Fresnel number
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In
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...
, in particular scalar diffraction theory, the Fresnel number (), named after the physicist
Augustin-Jean Fresnel Augustin-Jean Fresnel (10 May 1788 – 14 July 1827) was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Isaac Newton, Newton's c ...
, is a
dimensionless number Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into unit of measurement, units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that a ...
relating to the pattern a beam of light forms on a surface when projected through an
aperture In optics, the aperture of an optical system (including a system consisting of a single lens) is the hole or opening that primarily limits light propagated through the system. More specifically, the entrance pupil as the front side image o ...
.


Definition

For an
electromagnetic wave In physics, electromagnetic radiation (EMR) is a self-propagating wave of the electromagnetic field that carries momentum and radiant energy through space. It encompasses a broad spectrum, classified by frequency or its inverse, wavelength, ...
passing through an
aperture In optics, the aperture of an optical system (including a system consisting of a single lens) is the hole or opening that primarily limits light propagated through the system. More specifically, the entrance pupil as the front side image o ...
and hitting a screen, the Fresnel number ''F'' is defined as : F = \frac where : a is the characteristic size (e.g.
radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...
) of the aperture : L is the distance of the screen from the aperture : \lambda is the incident
wavelength In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same ''phase (waves ...
. Conceptually, it is the number of half- period zones in the
wavefront In physics, the wavefront of a time-varying ''wave field (physics), field'' is the set (locus (mathematics), locus) of all point (geometry), points having the same ''phase (waves), phase''. The term is generally meaningful only for fields that, a ...
amplitude, counted from the center to the edge of the aperture, as seen from the observation point (the center of the imaging screen), where a half-period zone is defined so that the wavefront
phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform *Phase space, a mathematica ...
changes by \pi when moving from one half-period zone to the next. An equivalent definition is that the Fresnel number is the difference, expressed in half-wavelengths, between the ''slant'' distance from the observation point to the ''edge'' of the aperture and the ''orthogonal'' distance from the observation point to the ''center'' of the aperture.


Application

The Fresnel number is a useful concept in
physical optics In physics, physical optics, or wave optics, is the branch of optics that studies Interference (wave propagation), interference, diffraction, Polarization (waves), polarization, and other phenomena for which the ray approximation of geometric opti ...
. The Fresnel number establishes a coarse criterion to define the near and far field approximations. Essentially, if Fresnel number is small – less than roughly 1 – the beam is said to be in the ''far field''. If Fresnel number is larger than 1, the beam is said to be ''near field''. However this criterion does not depend on any actual measurement of the wavefront properties at the observation point. The angular spectrum method is an exact propagation method. It is applicable to all Fresnel numbers. A good approximation for the propagation in the near field is
Fresnel diffraction In optics, the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff's diffraction formula, Kirchhoff–Fresnel diffraction that can be applied to the propagation of waves in the near and far field, near fi ...
. This approximation works well when at the observation point the distance to the aperture is bigger than the aperture size. This propagation regime corresponds to \ F \sim 1. Finally, once at the observation point the distance to the aperture is much bigger than the aperture size, propagation becomes well described by
Fraunhofer diffraction In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when plane waves are incident on a diffracting object, and the diffraction pattern is viewed at a sufficiently long distance (a distance satisfying Fraunhofer ...
. This propagation regime corresponds to \ F \ll 1. The reason why the angular spectrum method is not used in all cases, is that for large propagation distances it burdens a larger computation time than the other methods. Depending on the specific problem, any memory size of computers is too small to solve the problem.


