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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.^[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 ${\hat {z}}$ axis with wave number $k=2\pi /\lambda$ , the Rayleigh length is given by^[2]

z_{\mathrm {R} }={\frac {\pi w_{0}^{2}}{\lambda }}={\frac {1}{2}}kw_{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; $w_{0}\geq 2\lambda /\pi$ .^[3]

The radius of the beam at a distance $z$ from the waist is^[4]

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^[1]

\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

^ ^a ^b Siegman, A. E. (1986). Lasers. University Science Books. pp. 664–669. ISBN 0-935702-11-3.
^ ^a ^b Damask, Jay N. (2004). Polarization Optics in Telecommunications. Springer. pp. 221–223. ISBN 0-387-22493-9.
^ Siegman (1986) p. 630.
^ Meschede, Dieter (2007). Optics, Light and Lasers: The Practical Approach to Modern Aspects of Photonics and Laser Physics. Wiley-VCH. pp. 46–48. ISBN 978-3-527-40628-9.

Rayleigh length RP Photonics Encyclopedia of Optics

[Siegman1986-1] Siegman, A. E. (1986). Lasers. University Science Books. pp. 664–669. ISBN 0-935702-11-3.

[Damask-2] 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 978-3-527-40628-9.

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[2]

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Explanation

Related quantities

See also

References