Fractional Fourier transform applied to spatial filtering in the Fresnel domain

Granieri, Sergio; Trabocchi, Osvaldo; Sicre, Enrique E.

doi:10.1016/0030-4018(95)00348-c

Cited by 32 publications

(8 citation statements)

References 11 publications

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“…15,16 A number of algorithms are also specifically devised for FrFT's [17][18][19][20][21] whose properties have been intensively investigated both mathematically [3][4][5]22 and physically [23][24][25] in terms of their applications to optical beam propagation, 6 imaging, [24][25][26] diffraction, 27 and signal and image processing. [28][29][30][31][32][33][34][35][36] The relations among FnT's, DFT's, and FrFT's are also well established. 32,37,38 Our aim in this paper is to develop an efficient algorithm that unifies the evaluations of these formulas.…”

Section: Introductionmentioning

confidence: 94%

Fast algorithm for chirp transforms with zooming-in ability and its applications

et al. 2000

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A general fast numerical algorithm for chirp transforms is developed by using two fast Fourier transforms and employing an analytical kernel. This new algorithm unifies the calculations of arbitrary real-order fractional Fourier transforms and Fresnel diffraction. Its computational complexity is better than a fast convolution method using Fourier transforms. Furthermore, one can freely choose the sampling resolutions in both x and u space and zoom in on any portion of the data of interest. Computational results are compared with analytical ones. The errors are essentially limited by the accuracy of the fast Fourier transforms and are higher than the order 10 Ϫ12 for most cases. As an example of its application to scalar diffraction, this algorithm can be used to calculate near-field patterns directly behind the aperture, 0

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Section: Introductionmentioning

confidence: 94%

Fast algorithm for chirp transforms with zooming-in ability and its applications

et al. 2000

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“…Then the diffracted amplitude distribution is modulated by converging spherical wave with centre on a distance R along optical axis 9 . It allows us to generalize forming of the periodic optical elements images in Fresnel diffraction domain on the FrFT domain.…”

Section: Theorymentioning

confidence: 99%

Self-images of periodic phase elements in the fractional Fourier transform domain

Shovgenyuk

Kozlovskii

2006

SPIE Proceedings

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General conditions of periodic phase elements self-images forming (Talbot effect) in the fractional Fourier transform (FrFT) domain is given. Analytical solution of the FrFT images intensity distribution for the different forms (binary, linear, parabolic and others) of periodic elements low-level cell profiles is presented. Intensity difference I ∆ measuring of the FrFT periodic self-image allow to determine the phase difference ϕ ∆ of periodic elements low-level cell profile.

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“…The fractional Fourier transform, which is an extension of the conventional Fourier transform to the fractional order has been introduced into the mathematics literature by Namias[1] in 1980; recently, Mendlovic, Ozaktas and others authors [2][3][4][5][6] introduced a new tool for image analysis in optics; since then, its properties, optical implementation and applications have been studied extensively. An operational definition of fractional Fourier transform in optics and the interpretation of fractional order Fourier transform as the mathematical representation of Fresnel diffraction was stated.…”

Section: Introductionmentioning

confidence: 99%

Fractional order Fourier transform as a tool for analyzing multi-element optical system

Torres

2003

SPIE Proceedings

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The ABCD matrix formalism, the Collins formula and the complex amplitude distributions on two spherical surfaces of given curvature and spacing are adapted to the mathematical expression of fractional order Fourier transform. This result provides a general expression as a tool for analyzing complicated systems involving several lenses and mirrors separated by arbitrary distances; for this class of system it is sufficient to specify the ray transfer matrix and the order of fractional Fourier transform to characterize the system completely.

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Fractional Fourier transform applied to spatial filtering in the Fresnel domain

Cited by 32 publications

References 11 publications

Fast algorithm for chirp transforms with zooming-in ability and its applications

Fast algorithm for chirp transforms with zooming-in ability and its applications

Self-images of periodic phase elements in the fractional Fourier transform domain

Fractional order Fourier transform as a tool for analyzing multi-element optical system

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