Rayleigh length

Gaussian beam width w(z) as a function of the axial distance z. w₀: beam waist; b: confocal parameter; z_R: Rayleigh length; Θ: 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 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 λ is the wavelength and w₀ 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) = w₀, by definition. At distance z_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

• Beam divergence
• Beam parameter product
• Gaussian function
• Electromagnetic wave equation
• John Strutt, 3rd Baron Rayleigh
• Robert Strutt, 4th Baron Rayleigh
• Depth of field

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. 

• Rayleigh length RP Photonics Encyclopedia of Optics