---
title: "Laser Startup from Noise Interactive Lab | Sandbox Physics"
description: "How does a laser grow from random noise into one dominant mode? Spontaneous emission seeds every cavity mode with a little noise."
canonical_url: "https://sandboxphysics.com/labs/laser-startup-from-noise/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/laser-startup-from-noise/"
last_modified: "2026-10-28"
---

Optics 103 · Mode competition

# Laser Startup from Noise

**How does a laser grow from random noise into one dominant mode?** Spontaneous emission seeds every cavity mode with a little noise. Above threshold, gain beats cavity loss, and the mode with the most net gain grows exponentially, faster than the rest. As it grows it saturates the gain, pulling it down until gain just equals loss, and that mode ends up dominating.

The page opens on a recorded start-up from pump turn-on. Every longitudinal mode starts from spontaneous noise; watch them grow together on the camera until one is left.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Mean-field rate equations for a homogeneously broadened four-level gain medium in a linear two-mirror cavity. All longitudinal modes share one inversion, and the gain and the spontaneous emission into each mode follow one Lorentzian line. Each mode gains photons by stimulated emission in proportion to the photons already in it plus one; the extra one is the spontaneous seed. Photons leave with the photon lifetime set by the round-trip time, the output coupler and the other round-trip loss. Inversion is given in units of the threshold inversion of a mode at line centre, and the pump in units of that mode's threshold pump. The spontaneous fraction sets how many ions that threshold needs, so it also sets the photon numbers and output power; the power uses a 1064 nanometre photon energy. Each record starts with no inversion and no photons when the pump turns on, and runs for six upper-level lifetimes. Spontaneous emission enters as its mean value: the model has no individual random events, optical phase or linewidth. The single-mode laser start-up Lab shows discrete spontaneous events. Spatial hole burning is left out. In a real standing-wave cavity it lets a few more modes survive; a unidirectional ring cavity removes it. The offset of the nearest mode from line centre is a sub-wavelength mirror shift and stays fixed while the mirror spacing changes. There are no thermal effects, transverse modes or polarization. Analytic values are single-mode limits for the best mode, shown separately from the record. The ringing frequency from the record uses only peaks within twenty percent of the clamped photon number. The scene uses one display scale for mirror spacing; components, the grating dispersion and the camera size are schematic, and brightness is logarithmic on a fixed scale.

Physics tutorial

## Watch noise turn into one laser mode

**Background** A laser cavity supports many longitudinal modes, spaced by the speed of light over twice the mirror spacing. When the pump turns on, spontaneous emission puts a little light into every one of them.

**Why it matters** Nobody chooses which mode lases. The record shows the modes growing from the same noise, the inversion stopping at a threshold, and all but one mode fading out.

Start with the essentials

- **Focus question**: Why does one mode win, and how long does it take?
- **One-sentence intuition**: All modes draw on one shared inversion. Once the best mode clamps it at that mode's threshold, every other mode has less gain than loss, so it decays back to its spontaneous glow.

### Core mathematical model

#### One inversion, one equation per mode

$$
\begin{aligned}\frac{dx}{dt}&=\frac{r-x}{\tau_L}-\frac{x}{\tau_p N_{\rm th}}\sum_k \ell_k n_k\\\frac{dn_k}{dt}&=\frac{x\,\ell_k\,(n_k+1)-n_k}{\tau_p}\end{aligned}
$$

Here x is the inversion over the threshold of a line-centre mode, r the pump over that mode's threshold pump, and the plus one is spontaneous emission into the mode.

#### The gain line and the mode comb

$$
\begin{aligned}\ell_k&=\frac{1}{1+\left[2(k+\delta)\,\Delta\nu_{\rm FSR}/\Delta\nu_g\right]^2}\\\Delta\nu_{\rm FSR}&=\frac{c}{2L}\end{aligned}
$$

Mode k sits k spacings from the mode nearest line centre, which is offset by delta spacings. A longer cavity packs more modes under the same line.

#### Photon lifetime

$$
\tau_p=\frac{2L/c}{-\ln\left[(1-T)(1-\alpha_{\rm i})\right]}
$$

The round-trip time divided by the logarithmic round-trip loss of the output coupler and the other losses.

#### Threshold and the first crossing

$$
\begin{aligned}x_{\rm th}&=\frac{1}{\ell_0}\\t_{\rm th}&=\tau_L\ln\frac{r\ell_0}{r\ell_0-1}\end{aligned}
$$

Before the light is strong the inversion rises as r times one minus the exponential decay. The best mode reaches threshold at this time, and the first spike comes a little later.

