Rayleigh length

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Rayleigh length

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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 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.

Contents

1 Explanation
2 Related quantities
3 See also
4 References

Explanation

For more details on this topic, see Gaussian beam.

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

$z_\mathrm{R} = \frac{\pi w_0^2}{\lambda} ,$

where $\lambda$ is the wavelength and $w_0$ is the beam waist, the radial size of the beam at its narrowest point.

The radius of the beam at a distance $z$ from the waist is ^[3]

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

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]

$\Theta_{\mathrm{div}} = 2\frac{w_0}{z_R}.$

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

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

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.
^ 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.

Rayleigh length RP Photonics Encyclopedia of Optics

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