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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 17, Iss. 12 — Dec. 1, 2000
  • pp: 2330–2338

Fractionalization of the linear cyclic transforms

T. Alieva and M. L. Calvo  »View Author Affiliations

JOSA A, Vol. 17, Issue 12, pp. 2330-2338 (2000)

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In this study the general algorithm for the fractionalization of the linear cyclic integral transforms is established. It is shown that there are an infinite number of continuous fractional transforms related to a given cyclic integral transform. The main properties of the fractional transforms used in optics are considered. As an example, two different types of fractional Hartley transform are introduced, and the experimental setups for their optical implementation are proposed.

© 2000 Optical Society of America

OCIS Codes
(070.0070) Fourier optics and signal processing : Fourier optics and signal processing
(070.2590) Fourier optics and signal processing : ABCD transforms
(070.6020) Fourier optics and signal processing : Continuous optical signal processing

Original Manuscript: February 22, 2000
Revised Manuscript: June 21, 2000
Manuscript Accepted: June 23, 2000
Published: December 1, 2000

T. Alieva and M. L. Calvo, "Fractionalization of the linear cyclic transforms," J. Opt. Soc. Am. A 17, 2330-2338 (2000)

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  1. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation,” J. Opt. Soc. Am. A 10, 1875–1881 (1993). [CrossRef]
  2. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformation in optics,” in Progress in Optics, XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998), pp. 263–342.
  3. H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999). [CrossRef]
  4. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional Hilbert transform,” Opt. Lett. 21, 281–283 (1996). [CrossRef] [PubMed]
  5. A. W. Lohmann, E. Tepichin, J. G. Ramirez, “Optical implementation of the fractional Hilbert transform for two-dimensional objects,” Appl. Opt. 36, 6620–6626 (1997). [CrossRef]
  6. L. Yu, Y. Lu, X. Zeng, M. Huang, M. Chen, W. Huang, Z. Zhu, “Deriving the integral representation of a fractional Hankel transform from a fractional Fourier transform,” Opt. Lett. 23, 1158–1160 (1998). [CrossRef]
  7. B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. 30, 25–31 (1982). [CrossRef]
  8. C. C. Shih, “Fractionalization of Fourier transform,” Opt. Commun. 118, 495–498 (1995). [CrossRef]
  9. S. Liu, J. Jiang, Y. Zhang, J. Zhang, “Generalized fractional Fourier transforms,” J. Phys. A 30, 973–981 (1997). [CrossRef]
  10. T. Alieva, M. J. Bastiaans, “Powers of transfer matrices determined by means of eigenfunctions,” J. Opt. Soc. Am. A 16, 2413–2418 (1999). [CrossRef]
  11. D. A. Linden, “A discussion of sampling theorems,” Proc. IRE 47, 1219–1226 (1959). [CrossRef]
  12. R. P. Kanwal, Linear Integral Equations: Theory and Techniques (Academic, New York, 1971), Chaps. 8 and 9.
  13. T. Alieva, A. Barbe, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, L211–L215 (1997). [CrossRef]
  14. R. N. Bracewell, H. Bartelt, A. W. Lohmann, N. Streibl, “Optical synthesis of the Hartley transform,” Appl. Opt. 24, 1401–1402 (1985). [CrossRef] [PubMed]

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