#### Clamped steady state

$$
\begin{aligned}x\,\ell_0&\to1\\n_0&=\frac{r-1/\ell_0}{\beta}\\n_{k\neq0}&=\frac{\ell_k}{\ell_0-\ell_k}\end{aligned}
$$

Extra pump only adds photons to the winner. Each losing mode keeps a spontaneous level set by how close its gain is to the winner's.

#### Relaxation oscillation

$$
\begin{aligned}\omega_R^2&=\frac{r\ell_0-1}{\tau_L\tau_p}-\gamma_R^2\\\gamma_R&=\frac{r\ell_0}{2\tau_L}\end{aligned}
$$

After the first spike, the photons and the inversion trade energy and ring down at this rate. The record's late peaks are compared with this small-signal result.

#### Why the losers fade

$$
\frac{d\ln n_k}{dt}\approx-\frac{1-\ell_k/\ell_0}{\tau_p}
$$

Once the inversion is clamped, a mode whose gain is almost as high as the winner's loses almost nothing per round trip and fades slowly.

### Common difficulties

#### Clamping does not mean the pump stops mattering

**Typical misconception** Once the gain is clamped, pumping harder changes nothing.

**Better mental model** The inversion stays at threshold, but the extra pump goes into photons in the winning mode, so the output power keeps rising.

#### Homogeneous gain alone is not the whole story

**Typical misconception** Every laser with a homogeneous gain line runs on one mode.

**Better mental model** This model shares one inversion among all modes. In a real standing-wave cavity, spatial hole burning leaves unused inversion between the standing-wave peaks, so extra modes can survive.

#### The first spike is not the steady power

**Typical misconception** The height of the first spike is what the laser will deliver.

**Better mental model** The inversion overshoots threshold before the light catches up, so the first spike is far above the steady power and is followed by ringing.

### Run the experiment

1. 01

   #### Watch the record start from noise

   Keep the default preset, Noise grows into one mode, and watch the line camera through one replay.

   **What to observe:** The gain medium brightens first. Then many faint spots on the camera grow at once, a spike appears, and soon one spot is left.

2. 02

   #### Turn the pump down below threshold

   Load Below threshold: only glow.

   **What to observe:** The inversion settles at the pump level without ever reaching threshold. Every mode keeps only a weak glow, shaped by the gain line.

3. 03

   #### Broaden the gain line and lengthen the cavity

   Load Broad gain line: modes linger, or drag the output coupler outward.

   **What to observe:** More modes fit under the line and their gains are closer together, so several modes are still lasing at the end of the record.

4. 04

   #### Put line centre between two modes

   Load Two modes almost tied and compare the two strongest modes in the growth plot.

   **What to observe:** Two modes grow almost equally. The slightly better one wins, but the suppression at the end is far below the steady value.

Physical conclusion

Threshold, clamping and the winner are set by the gain line and the mode positions. How fast the losers fade depends on how close their gain is to the winner's.

## Related experiments and articles

Continue exploring

### From a seed to laser light

Does a smooth startup curve describe what happens in every individual run?

1. [You are here **Laser Startup from Noise** Follow the mean start-up of many modes and see one win.](https://sandboxphysics.com/labs/laser-startup-from-noise/)
2. [Lab **Laser Noise Startup Workbench** Repeat stochastic seeding and inspect the recorded photon history.](https://sandboxphysics.com/labs/laser-noise-startup/)

Probes and measurement · Reading path

### Controlled light and precision measurement

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How an ammonia beam became a microwave oscillator**](https://sandboxphysics.com/stories/probe-evidence/maser-inversion-and-feedback/)
2. [Lab **Absorption, Spontaneous & Stimulated Emission** Watch a probe beam grow, then fade, in a prepared gas](https://sandboxphysics.com/labs/absorption-spontaneous-stimulated-emission/)
3. [Lab **Pumping & Population Inversion** Pump three gases with one lamp and see which one amplifies](https://sandboxphysics.com/labs/pumping-population-inversion/)
4. [Lab **Passive Optical Resonator** Compare stored light with self-sustained oscillation](https://sandboxphysics.com/labs/passive-optical-resonator/)
5. [Lab **Stable Resonator & Transverse Modes** Curve two mirrors and size an iris until one beam shape is left](https://sandboxphysics.com/labs/stable-resonator-transverse-modes/)
6. [Lab **Build & Debug Your First Laser** Test gain against cavity loss](https://sandboxphysics.com/labs/build-debug-laser/)
7. [Lab **Laser Noise Startup Workbench** Follow a cavity mode from its first seed](https://sandboxphysics.com/labs/laser-noise-startup/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)