Rayleigh range

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

Explanation

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

:$z_\mathrm = \frac = \frac k w_0^2$
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; $w_0 \ge 2\lambda/\pi$.

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

:$w(z) = w_0 \, \sqrt .$

The minimum value of $w(z)$ occurs at $w(0) = w_0$, by definition. At distance $z_\mathrm$ from the beam waist, the beam radius is increased by a factor $\sqrt$ 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

:$\Theta_ \simeq 2\frac.$

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

:$D = 2\,w_0 \simeq \frac$.

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

{{reflist

Rayleigh length
''RP Photonics Encyclopedia of Optics''

Optical quantities
Laser science