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# Rayleigh range

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### Rayleigh range

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 $\hat\left\{z\right\}$ axis, the Rayleigh length is given by [2]

$z_\mathrm\left\{R\right\} = \frac\left\{\pi w_0^2\right\}\left\{\lambda\right\} ,$

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]

$w\left(z\right) = w_0 \, \sqrt\left\{ 1+ \left\{\left\left( \frac\left\{z\right\}\left\{z_\mathrm\left\{R\right\}\right\} \right\right)\right\}^2 \right\} .$

The minimum value of $w\left(z\right)$ occurs at $w\left(0\right) = w_0$, by definition. At distance $z_\mathrm\left\{R\right\}$ from the beam waist, the beam radius is increased by a factor $\sqrt\left\{2\right\}$ 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]

$\Theta_\left\{\mathrm\left\{div\right\}\right\} \simeq 2\frac\left\{w_0\right\}\left\{z_R\right\}.$

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

$D = 2\,w_0 \simeq \frac\left\{4\lambda\right\}\left\{\pi\, \Theta_\left\{\mathrm\left\{div\right\}\right\}\right\}$.

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.

## References

• Rayleigh length RP Photonics Encyclopedia of Optics
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