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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 10, Iss. 10 — Oct. 1, 1993
  • pp: 2181–2186

Image rotation, Wigner rotation, and the fractional Fourier transform

Adolf W. Lohmann  »View Author Affiliations


JOSA A, Vol. 10, Issue 10, pp. 2181-2186 (1993)
http://dx.doi.org/10.1364/JOSAA.10.002181


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Abstract

In this study the degree p = 1 is assigned to the ordinary Fourier transform. The fractional Fourier transform, for example with degree P = 1/2, performs an ordinary Fourier transform if applied twice in a row. Ozaktas and Mendlovic [ “ Fourier transforms of fractional order and their optical implementation,” Opt. Commun. (to be published)] introduced the fractional Fourier transform into optics on the basis of the fact that a piece of graded-index (GRIN) fiber of proper length will perform a Fourier transform. Cutting that piece of GRIN fiber into shorter pieces corresponds to splitting the ordinary Fourier transform into fractional transforms. I approach the subject of fractional Fourier transforms in two other ways. First, I point out the algorithmic isomorphism among image rotation, rotation of the Wigner distribution function, and fractional Fourier transforming. Second, I propose two optical setups that are able to perform a fractional Fourier transform.

© 1993 Optical Society of America

History
Original Manuscript: November 9, 1992
Revised Manuscript: April 15, 1993
Manuscript Accepted: April 20, 1993
Published: October 1, 1993

Citation
Adolf W. Lohmann, "Image rotation, Wigner rotation, and the fractional Fourier transform," J. Opt. Soc. Am. A 10, 2181-2186 (1993)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-10-10-2181


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References

  1. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical implementation,” Opt. Commun. (to be published).
  2. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978); “Wigner distribution function and its application to first-order optics,”J. Opt. Soc. Am. 69, 1710–1716 (1979). [CrossRef]
  3. H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980). [CrossRef]
  4. K.-H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D-signals,” Opt. Commun. 42, 310–314 (1982). [CrossRef]
  5. A. W. Lohmann, N. Streibl, “Map transformations by optical anamorphic processing,” Appl. Opt. 22, 780–783 (1983). [CrossRef] [PubMed]
  6. A. W. Lohmann, D. Mendlovic, “An optical transform with odd cycles,” Opt. Commun. 93, 25–26 (1992). [CrossRef]
  7. A. W. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2112 (1992). [CrossRef]
  8. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,”J. Inst. Math. Appl. 25, 241–265 (1980). [CrossRef]
  9. A. C. McBride, F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987). [CrossRef]
  10. B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982). [CrossRef]

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