# Rayleigh length

In optics and especially laser science, the Rayleigh length or Rayleigh range, $z_{\mathrm {R} }$ , 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. 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. 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

## Explanation

For a Gaussian beam propagating in free space along the ${\hat {z}}$  axis with wave number $k=2\pi /\lambda$ , the Rayleigh length is given by 

$z_{\mathrm {R} }={\frac {\pi w_{0}^{2}}{\lambda }}={\frac {1}{2}}kw_{0}^{2}$

where $\lambda$  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; $w_{0}\geq 2\lambda /\pi$ .

The radius of the beam at a distance $z$  from the waist is 

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

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

$\Theta _{\mathrm {div} }\simeq 2{\frac {w_{0}}{z_{R}}}.$

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.