Rayleigh length explained

In optics and especially laser science, the Rayleigh length or Rayleigh range,

z _{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. The Rayleigh length is particularly important when beams are modeled as Gaussian beams.

Explanation

For a Gaussian beam propagating in free space along the

\hat{z} axis with wave number k = 2 \pi/ λ , the Rayleigh length is given by^[2] z _{R} = (\pi w ^{2} _{0})/(λ) = (1)/(2) k w ^{2} _{0} where λ is the wavelength (the vacuum wavelength divided by n , the index of refraction) 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 2λ /\pi .^[3]

The radius of the beam at a distance

z from the waist is^[4] w(z) = w _{0} \sqrt{ 1+ {\left( (z)/(z _{R}) \right)} ^{2} } .

The minimum value of

w(z) occurs at w( 0 ) = w _{0} , 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 _{div} \simeq 2 (w _{0})/(z _{R}) .

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

D = 2 w _{0} \simeq (4λ)/(\pi \Theta _{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

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

References

• Rayleigh length RP Photonics Encyclopedia of Optics

Notes and References

1. Book: Siegman, A. E. . Lasers . registration . University Science Books . 1986 . 664–669 . 0-935702-11-3 .
2. Book: Damask, Jay N. . Polarization Optics in Telecommunications . limited . . 2004 . 221–223 . 0-387-22493-9 .
3. Siegman (1986) p. 630.
4. Book: Meschede, Dieter . Optics, Light and Lasers: The Practical Approach to Modern Aspects of Photonics and Laser Physics . limited . Wiley-VCH . 2007 . 46–48 . 978-3-527-40628-9 .