Rayleigh length

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title: "Rayleigh length" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["optical-quantities", "laser-science"] description: "Concept in laser optics" topic_path: "general/optical-quantities" source: "https://en.wikipedia.org/wiki/Rayleigh_length" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Concept in laser optics ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/5/53/GaussianBeamWaist.svg" caption="Gaussian beam width w(z) as a function of the axial distance z. w_0: beam waist; b: confocal parameter; z_\mathrm{R}: Rayleigh length; \Theta: total angular spread"] ::

In optics and especially laser science, the Rayleigh length or Rayleigh range, z_\mathrm{R}, is the distance along the propagation direction of a beam from the waist to the place where the area of the cross section is doubled. A related parameter is the confocal parameter, b, which is twice the Rayleigh length. The Rayleigh length is particularly important when beams are modeled as Gaussian beams.

Explanation

For a Gaussian beam propagating in free space along the \hat{z} axis with wave number k = 2\pi/\lambda, the Rayleigh length is given by

:z_\mathrm{R} = \frac{\pi w_0^2}{\lambda} = \frac{1}{2} k w_0^2 where \lambda is the wavelength (the vacuum wavelength divided by n, the index of refraction) and w_0 is the beam waist, the radial size of the beam at its narrowest point. This equation and those that follow assume that the waist is not extraordinarily small; w_0 \ge 2\lambda/\pi.

The radius of the beam at a distance z from the waist is

:w(z) = w_0 , \sqrt{ 1+ {\left( \frac{z}{z_\mathrm{R}} \right)}^2 } .

The minimum value of w(z) occurs at w(0) = w_0, by definition. At distance z_\mathrm{R} from the beam waist, the beam radius is increased by a factor \sqrt{2} and the cross sectional area by 2.

Related quantities

The total angular spread of a Gaussian beam in radians is related to the Rayleigh length by

:\Theta_{\mathrm{div}} \simeq 2\frac{w_0}{z_R}.

The diameter of the beam at its waist (focus spot size) is given by

:D = 2,w_0 \simeq \frac{4\lambda}{\pi, \Theta_{\mathrm{div}}}.

These equations are valid within the limits of the paraxial approximation. For beams with much larger divergence the Gaussian beam model is no longer accurate and a physical optics analysis is required.

References

• Rayleigh length RP Photonics Encyclopedia of Optics

References

1. Siegman, A. E.. (1986). "Lasers". University Science Books.
2. Damask, Jay N.. (2004). "Polarization Optics in Telecommunications". [[Springer Science+Business Media.
3. Siegman (1986) p. 630.
4. Meschede, Dieter. (2007). "Optics, Light and Lasers: The Practical Approach to Modern Aspects of Photonics and Laser Physics". Wiley-VCH.

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