Rayleigh length



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


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

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.