---
title: "Gaussian Beam & Beam Quality Interactive Lab | Sandbox Physics"
description: "Can a lens make a poor beam diffraction limited? No."
canonical_url: "https://sandboxphysics.com/labs/gaussian-beam-quality/"
language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/gaussian-beam-quality/"
last_modified: "2026-10-27"
---

L12 · Propagation, focus and measurement

# Gaussian Beam & Beam Quality

**Can a lens make a poor beam diffraction limited?** No. An ideal lens can make the beam smaller or less divergent, but shrinking one enlarges the other: the product of waist size and divergence stays fixed, and that product is the beam quality. To measure it, record the beam radius at several distances, locate the waist and work out the quality factor.

Send a beam through free space, a focusing lens or a beam expander and follow its horizontal and vertical widths. Move a profiler along the beam, record how the radius changes, and recover the beam quality from the measured second-moment radii.

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

The beam travels in air. The model uses scalar paraxial propagation and ideal thin lenses with unlimited aperture. Axial position zero is the source's horizontal waist. The horizontal and vertical axes are fixed principal axes, and the model does not couple them. The radius is twice the root-mean-square width; for a Gaussian this is the radius where intensity falls to one over e squared. Sources with beam quality one are coherent Gaussian fields and have a physical complex beam parameter. Sources with a higher quality factor use a Gaussian Schell moment model: the intensity keeps the same Gaussian shape, but the angular spread is larger. They are not drawn as a coherent Gaussian with an arbitrary quality factor. Moving the vertical source waist along the axis models simple astigmatism. The covariances propagate through the same ray matrices as the apparatus, and ideal optics keep the quality of each axis fixed. The profiler is a movable sampling plane that does not disturb the beam; it is not an opaque stop. Plots are normalized to their peak. Records hold ideal profile radii, not noisy experimental images. Only planes after the last enabled lens enter the fit. The automatic 13-plane scan adapts to the output caustic and may run past the edge of the apparatus view. Three distinct planes are enough for this noiseless quadratic fit, but not for an ISO-compliant measurement. Finite apertures, aberrations, partial-coherence reconstruction, general rotated astigmatism and detector noise are left out. An output waist inferred upstream of the last lens is virtual: it is extrapolated back from the segment after that lens.

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 lens changes focus, not the phase-space area

**Background** A Gaussian beam spreads out from its narrowest point, the waist, because of diffraction. A lens bends the wavefront and can form a smaller waist, but a lossless lens leaves the determinant of the position–angle covariance unchanged.

**Why it matters** A single spot at one plane cannot tell a narrow, high-quality beam from a poor beam caught near its focus. To measure quality, you have to follow the beam as it propagates.

Start with the essentials

- **Focus question**: Can a lens reduce divergence without improving the beam quality factor?
- **One-sentence intuition**: Ideal optics can trade beam size for divergence, but the product of the two, taken from second moments, stays fixed and sets the beam quality; radii measured at several planes locate the waist and recover that product.

### Core mathematical model

#### Coherent Gaussian field

$$
\begin{aligned}q&=z-z_0+iz_R\\z_R&=\pi w_0^2/\lambda\\w^2(z)&=w_0^2\left[1+\left(\frac{z-z_0}{z_R}\right)^2\right]\end{aligned}
$$

This complex beam parameter describes a Gaussian field whose beam quality is one. Its real part is zero at the waist.

#### Free propagation and thin lenses

$$
\begin{aligned}P(d)&=\begin{pmatrix}1&d\\0&1\end{pmatrix}\\F(f)&=\begin{pmatrix}1&0\\-1/f&1\end{pmatrix}\\q^{\prime}&=(Aq+B)/(Cq+D)\end{aligned}
$$

The same ray matrices describe the apparatus and carry the beam's second moments through it. A lens with a positive focal length converges the beam; a negative one diverges it.

#### Second moments and conserved quality

$$
\begin{aligned}\Sigma&=\begin{pmatrix}\langle x^2\rangle&\langle x\theta\rangle\\\langle x\theta\rangle&\langle\theta^2\rangle\end{pmatrix}\\\Sigma^{\prime}&=T\Sigma T^{\mathsf T}\\M_x^2&=\frac{4\pi}{\lambda}\sqrt{\det\Sigma}\end{aligned}
$$

The beam is centered on the axis here. An ideal optical matrix has determinant one, so it keeps the covariance determinant and the quality along each principal axis unchanged.

