Home   >   CSC-OpenAccess Library   >    Manuscript Information
Sine and Cosine Fresnel Transforms
Habib Hamam
Pages - 80 - 88     |    Revised - 31-03-2015     |    Published - 30-04-2015
Volume - 9   Issue - 2    |    Publication Date - March / April 2015  Table of Contents
MORE INFORMATION
KEYWORDS
Diffraction, Fresnel Transform, Fractional Fourier Transform, Cosine Transform, Sine Transform.
ABSTRACT
Two novel transforms, related together and called Sine and Cosine Fresnel Transforms, as well as their optical implementation are presented. Each transform combines both backward and forward light propagation in the framework of the scalar diffraction approximation. It has been proven that the Fresnel transform is the optical version of the fractional Fourier transform. Therefore the former has the same properties as the latter. While showing properties similar to those of the Fresnel transform and therefore of the fractional Fourier transform, each of the Sine and Cosine Fresnel transforms provides a real result for a real input distribution. This enables saving half of the quantity of information in the complex plane. Because of parallelism, optics offers high speed processing of digital signals. Speech signals should be first represented by images through special light modulators for example. The Sine and Cosine Fresnel transforms may be regarded respectively as the fractional Sine and Cosine transforms which are more general than the Cosine transform used in information processing and compression.
1 Google Scholar 
2 CiteSeerX 
3 refSeek 
4 Scribd 
5 SlideShare 
6 PdfSR 
Bhandari and P. Marziliano, "Sampling and reconstruction of sparse signals in fractional Fourier domain," IEEE Signal Processing Letters, 17, pp 221–224, 2010.
D. H. Bailey and P. N. Swarztrauber, "The fractional Fourier transform and applications," SIAM Review, vol 33, pp 389-404, 1991.
D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation. I,” J. Opt. Soc. Am. A, vol 10, pp 1875–1881, 1993.
H. Hamam and J. L. de Bougrenet de la Tocnaye. “Efficiency of programmable quantized diffractive phase elements” Pure and Applied Optics, vol 5, pp 389-403, 1996.
H. Hamam. “Talbot imaging and unification”, Applied Optics - Information Processing. vol 42, pp 7052-7059, 2003.
Hong-yi Fan and Li-yun Hu, “Optical transformation from chirplet to fractional Fourier transformation kernel” (2009), http://www.arxiv.org/abs/0902.1800. [Feb. 9, 2015]
J. Azaña and S. Gupta, “Complete family of periodic Talbot filters for pulse repetition rate multiplication”, Opt. Express, vol 14, pp 4270-4279, 2006.
J. W. Goodman, “Introduction to Fourier optics”, Roberts & Company Publishers; 3rd Revised Ed, 2005.
M. Born and E. Wolf , “Principles of Optics”, Pergamon Press, New-York, 1964.
P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. , vol 19, pp 1388 – 1390 (1994).
Papoulis, "Pulse compression, fiber communications, and diffraction: a unified approach," J. Opt. Soc. Amer. A, vol 11, pp 3-13, 1994.
V. Namias, “The fractional order Fourier transform and its application to quantum mechanics”, J. Inst. Appl. Math. , vol 25, pp 241–265, 1980.
Professor Habib Hamam
University of Moncton - Canada
Habib.Hamam@umoncton.ca