---
title: "Laser Linewidth & Phase Noise Interactive Lab | Sandbox Physics"
description: "How does phase noise become a laser linewidth? The phase drifts randomly, so the longer you wait, the less the field keeps its earlier phase."
canonical_url: "https://sandboxphysics.com/labs/laser-linewidth-phase-noise/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/laser-linewidth-phase-noise/"
last_modified: "2026-10-27"
---

L30 · Follow a phase until a line appears

# Laser Linewidth & Phase Noise

**How does phase noise become a laser linewidth?** The phase drifts randomly, so the longer you wait, the less the field keeps its earlier phase. That loss of coherence spreads the spectrum into a line of finite width. What you measure depends on the kind of noise, on the reference laser you beat against and on how long you observe; reading the width off the highest point of one spectrum is not enough.

Let the laser phase take a reproducible random walk, then look at the beat spectrum, frequency-noise density and coherence that come out of it. Change the photon lifetime, output power, technical noise and observation time, and compare.

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

This teaching model follows only the phase of a single-mode laser in a good cavity, with fixed amplitude and an assumed wavelength of one thousand sixty-four nanometers. The gain medium is ideal and fully inverted: spontaneous emission into the mode equals the photon loss rate, and output loss is half of the total loss. Output power and photon lifetime set the mean photon number in the cavity. The Lorentzian FWHM follows a Schawlow–Townes-type convention: the loss rate divided by four pi times that photon number. This ideal scaling is not a general formula for semiconductor, bad-cavity or multimode lasers. The model leaves out Henry linewidth enhancement, incomplete inversion, nonorthogonal modes, amplitude noise and thermal laser dynamics. The tested laser and the reference each get independent white phase diffusion, so their Lorentzian widths add in the beat. Extra technical frequency noise is a stationary Ornstein–Uhlenbeck process with a set RMS and correlation time. At each step, the noise value and its time integral are drawn together from their exact Gaussian covariance. Sinusoidal vibration is integrated analytically. Seeded pseudorandom Gaussian steps make each run reproducible; the result is a simulation, not measured data, and it does not follow individual spontaneous photons. An ideal coherent optical hybrid records both quadratures at one megasample per second, with a nominal positive beat of fifty kilohertz. Phase unwrapping is perfect in the model; real receiver phase slips, shot noise and anti-aliasing electronics are not simulated. The optical spectrum applies a Hann window to the captured complex samples. The one-sided frequency-noise PSD averages four nonoverlapping, mean-removed Hann windows of the recorded phase increments, so its bins are four times wider. Every diagnostic uses the same frozen record. The symmetric half-power span is measured on this finite record; it is not a Lorentzian FWHM fit. Field coherence is a lag average over the finite record and is statistically uncertain at long lags. The green coherence curve is a separate reference for white noise only. The continuous frequency-noise reference includes the stationary correlated noise but not the vibration line or finite sampling, so compare it only well below the Nyquist frequency. Changing the observation length changes the resolution and the statistical scatter, and different realizations differ even with identical source settings. Spatial layout and beam colors are schematic.

