L06 · Gain and feedback

Laser Noise Startup Workbench

Pump an effective four-level medium inside a lossy cavity. Watch actual random spontaneous mode-seeding events, the turn-on delay, gain clamping and relaxation oscillations.

Model time $\text{—}$

Elapsed round trips $\text{—}$

Upper fraction / inversion $\text{—}$

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

Mean cavity mode photons $\text{—}$

Useful output power $\text{—}$

Current stimulated gain rate $\text{—}$

Total cavity loss rate $\text{—}$

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

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

Accepted spontaneous mode events $\text{—}$

L06 · Experiment

Where does the first cavity photon come from, and when can it grow?

Logarithmic photon traceTurn off to inspect linear output power and relaxation oscillations.Pump onChange live and retain stored excitation.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.

Hybrid single-mode rate model: Poisson spontaneous photons seed a classical mean photon population. One-way effective four-level pump; lower laser state empties instantly. Fixed one billion active emitters, 2 cm gain length, 1064 nm photons and spontaneous mode fraction 0.00000001. Cross section and lifetime are illustrative, not a material fit. Loss is round-trip logarithmic attenuation. Transverse physics, competing modes, optical phase and quantum photon statistics are omitted; the drawn standing wave is schematic.

Start with the essentials

Focus question: Can a pumped above-threshold medium remain exactly dark when spontaneous seeding is disabled?
One-sentence intuition: The threshold compares unsaturated gain with complete cavity loss. Finite spontaneous seeding smooths the transition; above threshold, saturation clamps gain near loss.

Core mathematical model

Coupled reservoir and photon flow

\dot n=R(1-n)-\frac{n}{\tau_u}-G n\frac{S_c}{N_a},\qquad\dot S_c=(Gn-\kappa)S_c+\beta\frac{N_a n}{\tau_u}

The displayed equations are the mean drift. In the simulation, the selected spontaneous contribution uses Poisson jumps: each accepted event removes one upper emitter and adds one mode photon. The remaining radiation is continuous.

Geometry, spectral overlap and loss

G=\frac{2\sigma_0N_a}{AT_R[1+(\Delta\nu/\gamma)^2]},\quad\kappa=-\frac{\ln[R_b(1-T)(1-\ell)]}{T_R}

The gain medium is traversed twice per linear-cavity round trip. The active-emitter count is fixed at one billion; changing area therefore changes density and photon overlap. Cavity time is a vacuum-equivalent optical length.

Ideal zero-noise threshold

n_{\mathrm{th}}=\kappa/G,\quad n_0=\frac{R\tau_u}{1+R\tau_u},\quad R_{\mathrm{th}}=\frac{n_{\mathrm{th}}}{\tau_u(1-n_{\mathrm{th}})}

The pump threshold is unreachable when the required fraction is at least one. This is an ideal zero-noise threshold, not a sharp discontinuity in a finite spontaneously seeded trajectory.

Gain clamping and calibrated output

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

The first two relations neglect the small spontaneous source above threshold. Output loss is the logarithmic attenuation from the output mirror divided by round-trip time; power is an averaged envelope readout.

Common difficulties

The stochastic events are only seeding

Typical misconceptionThe photon-number trace is an exact quantum counting simulation.

Better mental modelSpontaneous cavity events are discrete, but stimulated growth and loss use a positive classical envelope model. It does not calculate photon correlations, phase diffusion or linewidth.

Positive gain is insufficient

Typical misconceptionAny positive inversion guarantees oscillation.

Better mental modelUnsaturated gain must overcome complete feedback loss. Strong output coupling, internal loss or spectral mismatch can make the threshold unreachable.

Run the experiment

01
Separate noise from oscillation
Compare Below threshold and Above threshold using Advance 10 μs. Read the event count and output even in the weak state.
What to observe: Below threshold, spontaneous photons can leave the cavity but do not sustain a macroscopic mode. Above threshold, the mode grows and depletes stored inversion.
02
Test the exact dark state
Choose Exact dark state and advance 10 μs. Inject one photon, then advance again.
What to observe: Without a noise source, an exactly zero classical photon population remains zero. A single injected photon supplies the missing seed.
03
Observe relaxation
Choose Relaxation oscillation and advance. Compare the upper fraction with the loss threshold and inspect the linear output trace.
What to observe: Population storage, photon buildup and depletion produce overshoot and damped exchange; photon decay and upper-state recovery have different timescales.
04
Break spectral overlap
Choose Above threshold, increase mode detuning, then advance again. Turn the pump off to observe decay.
What to observe: Detuning reduces gain overlap while mirror loss remains. The state is preserved when rates change; each preset and random-seed change starts a new preparation.