Rayleigh length

File:GaussianBeamWaist.svg

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; $\mathrm{\Theta }$: total angular spread

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.

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

$z_{\mathrm{R}}=\frac{\pi w_0^2}{\lambda }=\frac{1}{2}k{w}_0^2$

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

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

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

${\mathrm{\Theta }}_{\mathrm{d}\mathrm{i}\mathrm{v}}\simeq 2\frac{w_0}{z_R}.$

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

$D=2w_0\simeq \frac{4\lambda }{\pi {\mathrm{\Theta }}_{\mathrm{d}\mathrm{i}\mathrm{v}}}$.

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