Laser foundations · Light, gain and feedback

[L01 · Compare sources](https://sandboxphysics.com/labs/laser-source-comparison/) [L02 · Count transitions](https://sandboxphysics.com/labs/light-matter-transitions/) [L03 · Build inversion](https://sandboxphysics.com/labs/population-inversion/) [L04 · Amplify once](https://sandboxphysics.com/labs/single-pass-amplifier/) [L05 · Add feedback](https://sandboxphysics.com/labs/passive-optical-resonator/) [L06 · Start from noise](https://sandboxphysics.com/labs/laser-noise-startup/) [L07 · Build and debug](https://sandboxphysics.com/labs/build-debug-laser/) [L08 · Measure and diagnose](https://sandboxphysics.com/labs/laser-diagnostic-bench/) [L09 · Select axial modes](https://sandboxphysics.com/labs/gain-longitudinal-modes/) [L10 · Confine the beam](https://sandboxphysics.com/labs/stable-resonator-geometry/) [L11 · Select spatial modes](https://sandboxphysics.com/labs/transverse-laser-modes/) [L12 · Propagate and measure](https://sandboxphysics.com/labs/gaussian-beam-quality/) [L13 · Align and match](https://sandboxphysics.com/labs/alignment-mode-matching/) [L14 · Optimize extraction](https://sandboxphysics.com/labs/output-coupler-optimization/) [L15 · Drive the dynamics](https://sandboxphysics.com/labs/laser-dynamics-relaxation/) [L16 · Store and release](https://sandboxphysics.com/labs/q-switched-giant-pulse/) [L17 · Lock phases into pulses](https://sandboxphysics.com/labs/mode-locking-frequency-comb/) [L18 · Calibrate the comb](https://sandboxphysics.com/labs/optical-frequency-comb/) [L19 · Inject carriers](https://sandboxphysics.com/labs/semiconductor-laser-diode/) [L20 · Compare gain media](https://sandboxphysics.com/labs/laser-medium-zoo/) [L21 · Distributed gain](https://sandboxphysics.com/labs/fiber-fbg-laser/) [L22 · Choose feedback](https://sandboxphysics.com/labs/laser-feedback-architectures/) [L23 · Balance ultrafast pulses](https://sandboxphysics.com/labs/ti-sapphire-oscillator/) [L24 · Stretch and compress](https://sandboxphysics.com/labs/chirped-pulse-amplification/) [L25 · Trap and extract](https://sandboxphysics.com/labs/regenerative-multipass-amplifier/) [L26 · Convert frequencies](https://sandboxphysics.com/labs/nonlinear-frequency-conversion/) [L27 · Scatter and cascade](https://sandboxphysics.com/labs/raman-brillouin-lasers/) [L28 · Synchronize a slave](https://sandboxphysics.com/labs/injection-locking/) [L29 · Lock to a reference](https://sandboxphysics.com/labs/pdh-frequency-lock/) [L30 · Measure phase noise](https://sandboxphysics.com/labs/laser-linewidth-phase-noise/) [L31 · Scatter for feedback](https://sandboxphysics.com/labs/random-laser/) [L32 · Diagnose thermal scaling](https://sandboxphysics.com/labs/thermal-lensing/) [L33 · Combine coherent fields](https://sandboxphysics.com/labs/coherent-beam-combining/) [L34 · Sustain a Kerr pulse](https://sandboxphysics.com/labs/microresonator-kerr-comb/) [L35 · Return a delayed field](https://sandboxphysics.com/labs/external-feedback-chaos/) [L36 · Scan without mode hops](https://sandboxphysics.com/labs/tunable-laser-scan/)

Physical basis: [MIT: coherence and spectrum](https://ocw.mit.edu/courses/res-6-006-video-demonstrations-in-lasers-and-optics-spring-2008/resources/coherence-length-and-source-spectrum/) · [Rate equations](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/resources/chapter2/) · [Pumping and gain media (§7.2)](https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/bf3bdeb971819ddc9c68719d28bb2745_classnotes.pdf) · [Laser dynamics](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/resources/chapter4/)

Physics tutorial

## A spectral line is the memory of a phase

**Background** Spontaneous emission and technical frequency noise keep changing the phase of a laser's light, even when its amplitude stays fixed.

**Why it matters** You analyze one simulated I/Q record twice: as a phase that wanders in time, and as a spectrum in frequency.

Start with the essentials

- **Focus question**: How does phase noise become a laser linewidth?
- **One-sentence intuition**: The shape of the line depends on the kind of noise, on the reference laser and on how long you observe; the tallest FFT bin alone does not tell you.

