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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
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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
528:
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
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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.
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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.
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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.
394:. 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
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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
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416:. 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
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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.
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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.
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and by its comparison with the input/output propagation distance. If the ratio between input/output propagation distance and
Rayleigh length returns
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Finally, once at the observation point the distance to the aperture is much bigger than the aperture size, propagation becomes well described by
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108:
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and fixes the best propagation method (among angular spectrum, Fresnel and
Fraunhofer diffraction) by looking at the behavior of a
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402:. However this criterion does not depend on any actual measurement of the wavefront properties at the observation point.
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allowing to define far and near field conditions, consists to measure the actual wavefront surface curvature for an
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An equivalent definition is that the
Fresnel number is the difference, expressed in half-wavelengths, between the
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Krist, J.E. (September 2007). "PROPER: An optical propagation library for IDL". In Kahan, Mark A (ed.).
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the ratio between input/output propagation distance and
Gaussian pilot beam Rayleigh range yields
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Dimensionless number describing the pattern a light beam through an aperture forms on a surface
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perfect lens having
Fresnel number equal to 100. Adopted wavelength for propagation is 1
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521:– the amount of wavefront curvature is high. This concept applies equivalently close to the
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Near/far field approximations are fixed by the analytical calculation of the Gaussian beam
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relating to the pattern a beam of light forms on a surface when projected through an
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505:. In this case the wavefront is planar at the aperture position, when the beam is
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is an exact propagation method. It is applicable to all Fresnel numbers.
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783:(7th Expanded ed.). Cambridge University Press. p. 486.
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piloted from the aperture position and the observation position.
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A good approximation for the propagation in the near field is
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398:. If Fresnel number is larger than 1, the beam is said to be
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Proceedings Optical Modeling and Performance Predictions III
513:. In detail, within a certain distance from the aperture –
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Aperture real amplitude as estimated at focus of a half
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Jenkins, Francis Arthur; White, Harvey Elliott (1957).
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331:when moving from one half-period zone to the next.
49:. Unsourced material may be challenged and removed.
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269:is the distance of the screen from the aperture
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210:{\displaystyle F={\frac {a^{2}}{L\lambda }}}
788:Lawrence, G.N. (1992). "Optical Modeling".
346:distance from the observation point to the
338:distance from the observation point to the
390:The Fresnel number is a useful concept in
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452:. This propagation regime corresponds to
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165:and hitting a screen, the Fresnel number
109:Learn how and when to remove this message
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299:Conceptually, it is the number of half-
805:(3rd ed.). New York: McGraw-Hill.
790:Applied Optics and Optical Engineering
741:(3rd ed.). New York: McGraw-Hill.
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750:. Vol. 6675, art. 66750P.
47:adding citations to reliable sources
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817:Coyote's Guide to IDL Programming
243:is the characteristic size (e.g.
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34:needs additional citations for
803:Introduction to Fourier optics
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137:), named after the physicist
779:Born, M.; Wolf, E. (2000).
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666:Jenkins & White (1957)
533:. This criterion is named
497:Another criterion called
438:{\displaystyle \ F\sim 1}
127:scalar diffraction theory
474:{\displaystyle \ F\ll 1}
342:of the aperture and the
284:{\displaystyle \lambda }
493:The Gaussian pilot beam
486:angular spectrum method
407:angular spectrum method
801:Goodman, J.W. (2005).
737:Fundamentals of optics
693:Born & Wolf (2000)
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324:{\displaystyle \pi }
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159:electromagnetic wave
143:dimensionless number
43:improve this article
756:2007SPIE.6675E..0PK
616:Fresnel diffraction
611:Fraunhofer distance
535:Gaussian pilot beam
499:Gaussian pilot beam
484:The reason why the
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32:This article
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41:Please help
36:verification
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832:Diffraction
354:Application
311:changes by
99:August 2021
653:References
646:Zone plate
507:collimated
400:near field
344:orthogonal
293:wavelength
153:Definition
69:newspapers
772:119742001
556:≤
511:diverging
466:≪
430:∼
396:far field
319:π
305:wavefront
279:λ
202:λ
826:Category
604:See also
531:aliasing
163:aperture
147:aperture
129:, the
752:Bibcode
220:where
157:For an
141:, is a
83:scholar
796:: 125.
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348:center
301:period
245:radius
123:optics
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768:S2CID
523:focus
336:slant
309:phase
90:JSTOR
76:books
580:>
405:The
364:inch
340:edge
62:news
760:doi
121:In
45:by
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673:^
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368:μm
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583:1
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257:L
231:a
199:L
193:2
189:a
183:=
180:F
167:F
135:F
133:(
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87:·
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