# Fresnel Number

Definition: a parameter determining the regime of diffraction effects

German: Fresnel-Zahl

Categories: general optics, optical resonators

Author: Dr. RĂ¼diger Paschotta

Originally, the Fresnel number was introduced in the context of *diffraction* *theory* for beam propagation. If a light wave first passes through an aperture of size (e.g. radius) <$a$> and then propagates over a distance <$L$> to a screen, the situation is characterized with the *Fresnel number*

where <$\lambda$> is the wavelength.

For values of the Fresnel number well below 1, *Fraunhofer diffraction* occurs where the screen essentially shows the far-field diffraction pattern of the aperture, which is closely related to the spatial Fourier transform of the complex amplitude distribution of the light field after the aperture.

Fresnel numbers around 1 or larger characterize the situation of *Fresnel diffraction* (or near-field diffraction), where the mathematical description is more complicated. For not too large Fresnel numbers and diffraction angles, the *Fresnel approximation* can be used.

## Fresnel Number of a Resonator

The concept of the Fresnel number has also been applied to optical resonators (cavities), in particular to laser resonators [1]. One again uses the equation

$${N_{\rm{F}}} = \frac{{{a^2}}}{{L\;\lambda }}$$where <$a$> is now the radius of the end mirrors, and <$L$> is the resonator length.

A large Fresnel number (well above 1) of a resonator (cavity) means that diffraction losses at the end mirrors are small for typical mode sizes (i.e. not near a stability limit of the resonator, where mode sizes can diverge). This is the usual situation in a stable laser resonator. Conversely, a small Fresnel number means that diffraction losses can be significant – particularly for higher-order modes, so that diffraction-limited operation may be favored.

Most stable laser resonators have a fairly large Fresnel number, whereas small Fresnel numbers occur in unstable resonators, which are sometimes applied in high-power lasers.

The Fresnel number is also important for the analysis of the modes of (plane) Fabry–Pérot interferometers, which extend to the edges of the mirrors.

### Bibliography

[1] | A. E. Siegman, Lasers, University Science Books, Mill Valley, CA (1986) |

See also: optical resonators, laser resonators, Fabry–Pérot interferometers

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