Rayleigh length

Concept in laser optics
Gaussian beam width w ( z ) {\displaystyle w(z)} as a function of the axial distance z {\displaystyle z} . w 0 {\displaystyle w_{0}} : beam waist; b {\displaystyle b} : confocal parameter; z R {\displaystyle z_{\mathrm {R} }} : Rayleigh length; Θ {\displaystyle \Theta } : total angular spread

In optics and especially laser science, the Rayleigh length or Rayleigh range, z R {\displaystyle 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 z ^ {\displaystyle {\hat {z}}} axis with wave number k = 2 π / λ {\displaystyle k=2\pi /\lambda } , the Rayleigh length is given by[2]

z R = π w 0 2 λ = 1 2 k w 0 2 {\displaystyle z_{\mathrm {R} }={\frac {\pi w_{0}^{2}}{\lambda }}={\frac {1}{2}}kw_{0}^{2}}

where λ {\displaystyle \lambda } is the wavelength (the vacuum wavelength divided by n {\displaystyle n} , the index of refraction) and w 0 {\displaystyle 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 ≥ 2 λ / π {\displaystyle w_{0}\geq 2\lambda /\pi } .[3]

The radius of the beam at a distance z {\displaystyle z} from the waist is[4]

w ( z ) = w 0 1 + ( z z R ) 2 . {\displaystyle w(z)=w_{0}\,{\sqrt {1+{\left({\frac {z}{z_{\mathrm {R} }}}\right)}^{2}}}.}

The minimum value of w ( z ) {\displaystyle w(z)} occurs at w ( 0 ) = w 0 {\displaystyle w(0)=w_{0}} , by definition. At distance z R {\displaystyle z_{\mathrm {R} }} from the beam waist, the beam radius is increased by a factor 2 {\displaystyle {\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]

Θ d i v ≃ 2 w 0 z R . {\displaystyle \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 ≃ 4 λ π Θ d i v {\displaystyle 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

References

  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 978-3-527-40628-9.
  • Rayleigh length RP Photonics Encyclopedia of Optics