Gaussian pilot beam

Another criterion called ''Gaussian pilot beam'' allowing to define far and near field conditions, consists to measure the actual wavefront surface curvature for an unaberrated system. In this case the wavefront is planar at the aperture position, when the beam is
collimated A collimated beam of light or other electromagnetic radiation has parallel rays, and therefore will spread minimally as it propagates. A laser beam is an archetypical example. A perfectly collimated light beam, with no divergence, would not disp ...
, or at its focus when the beam is converging/ diverging. In detail, within a certain distance from the aperture – ''the near field'' – the amount of wavefront curvature is low. Outside this distance – '' the far field '' – the amount of wavefront curvature is high. This concept applies equivalently close to the
focus Focus (: foci or focuses) may refer to: Arts * Focus or Focus Festival, former name of the Adelaide Fringe arts festival in East Australia Film *Focus (2001 film), ''Focus'' (2001 film), a 2001 film based on the Arthur Miller novel *Focus (2015 ...
. This criterion, firstly described by G.N. Lawrence and now adopted in propagation codes like PROPER, allows one to determine the realm of application of near and far field approximations taking into account the actual wavefront surface shape at the observation point, to sample its phase without
aliasing In signal processing and related disciplines, aliasing is a phenomenon that a reconstructed signal from samples of the original signal contains low frequency components that are not present in the original one. This is caused when, in the ori ...
. This criterion is named ''Gaussian pilot beam'' and fixes the best propagation method (among angular spectrum, Fresnel and Fraunhofer diffraction) by looking at the behavior of a
Gaussian beam In optics, a Gaussian beam is an idealized beam of electromagnetic radiation whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or ...
piloted from the aperture position and the observation position. Near/far field approximations are fixed by the analytical calculation of the Gaussian beam
Rayleigh length In optics and especially laser science, the Rayleigh length or Rayleigh range, z_\mathrm, is the distance along the propagation direction of a beam from the waist to the place where the area of the cross section is doubled. A related paramete ...
and by its comparison with the input/output propagation distance. If the ratio between input/output propagation distance and Rayleigh length returns \le 1 the surface wavefront maintains itself nearly flat along its path, which means that no sampling rescaling is requested for the phase measurement. In this case the beam is said to be near field at the observation point and angular spectrum method is adopted for the propagation. On the contrary, once the ratio between input/output propagation distance and Gaussian pilot beam Rayleigh range yields > 1 the surface wavefront gets curvature along the path. In this case a rescaling of the sampling is mandatory for a measurement of the phase preventing aliasing. The beam is said to be far field at the observation point and Fresnel diffraction is adopted for the propagation. Fraunhofer diffraction returns then to be an asymptotic case that applies only when the input/output propagation distance is large enough to consider the quadratic phase term, within the Fresnel diffraction integral, negligible irrespectively to the actual curvature of the wavefront at the observation point. As the figures explain, the Gaussian pilot beam criterion allows describing the diffractive propagation for all the near/far field approximation cases set by the coarse criterion based on Fresnel number.


See also

* Fraunhofer distance *
Fresnel diffraction In optics, the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff's diffraction formula, Kirchhoff–Fresnel diffraction that can be applied to the propagation of waves in the near and far field, near fi ...
* Fresnel imager *
Fresnel integral 250px, Plots of and . The maximum of is about . If the integrands of and were defined using instead of , then the image would be scaled vertically and horizontally (see below). The Fresnel integrals and are two transcendental functions n ...
*
Fresnel zone A Fresnel zone ( ), named after physicist Augustin-Jean Fresnel, is one of a series of confocal prolate ellipsoidal regions of space between and around a transmitter and a receiver. The size of the calculated Fresnel zone at any particular di ...
*
Near and far field The near field and far field are regions of the electromagnetic (EM) field around an object, such as a transmitting antenna, or the result of radiation scattering off an object. Non-radiative ''near-field'' behaviors dominate close to the an ...
*
Talbot effect The Talbot effect is a diffraction effect first observed in 1836 by Henry Fox Talbot. When a plane wave is incident upon a periodic diffraction grating, the image of the grating is repeated at regular distances away from the grating plane. The re ...
* Zone plate


References


Bibliography

* * * * *


External links


Coyote's Guide to IDL Programming
{{DEFAULTSORT:Fresnel Number Diffraction