Conceptually, it is the number of half-period zones in the wavefront 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 changes by when moving from one half-period zone to the next.[1]
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
Aperture real amplitude as estimated at focus of a half inch perfect lens having Fresnel number equal to 100. Adopted wavelength for propagation is 1 µm.
Aperture real amplitude as estimated at focus of a half inch perfect lens having Fresnel number equal to 1. Adopted wavelength for propagation is 1 µm.
Aperture real amplitude as estimated at focus of a half inch perfect lens having Fresnel number equal to 0.01. Adopted wavelength for propagation is 1 µm.
The Fresnel number is a useful concept in physical optics. 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 approximation method. This approximation works well when at the observation point the distance to the aperture is of the same order as the aperture size. This propagation regime satisfies .
The correct approximation for the propagation in the near field is Fresnel diffraction. This approximation works well when at the observation point the distance to the aperture is bigger than the aperture size. This propagation regime verifies .
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. This propagation regime verifies .
The 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, or at its focus when the beam is converging/diverging.[2] 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.[3]
This criterion, firstly described by G.N. Lawrence[4] and now adopted in propagation codes like PROPER,[2] 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. 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 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 and by its comparison with the input/output propagation distance. If the ratio between input/output propagation distance and Rayleigh length returns 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. At contrary, once the ratio between input/output propagation distance and Gaussian pilot beam Rayleigh range returns 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.[5]
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.
Jenkins, Francis Arthur; White, Harvey Elliott (1957). New York: McGraw-Hill 3rd (ed.). Fundamentals of optics. New York, McGraw-Hill.
Krist, J.E. (September 2007). "PROPER: An optical propagation library for IDL". In Kahan, Mark A (ed.). Optical Modeling and Performance Predictions III. Vol. 6675. pp. 66750P. Bibcode:2007SPIE.6675E..0PK. doi:10.1117/12.731179. S2CID 119742001.
Born, M.; Wolf, E. (2000). Principles of optics (7th Expanded ed.). Cambridge University Press. p. 486.
Goodman, J.W. (2005). New York: McGraw-Hill 3rd (ed.). Introduction to Fourier optics.
External links
Coyote's Guide to IDL Programming
January 27, 2023
fresnel, number, this, article, needs, additional, citations, verification, please, help, improve, this, article, adding, citations, reliable, sources, unsourced, material, challenged, removed, find, sources, news, newspapers, books, scholar, jstor, august, 20. This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Fresnel number news newspapers books scholar JSTOR August 2021 Learn how and when to remove this template message The Fresnel number F named after the physicist Augustin Jean Fresnel is a dimensionless number occurring in optics in particular in scalar diffraction theory Contents 1 Definition 2 Application 2 1 The Gaussian pilot beam 3 See also 4 References 5 Bibliography 6 External linksDefinition EditFor an electromagnetic wave passing through an aperture and hitting a screen the Fresnel number F is defined as F a 2 L l displaystyle F frac a 2 L lambda where a displaystyle a is the characteristic size e g radius of the aperture L displaystyle L is the distance of the screen from the aperture l displaystyle lambda is the incident wavelength Conceptually it is the number of half period zones in the wavefront 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 changes by p displaystyle pi when moving from one half period zone to the next 1 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 Edit Aperture real amplitude as estimated at focus of a half inch perfect lens having Fresnel number equal to 100 Adopted wavelength for propagation is 1 µm Aperture real amplitude as estimated at focus of a half inch perfect lens having Fresnel number equal to 1 Adopted wavelength for propagation is 1 µm Aperture real amplitude as estimated at focus of a half inch perfect lens having Fresnel number equal to 0 01 Adopted wavelength for propagation is 1 µm The Fresnel number is a useful concept in physical optics 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 approximation method This approximation works well when at the observation point the distance to the aperture is of the same order as the aperture size This propagation regime satisfies F 1 displaystyle F gg 1 The correct approximation for the propagation in the near field is Fresnel diffraction This approximation works well when at the observation point the distance to the aperture is bigger than the aperture size This propagation regime verifies F 1 displaystyle 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 This propagation regime verifies F 1 displaystyle F ll 1 The Gaussian pilot beam Edit 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 or at its focus when the beam is converging diverging 2 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 3 This criterion firstly described by G N Lawrence 4 and now adopted in propagation codes like PROPER 2 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 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 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 and by its comparison with the input output propagation distance If the ratio between input output propagation distance and Rayleigh length returns 1 displaystyle leq 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 At contrary once the ratio between input output propagation distance and Gaussian pilot beam Rayleigh range returns gt 1 displaystyle gt 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 5 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 EditFraunhofer distance Fresnel diffraction Fresnel imager Fresnel integral Fresnel zone Near and far field Talbot effect Zone plateReferences Edit Jenkins amp White 1957 a b Krist 2007 Born amp Wolf 2000 Lawrence 1992 Goodman 2005 Bibliography EditJenkins Francis Arthur White Harvey Elliott 1957 New York McGraw Hill 3rd ed Fundamentals of optics New York McGraw Hill Krist J E September 2007 PROPER An optical propagation library for IDL In Kahan Mark A ed Optical Modeling and Performance Predictions III Vol 6675 pp 66750P Bibcode 2007SPIE 6675E 0PK doi 10 1117 12 731179 S2CID 119742001 Born M Wolf E 2000 Principles of optics 7th Expanded ed Cambridge University Press p 486 Lawrence G N 1992 Optical Modeling Applied Optics and Optical Engineering 11 125 Goodman J W 2005 New York McGraw Hill 3rd ed Introduction to Fourier optics External links EditCoyote s Guide to IDL Programming Retrieved from https en wikipedia org w index php title Fresnel number amp oldid 1110672048, wikipedia, wiki, book, books, library,
