What is the Rayleigh range?

The Rayleigh range z_R is the propagation distance from the beam waist at which the beam radius increases from w₀ to w₀·√2, meaning the cross-sectional area doubles. It sets the scale of the depth of focus: a tight focus has a short Rayleigh range, while a well-collimated beam has a long one.

What is beam waist (w₀)?

The beam waist w₀ is the minimum 1/e² intensity radius of the Gaussian beam, located at the focal plane (z = 0). It is the tightest point in the beam path. A smaller w₀ produces a tighter focus but also a shorter Rayleigh range and faster divergence.

What is the confocal parameter?

The confocal parameter b = 2·z_R is the total depth of focus — the axial distance over which the beam remains within √2 of its minimum radius. It equals twice the Rayleigh range. In resonator design, it also represents the resonator length at which the beam fills the mirrors most efficiently.

How is the half-angle divergence derived?

In the far field (z >> z_R) the beam radius grows linearly: w(z) ≈ w₀·z/z_R. The half-angle divergence θ = w₀/z_R = λ/(π·w₀) describes the cone angle. A smaller waist gives a larger divergence angle — a fundamental consequence of diffraction.

What does beam radius at distance z mean?

w(z) = w₀·√(1+(z/z_R)²) gives the 1/e² intensity radius at any axial position z from the waist. When |z| = z_R the radius is w₀·√2 ≈ 1.414·w₀; in the far field it grows linearly with z.

Does this assume a perfect Gaussian (M² = 1) beam?

Yes. The formulas here are for an ideal TEM₀₀ Gaussian beam with M² = 1. For real beams with M² > 1, multiply z_R by 1/M² or multiply the divergence angle by M². Your beam profiler datasheet should list M².

Rayleigh Range Calculator

Enter laser wavelength and beam waist to instantly compute Rayleigh range, confocal parameter, half-angle divergence, and beam size at any distance.

Beam Parameters

Wavelength (λ)

Common lasers: 405 nm (violet), 532 nm (green), 780 nm (diode), 1064 nm (Nd:YAG), 10600 nm (CO&sub2;)

Presets:

Beam Waist Radius (w₀)

1/e² intensity radius at the focal plane

Propagation Distance (z) — optional

Distance from waist to evaluate beam radius w(z)

Results

Enter values and click Calculate

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Summary

The Rayleigh range (z_R) is the axial distance from the beam waist at which the cross-sectional area of a Gaussian beam doubles — equivalently, the radius grows by a factor of √2. It defines the depth of focus and governs how quickly the beam diverges. This calculator applies the standard formula z_R = π·w₀²/λ and derives the full set of Gaussian beam parameters in one step.

How it works

Enter the laser wavelength (nm, µm, or m).
Enter the beam waist radius w₀ — the minimum 1/e² intensity radius at the focus (µm or mm).
Click Calculate to run all computations.
The tool applies z_R = π·w₀²/λ for Rayleigh range, b = 2·z_R for confocal parameter, θ = λ/(π·w₀) for half-angle divergence, and w(z) = w₀·√(1+(z/z_R)²) for beam radius at distance z.
Adjust the propagation distance z to see how beam radius evolves along the optical axis.

Use cases

Determine depth of focus for laser machining and material processing.
Design laser focusing optics and estimate spot size at a target distance.
Verify beam collimation quality for free-space optical communications.
Calculate divergence angle for laser pointer or diode laser modules.
Estimate confocal parameter for scanning microscopy and confocal imaging.
Assess near-field versus far-field propagation regimes in photonics labs.
Support M² beam quality analysis by providing ideal Gaussian reference values.
Assist students learning Gaussian beam optics and resonator design.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu