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Fresnel Number

Definition: a parameter determining the regime of diffraction effects

German: Fresnel-Zahl

Categories: general opticsgeneral optics, optical resonatorsoptical resonators


Cite the article using its DOI: https://doi.org/10.61835/uvi

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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

$${N_{\rm{F}}} = \frac{{{a^2}}}{{L\;\lambda }}$$

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.

More to Learn

Encyclopedia articles:


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

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