#### Radius, divergence and caustic

$$
\begin{aligned}w_\sigma&=2\sqrt{\langle x^2\rangle}\\w_\sigma^2(z)&=w_{\sigma0}^2+\Theta^2(z-z_0)^2\\M^2&=\pi w_{\sigma0}\Theta/\lambda\end{aligned}
$$

The radius is twice the root-mean-square width, and the divergence is a half-angle. The superscript is part of the quality factor's name, so do not square the slider value again.

#### Recover quality from recorded planes

$$
\begin{aligned}w_\sigma^2(z)&=a+bz+cz^2\\z_0&=-b/(2c)\\w_{\sigma0}^2&=a-b^2/(4c)\\M^2&=\frac{\pi}{\lambda}\sqrt{ac-b^2/4}\end{aligned}
$$

Fit the squared radius to a quadratic in position, using only the recorded profiles. You need at least three distinct planes to fix all the coefficients; a real measurement needs more planes and an estimate of the noise.

#### Keplerian beam expander

$$
\begin{aligned}d&=f_1+f_2\\|\mathcal M|&=f_2/f_1\\w_{\rm out}&\simeq|\mathcal M|w_{\rm in}\\\Theta_{\rm out}&\simeq\Theta_{\rm in}/|\mathcal M|\end{aligned}
$$

Two positive lenses spaced by the sum of their focal lengths enlarge a nearly collimated beam and divide its divergence by the same factor. A Gaussian beam of finite size departs slightly from this ray-optics magnification.

### Common difficulties

#### A small spot does not mean high quality

**Typical misconception** The beam with the smallest focal spot must have the best quality.

**Better mental model** Beam quality depends on both the waist radius and the far-field divergence. The focal spot size also depends on the source width, the wavelength and the lens power.

#### A moment model is not a coherent field

**Typical misconception** You can put any quality factor into the Gaussian beam parameter and still have the same pure field.

**Better mental model** Here, sources with a higher quality factor use a Gaussian Schell moment model. Their intensity can still be Gaussian, but their coherence and angular spread are different.

#### A teaching scan is not a certified measurement

**Typical misconception** Thirteen ideal samples make an ISO-compliant experimental measurement.

**Better mental model** This noiseless scan only demonstrates the fit. A real measurement needs prescribed sampling, calibration, background correction and an uncertainty estimate.

### Run the experiment

1. 01

   #### Watch free propagation

   Choose Free propagation and move the profiler to several planes. Record three distinct planes.

   **What to observe:** The spot grows as it moves away from the source waist, and the fitted beam quality comes out as one.

2. 02

   #### Focusing does not improve quality

   Compare the Focus a Gaussian and Same size, poorer beam quality presets, and scan 13 planes for each.

   **What to observe:** Two sources with the same radius can have different angular spreads. The lens reshapes the caustic, but each quality factor stays the same.

3. 03

   #### Separate the astigmatic axes

   Choose Two different waist planes, move the profiler and scan.

   **What to observe:** The horizontal and vertical waists sit at different positions. A spot that looks round at one plane does not mean the two axes share a waist.

4. 04

   #### Trade divergence for size

   Choose the Threefold Keplerian expander, compare the radii and divergences, and switch the second lens on and off.

   **What to observe:** The two lenses roughly triple the radius of a nearly collimated beam and cut its divergence to about a third. The beam quality does not change.

Physical conclusion

The complex beam parameter describes only coherent Gaussian fields. For poorer or simply astigmatic beams, propagate the second moments of each axis and fit radii recorded on a single free-space segment.

## Related experiments and articles

Continue exploring

### Beam shape and beam quality

A lens moves a waist. Can it also improve the intrinsic beam quality?

1. [Lab **Gaussian Beam & Gouy Phase** Locate the waist, divergence and Gouy phase of an ideal beam.](https://sandboxphysics.com/labs/gaussian-beam-gouy-phase/)
2. [You are here **Gaussian Beam & Beam Quality** Propagate imperfect beams through lenses and measure their caustics.](https://sandboxphysics.com/labs/gaussian-beam-quality/)

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)