Rayleigh length

Explanation

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

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

where ${\displaystyle \lambda }$ is the wavelength (the vacuum wavelength divided by ${\displaystyle n}$, the index of refraction) and ${\displaystyle w_0}$ 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; ${\displaystyle w_0\ge 2\lambda /\pi }$.

The radius of the beam at a distance ${\displaystyle z}$ from the waist is

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

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