seeds arrive as accepted Poisson events: each removes one excited emitter and adds one photon. Stimulated growth and losses are continuous averages.

Geometry, spectral overlap and loss

\begin{aligned}G&=\frac{2\sigma_0N_a}{AT_R}\\&\quad\times\frac{1}{1+(\Delta\nu/\gamma)^2}\\\kappa&=-\frac{\ln R_b}{T_R}\\&\quad-\frac{\ln(1-T)}{T_R}\\&\quad-\frac{\ln(1-\ell)}{T_R}\end{aligned}

In a linear cavity, light crosses the gain medium twice per round trip. The number of active emitters is fixed at one billion, so changing the mode area changes their density and their overlap with the photons. The round-trip time uses a vacuum-equivalent optical length.

Ideal zero-noise threshold

\begin{aligned}n_{\mathrm{th}}&=\kappa/G\\n_0&=\frac{R\tau_u}{1+R\tau_u}\\R_{\mathrm{th}}&=\frac{n_{\mathrm{th}}}{\tau_u(1-n_{\mathrm{th}})}\end{aligned}

If the required upper-level fraction is one or more, no finite pump rate reaches threshold. This is a zero-source steady-state reference. Finite seeding leaves a weak background below it; a single trajectory still contains discrete one-photon spontaneous jumps.

Gain clamping and envelope output

\begin{aligned}n_{\mathrm{ss}}&\simeq n_{\mathrm{th}}\\\frac{\kappa S_{c,\mathrm{ss}}}{N_a}&\simeq R(1-n_{\mathrm{th}})\\&\quad-\frac{n_{\mathrm{th}}}{\tau_u}\\P_{\mathrm{out}}&=h\nu\kappa_{\mathrm{out}}S_c\end{aligned}

The first two relations ignore the small spontaneous source above threshold. The output loss rate is the logarithmic attenuation at the output mirror divided by the round-trip time, and the displayed power is an average taken from the envelope.

Common difficulties

Only the seeding is random

Typical misconceptionThe photon-number trace is an exact quantum simulation that counts every photon.

Better mental modelThe spontaneous events in the mode are discrete, but stimulated growth and loss follow a classical envelope that stays positive. The model does not compute photon correlations, phase diffusion or linewidth.

Positive gain is not enough

Typical misconceptionAny positive inversion is enough to make the laser oscillate.

Better mental modelThe unsaturated gain has to beat all the losses in one trip around the cavity. Strong output coupling, internal loss or a mode far from the gain line can put threshold out of reach.

Run the experiment

01
Tell noise from lasing
Compare Below threshold and Above threshold. Each selection automatically acquires ten microseconds. Read the event count and the output, even when the light is weak.
What to observe: Below threshold, spontaneous photons leak out of the cavity, but they never build a macroscopic population in the mode. Above threshold, the mode grows and uses up the stored inversion.
02
Start from exact darkness
Choose Exact dark state, then use Inject one photon and evolve. The one-photon intermediate sample is retained before ten more microseconds are integrated.
What to observe: With no noise source, a classical photon population of exactly zero stays at zero. One injected photon provides the missing seed.
03
Watch relaxation oscillations
Choose Relaxation oscillation. Compare the acquired upper fraction with the loss threshold; Logarithmic photon trace in Advanced controls changes only the display scale.
What to observe: Stored population, photon buildup and depletion produce an overshoot, followed by a damped exchange of energy between the atoms and the photons. Photon decay and upper-level recovery happen on different timescales.
04
Move the mode off the gain line
Choose Above threshold and increase the mode offset. Return to Above threshold, switch Pump on off, and inspect the continued record.
What to observe: Detuning reduces gain without changing mirror loss. Numerical parameters prepare a new run. Switching pumping or seeding keeps the stored state and integrates ten more microseconds, so the subsequent decay remains in the same record.

Laser Noise Startup Workbench

Let a mode grow from one random event

Core mathematical model

Rate equations for stored excitation and cavity photons

Geometry, spectral overlap and loss

Ideal zero-noise threshold

Gain clamping and envelope output

Common difficulties

Only the seeding is random

Positive gain is not enough

Run the experiment

Tell noise from lasing

Start from exact darkness

Watch relaxation oscillations

Move the mode off the gain line

Related experiments and articles

From a seed to laser light

Controlled light and precision measurement