# Rayleigh Length

Definition: the distance from a beam waist where the mode radius increased by a factor square root of 2

German: Rayleigh-Länge

Units: m

Formula symbol: <$z_\textrm{R}$>

Author: Dr. Rüdiger Paschotta

The Rayleigh length (or *Rayleigh range*) of a laser beam is the distance from the beam waist (in the propagation direction) where the beam radius is increased by a factor of the square root of 2. For a circular beam, this means that the mode area is doubled at that point. Typically, the Rayleigh length is considered for Gaussian beams; it is determined by the Gaussian (<$1/e^2$>) beam waist radius <$w_0$> and the wavelength <$\lambda$>:

where the wavelength <$\lambda$> is the vacuum wavelength divided by the refractive index <$n$> of the material.

Older literature often uses the *confocal parameter* <$b$>, which is two times the Rayleigh length.

## Effective Rayleigh Length

For beams with imperfect beam quality and a given waist radius, the Rayleigh length is effectively decreased by the so-called *M*^{2} factor. This implies that such beams have a larger beam divergence for a given beam waist radius.

## Optimization of Beam Focusing

The (effective) Rayleigh length is a convenient quantity for calculations in the context of focused laser beams. Essentially, it determines the *depth of focus*.

When focusing a laser beam into a crystal, e.g. a pump beam into a nonlinear crystal for frequency doubling, it is often advisable to focus such that the Rayleigh length is of the order of the crystal length. One could achieve even higher optical intensities in the crystal with stronger focusing, but in that case only over a shorter length because of the strong beam divergence. Similarly, for end pumping off a laser crystal and will usually not want to make the (effective) Rayleigh length substantially shorter than the crystal length.

The article on laser beams contains a paragraph titled “Limitations for the Focusing of Laser Beams”, where the effective Rayleigh length is also discussed.

See also: Gaussian beams, laser beams, collimated beams, Gouy phase shift, beam quality, *M*^{2} factor, wavelength

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2020-04-15

How to calculate a high-order fiber mode's Rayleigh length? For example, consider a high power fiber laser output beam.

The author's answer:

Fiber mode as such does not have a Rayleigh length; by definition, its amplitude profile does not diverge. Outside the fiber (in free space) it does diverge, and there you could calculate an effective Rayleigh length for each mode. Higher-order modes would generally have shorter values of that length. The calculation could be based on numerical beam propagation, for example.