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.

Explanation[edit]

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. This equation and those that follow assume that the waist is not extraordinarily small; w_0 \ge 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[edit]

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.

See also[edit]

References[edit]

  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.