Rayleigh length

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

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%2B {\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}}}.

See also

References

  1. ^ a b Siegman, A. E. (1986). Lasers. University Science Books. pp. 664–669. ISBN 0935702113. 
  2. ^ a b Damask, Jay N. (2004). Polarization Optics in Telecommunications. Springer. pp. 221–223. ISBN 0387224939. 
  3. ^ Meschede, Dieter (2007). Optics, Light and Lasers: The Practical Approach to Modern Aspects of Photonics and Laser Physics. Wiley-VCH. pp. 46–48. ISBN 352740628X.