Rayleigh length

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 :\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. ==See also==