### Core mathematical model

#### Ideal quantum-limited linewidth

$$
\begin{aligned}\bar n&=\frac{P_{\rm out}\tau_p}{\eta_{\rm out}h\nu}\\\Delta\nu_q&=\frac{1}{4\pi\bar n\tau_p}\\\eta_{\rm out}&=\frac12\end{aligned}
$$

In this ideal gain medium, spontaneous emission adds one photon to the lasing mode per photon lifetime. The numerical factor rests on these assumptions and does not hold for every laser.

#### Phase diffusion and the beat linewidth

$$
\begin{aligned}d\phi&=2\pi f_b\,dt+\sqrt{2D}\,dW\\D&=\pi\Delta\nu_w\\\Delta\nu_b&=\Delta\nu_l+\Delta\nu_r\end{aligned}
$$

When the laser and the reference each have independent white phase noise, their linewidths add in the beat note between them.

#### Technical noise with a correlation time

$$
\begin{aligned}dX&=-\frac{X}{\tau_c}\,dt+\sqrt{\frac{2\sigma_f^2}{\tau_c}}\,dW\\\langle X(t)X(t+\tau)\rangle&=\sigma_f^2e^{-|\tau|/\tau_c}\end{aligned}
$$

This frequency noise is stationary and stays correlated for about one correlation time. The model advances the noise and the phase it accumulates together, so every spectrum comes from the same record.

#### White and colored frequency-noise density

$$
\begin{aligned}S_{\nu,w}(f)&=\Delta\nu_b/\pi\\S_{\nu,c}(f)&=\frac{4\sigma_f^2\tau_c}{1+(2\pi f\tau_c)^2}\\S_\nu&=S_{\nu,w}+S_{\nu,c}\end{aligned}
$$

The green curve is a one-sided continuous reference; it leaves out vibration lines and sampling effects. The measured PSD is computed from the recorded phase increments.

#### Coherence from phase diffusion

$$
\begin{aligned}|g^{(1)}(\tau)|&=e^{-\pi\Delta\nu_b|\tau|}\\\tau_{\rm coh}&=1/(\pi\Delta\nu_b)\end{aligned}
$$

The exponential is the ensemble average for white noise alone. A coherence averaged over lags in one finite record can deviate from it.

#### Spectra of the recorded signal

$$
\begin{aligned}z_n&=e^{i\phi_n}\\S_z(f)&\propto|\operatorname{DFT}(w_nz_n)|^2\\\delta f_{\rm opt}&=1/(N\Delta t)\\\delta f_{\rm noise}&=4\delta f_{\rm opt}\end{aligned}
$$

The optical spectrum uses one Hann window over the whole record. The frequency-noise PSD averages four shorter windows. The half-power span is measured on the spectrum itself; it is not a fitted FWHM.

### Common difficulties

#### One linewidth describes all noise

**Typical misconception** Every broad line is a Lorentzian, and the width around its tallest bin is the laser's intrinsic linewidth.

**Better mental model** Colored noise and vibration change the shape of the line. Before you quote a width, state how long you observed and which measure of width you used.

#### A beat measures only the tested laser

**Typical misconception** You can ignore the reference laser's noise when you read a beat note.

**Better mental model** The white-noise linewidths of two independent lasers add in the beat. Unless you know the reference is quiet, the beat cannot give you the linewidth of either laser on its own.

### Run the experiment

1. 01

   #### Change power and photon lifetime

   With the same seed, capture the Ideal quantum scaling, Brighter laser and Shorter photon lifetime presets.

   **What to observe:** Higher power slows the ideal phase diffusion, and a shorter photon lifetime speeds it up.

2. 02

   #### Add technical noise

   Capture the Correlated drift, Vibration sidebands and Noisy reference presets.

   **What to observe:** Each source leaves a different mark: drift makes the phase wander slowly, vibration adds discrete sidebands, and a noisy reference broadens the beat.

3. 03

   #### Change the record length

   Compare a short record with a long one, then press New noise realization.

   **What to observe:** The apparent spectral width changes with frequency resolution and from one random realization to the next.

Physical conclusion

Check the ideal white-noise FWHM against what the record shows: its noise density, its coherence and the half-power span of a finite record.

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)