Gaussian Beams: Waist, Rayleigh Range and M², Explained Interactively
Most lasers produce a beam whose brightness falls off from the center like a bell curve. That shape is called a Gaussian beam, and it is the starting point for almost every calculation in laser optics: how small you can focus a spot, how far a beam stays collimated, and how much power lands where you want it.
Even a perfect Gaussian beam cannot stay narrow forever. Diffraction makes it spread, and how fast depends on three things: the wavelength, the size of the beam at its narrowest point, and the beam quality factor M². The simulator below lets you change all three and see what happens.
How to use the simulator
- Start from a real laser. The presets include a helium-neon laser, a single-mode fiber being collimated, and a 1064 nm beam being focused by a lens, once with a good beam and once with a multimode beam.
- Side view. The white line is the beam radius where the intensity has dropped to 13.5% of the peak (the 1/e² radius). The dashed orange lines show the far-field divergence. The width is drawn much larger than the length so you can see the shape; the label in the corner says by how much.
- Cross-section. Drag across the side view, or use the slider, to look at the beam anywhere along its path. The teal circle is the 1/e² diameter and the dashed orange circle is the FWHM (full width at half maximum).
- Add a lens. Turn on the focusing lens to see how a lens creates a new waist, and how its size depends on the beam you send in.
The numbers that describe a beam
| Quantity | Formula | What it tells you |
|---|---|---|
| Beam radius, w(z) | w₀ √(1 + (z/z_R)²) | The 1/e² radius at distance z from the waist. About 86.5% of the power is inside this circle. |
| Waist, w₀ | — | The radius at the narrowest point. Here the wavefront is flat. |
| Rayleigh range, z_R | π w₀² / (M² λ) | How far from the waist the beam stays roughly the same size. At z_R the radius has grown by √2 and the area has doubled. |
| Depth of focus | 2 z_R | The length over which the beam counts as “in focus”. |
| Divergence, θ | M² λ / (π w₀) | The half-angle at which the beam spreads far from the waist. A smaller waist spreads faster. |
| Beam parameter product | w₀ θ = M² λ / π | Waist times divergence. Lenses can trade one for the other, but cannot shrink the product without throwing power away. |
| Wavefront radius, R(z) | z (1 + (z_R/z)²) | Flat at the waist, most curved at z = z_R, and close to z far away, like light from a point. |
| Gouy phase | arctan(z / z_R) | An extra phase shift that builds up through a focus, 180° in total. It matters in resonators and interferometers. |
| Peak intensity | 2P / (π w²) | Twice the average intensity inside the 1/e² circle. It is what damages optics and drives nonlinear effects. |
| FWHM diameter | 1.177 w | The width at half the peak intensity, often quoted on laser data sheets. |
What M² really means
M² compares a real beam to a perfect Gaussian with the same waist. A perfect TEM₀₀ beam has M² = 1. A beam with M² = 5 diverges five times faster than an ideal beam of the same waist, or, put the other way, it can only be focused to a spot five times larger at the same divergence.
That is why M² shows up everywhere in laser specifications. It is a single number that says how well a beam can be focused, how far it can be sent, and how efficiently it can be coupled into a fiber.
| Laser type | Typical M² |
|---|---|
| Helium-neon | under 1.1 |
| Single-mode fiber laser | 1.05 to 1.1 |
| Diode-pumped solid-state (DPSS) | 1.05 to 1.3 |
| Edge-emitting diode laser | about 1 to 1.5 in one axis, often 5 to 20 or more in the other |
| Multimode industrial laser | 5 to 50 or more |
| Excimer laser | 100 or more |
Typical ranges only; check the data sheet for a specific laser. The simulator uses the “embedded Gaussian” picture: it draws a Gaussian-shaped profile whose size follows the real beam’s M². Real multimode beams have more complicated profiles, but their size along the beam follows the same curve.
How small can you focus it?
When a collimated beam of diameter D goes through a lens of focal length f, the focused spot diameter is close to:
2w₀′ ≈ 4 M² λ f / (π D)
So the spot gets smaller with a shorter focal length, a bigger beam on the lens, a shorter wavelength and a lower M². Try the “Focusing a collimated beam” preset: a 5 mm beam at 1064 nm through a 100 mm lens focuses to about 33 µm with M² = 1.2. Switch to the multimode preset and the same lens gives about 405 µm.
There is a catch. The depth of focus shrinks with the square of the spot size, so halving the spot leaves you a quarter of the depth to work with.
How far does a collimated beam stay collimated?
No beam is perfectly collimated. Light leaving a single-mode fiber has a waist of only about 5 µm, so it spreads quickly. Place a lens one focal length away and the beam becomes much wider and almost parallel. In the fiber preset, an 11 mm lens turns the fiber’s 10.4 µm mode into a beam about 2.1 mm across, with a Rayleigh range of about 2.2 m. That is how far the beam stays close to its collimated size.
How M² is measured
The standard method (ISO 11146) focuses the beam with a lens and measures its diameter at several positions, some within one Rayleigh range of the new waist and some well beyond two Rayleigh ranges. The diameters are measured with the second-moment (“D4σ”) definition, which works for any beam shape. Fitting the hyperbola w(z) to those points gives the waist, the Rayleigh range and M².
Why pointing stability matters
A small, well-focused spot is only useful if it stays where you put it. When a mirror tilts by an angle δ, the reflected beam turns by 2δ. After a lens of focal length f, the focused spot moves by about 2δf. With the 100 mm lens above, a mirror drifting by 50 µrad moves the 33 µm spot by 10 µm, almost a third of its size. Over long free-space paths or when coupling into a single-mode fiber, drifts far smaller than that matter.
That is the problem Physiverse mounts are built for: monolithic flexure mounts that hold their angle through temperature changes and long experiments. See the product line.