# Rayleigh Length

Author: the photonics expert Dr. Rüdiger Paschotta

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

Units: m

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

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 <$\lambda$> is the wavelength in the medium, i.e., 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.

## More to Learn

Encyclopedia articles:

## Questions and Comments from Users

Here you can submit questions and comments. As far as they get accepted by the author, they will appear above this paragraph together with the author’s answer. The author will decide on acceptance based on certain criteria. Essentially, the issue must be of sufficiently broad interest.

Please do not enter personal data here. (See also our privacy declaration.) If you wish to receive personal feedback or consultancy from the author, please contact him, e.g. via e-mail.

By submitting the information, you give your consent to the potential publication of your inputs on our website according to our rules. (If you later retract your consent, we will delete those inputs.) As your inputs are first reviewed by the author, they may be published with some delay.

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.