An algorithm for efficient and accurate computation of the fractional Fourier transform is given. For signals with time-bandwidth product N, the presented algorithm computes the fractional transform in O(NlogN) time. A definition for the discrete fractional Fourier transform that emerges from our analysis is also discussed.

/pdf/digital-computation-of-the-fractional-fourier-transform-4o0rbk380l.pdf

Digital computation of the fractional Fourier transform

Signals and Systems, Alexander D. Poularikas, University of Alabama in Huntsville Fourier Transforms, Kenneth B. Howell, University of Alabama in Huntsville Sine and Cosine Transforms, Pat Yip, McMaster University, Ontario The Hartley Transform, Kraig J. Olejniczak, University of Arkansas, Fayetteville Laplace Transforms, Samuel Seely (deceased), Westbrook, Connecticut The Z-Transform, Alexander D. Poularikas, University of Alabama in Huntsville Hilbert Transforms, Stefan L. Hahn, Warsaw University of Technology Radon and Abel Transforms, Stanley Deans, University of South Florida The Hankel Transform, Robert Piessens , Katholieke Universieit Leuven, Belgium Wavelet Transform, Yunlong Sheng, Laval University, Quebec The Mellin Transform, Jacqueline Bertrand, Pierre Bertrand, Universite de Paris VII, and Jean-Philippe Ovarlez, ONERA/DES, France Mixed Time-Frequency Signal Transformations, G. Faye Boudreaux-Bartels, University of Rhode Island Fractional Fourier Transforms, Dr. Mustafa Abushagur, University of Alabama in Huntsvill, Dr. Ahmed M. Almanasrah, Photronix, Malaysia The Lapped Transforms, Ricardo L. de Queiroz, Xerox Corporation The Discrete Time and the Discrete Transforms, Alexander D. Poularikas, University of Alabama in Huntsville Discrete Time and the Discrete Transforms, Alexander D. Poularikas, University of Alabama in Huntsville Appendices, Alexander D. Poularikas, University of Alabama in Huntsville Index

The transforms and applications handbook

All optical fabrication of Surface Relief Grating (SRG) on azobenzene functionalized polymer films is reviewed in this article. The uniqueness of the single step erasable photofabrication of large amplitude surface relief structures on azo polymer films is mainly due to the photoisomerization and photoanisotropic behavior of the azobenzene group. The design and synthesis of several classes of suitable azobenzene functionalized polymers for the SRG formation is presented. Experimental studies are carried out in a systematic manner to understand the chemical and physical intricacies involved in the SRG formation process. Structural requirements and their limitations for the efficient fabrication of SRGs on azo functionalized polymer films are also discussed. Understanding the fundamental mechanism of SRG formation on the azo functionalized polymer films enables us to make use of these structures in a variety of applications.

Surface relief structures on azo polymer films

A brief introduction to the fractional Fourier transform and its properties is given. Its relation to phase-space representations (time- or space-frequency representations) and the concept of fractional Fourier domains are discussed. An overview of applications which have so far received interest are given and some potential application areas remaining to be explored are noted.

/pdf/the-fractional-fourier-transform-2prn7ao4oa.pdf

The fractional fourier transform

Publisher Summary This chapter describes the self-imaging phenomenon and its applications. The self-imaging phenomenon requires a highly spatially coherent illumination. It disappears when the lateral dimensions of the light source are increased. When the source is made spatially periodic and is placed at the proper distance in front of the periodic structure, a fringe pattern is formed in the space behind the structure. The chapter discusses the theoretical and applicational aspects of the self-imaging phenomenon—that is, the property of the Fresnel diffraction field of some objects illuminated by a spatially coherent light beam. The applications of self-imaging are summarized in four main groups—namely, (1) image processing and synthesis, (2) technology of optical elements, (3) optical testing, and (4) optical metrology. The chapter describes the double diffraction systems using spatially incoherent illumination. The first periodic structure plays the role of a periodic source composed of a multiple of mutually incoherent slits. Depending on whether the periods of two periodic structures are equal, the Lau or the generalized Lau effect is discussed. Various applications of incoherent double-grating systems are described in the fields of optical testing, image processing, and optical metrology. After examining some cases of coherent and incoherent illumination, the general issue of spatial periodicities of optical fields and its relevance to the replication of partially coherent fields in space is discussed.

I The Self-Imaging Phenomenon and its Applications

https://www.uv.es/~gpoei/articulos/(1999)_OC_164_fast_algor_for_Fresnel_y_FRT.pdf

Fast algorithms for free-space diffraction patterns calculation

The use of the ABCD matrix formalism to study fractional Fourier transforms (FRFTs) opens new perspectives on the optical implementation of the FRFTs and on the analysis and synthesis of optical systems. ABCD matrices of basic geometries implementing perfect and imperfect FRFTs are presented. The ABCD matrix method is used to demonstrate the cascading performances of FRFT elementary units and to support the established cascading rules. The matrix formalism is used to show that real and complex optical systems realize FRFTs of complex order.

Abcd matrix formalism of fractional fourier optics

The characteristics and properties of fractional Fourier transforms of an arbitrary degree, as implemented by a lens system, are presented. We describe the perfect imaging process and implement it by cascading a set of appropriate fractional-Fourier-transform elementary units of integer and/or fractional degree, forming generalized afocal systems.

Luís M. Bernardo

Papers

Fast algorithms for free-space diffraction patterns calculation

Abcd matrix formalism of fractional fourier optics

Fractional Fourier transforms and imaging

Fractional Fourier transforms and optical systems

Detection of colon cancer by terahertz techniques