# Rayleigh length

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Gaussian beam width ${\displaystyle w(z)}$ as a function of the axial distance ${\displaystyle z}$. ${\displaystyle w_{0}}$: beam waist; ${\displaystyle b}$: confocal parameter; ${\displaystyle z_{\mathrm {R} }}$: Rayleigh length; ${\displaystyle \Theta }$: total angular spread

In optics and especially laser science, the Rayleigh length or Rayleigh range 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.[1] A related parameter is the confocal parameter, b, which is twice the Rayleigh length,[2] the Rayleigh length is particularly important when beams are modeled as Gaussian beams.

## Explanation

For a Gaussian beam propagating in free space along the ${\displaystyle {\hat {z}}}$ axis, the Rayleigh length is given by [2]

${\displaystyle z_{\mathrm {R} }={\frac {\pi w_{0}^{2}}{\lambda }},}$

where ${\displaystyle \lambda }$ is the wavelength and ${\displaystyle w_{0}}$ 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; ${\displaystyle w_{0}\geq 2\lambda /\pi }$.[3]

The radius of the beam at a distance ${\displaystyle z}$ from the waist is [4]

${\displaystyle w(z)=w_{0}\,{\sqrt {1+{\left({\frac {z}{z_{\mathrm {R} }}}\right)}^{2}}}.}$

The minimum value of ${\displaystyle w(z)}$ occurs at ${\displaystyle w(0)=w_{0}}$, by definition. At distance ${\displaystyle z_{\mathrm {R} }}$ from the beam waist, the beam radius is increased by a factor ${\displaystyle {\sqrt {2}}}$ 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[1]

${\displaystyle \Theta _{\mathrm {div} }\simeq 2{\frac {w_{0}}{z_{R}}}.}$

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

${\displaystyle D=2\,w_{0}\simeq {\frac {4\lambda }{\pi \,\Theta _{\mathrm {div} }}}}$.

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

1. ^ a b Siegman, A. E. (1986). Lasers. University Science Books. pp. 664–669. ISBN 0-935702-11-3.
2. ^ a b Damask, Jay N. (2004). Polarization Optics in Telecommunications. Springer. pp. 221–223. ISBN 0-387-22493-9.
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. pp. 46–48. ISBN 3-527-40628-X.