Rayleigh length

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

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

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

• 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. Siegman, A. E. (1986). Lasers. University Science Books. pp. 664–669. ISBN 0-935702-11-3.
2. 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.

• Rayleigh length RP Photonics Encyclopedia of Optics