Fresnel Number
Author: the photonics expert Dr. Rüdiger Paschotta (RP)
Definition: a parameter determining the regime of diffraction effects
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DOI: 10.61835/uvi Cite the article: BibTex plain textHTML Link to this page LinkedIn
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.
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Bibliography
[1] | A. E. Siegman, Lasers, University Science Books, Mill Valley, CA (1986) |
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