OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 18, Iss. 10 — Oct. 1, 2001
  • pp: 2478–2485

Fractional Fourier transform for quasi-periodic Bloch functions

Chee Ching Chong, Apostolos Vourdas, and Cherif Bendjaballah  »View Author Affiliations


JOSA A, Vol. 18, Issue 10, pp. 2478-2485 (2001)
http://dx.doi.org/10.1364/JOSAA.18.002478


View Full Text Article

Acrobat PDF (807 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

The fractional Fourier transform (FRFT) for quasi-periodic Bloch functions is studied. An isomorphism between square-integrable functions on the real line and quasi-periodic Bloch functions is used to extend existing work on the fractional Fourier transform for the former functions to the latter. The properties of the FRFT for quasi-periodic Bloch functions are discussed, and various numerical examples are presented.

© 2001 Optical Society of America

OCIS Codes
(070.2590) Fourier optics and signal processing : ABCD transforms
(270.0270) Quantum optics : Quantum optics

Citation
Chee Ching Chong, Apostolos Vourdas, and Cherif Bendjaballah, "Fractional Fourier transform for quasi-periodic Bloch functions," J. Opt. Soc. Am. A 18, 2478-2485 (2001)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-18-10-2478


Sort:  Author  |  Year  |  Journal  |  Reset

References

  1. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
  2. A. C. McBride and F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
  3. D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. B Appl. Math. 38, 209–219 (1996).
  4. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transformations and their optical implementation. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
  5. H. M. Ozaktas, B. Barshan, D. Mendlovic, and L. Onural, “Convolution, filtering, and multiplexing in fractional domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
  6. A. W. Lohmann and B. H. Soffer, “Relationships between the Radon–Wigner and fractional Fourier transforms,” J. Opt. Soc. Am. A 11, 1798–1801 (1994).
  7. H. M. Ozakta and M. F. Erden, “Relationships among ray optical, Gaussian bearn, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
  8. D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
  9. D. Dragoman, “Fractional Wigner distribution function,” J. Opt. Soc. Am. A 13, 474–478 (1996).
  10. H. M. Ozaktas, M. A. Kutay, and D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron. Phys. 106, 239–291 (1999).
  11. S. Chountasis, A. Vourdas, and C. Bendjaballah, “Fractional Fourier operators and generalized Wigner functions,” Phys. Rev. A 60, 3467–3473 (1999).
  12. L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
  13. S. C. Pei, M. H. Yeh, and C. C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
  14. M. A. Man’ko, “Fractional Fourier transform in information processing, tomography of optical signal, and Green function of harmonic oscillator,” J. Russ. Laser Res. 20, 226–238 (1999).
  15. M. A. Man’ko, “Quasidistributions tomography, and fractional Fourier transform in signal analysis,” J. Russ. Laser Res. 21, 411–437 (2000).
  16. G. Cariolaro, T. Erseghe, and P. Kraniauskas, “Multiplicity of fractional Fourier transforms and their relationships,” IEEE Trans. Signal Process. 48, 227–241 (2000).
  17. L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981).
  18. M. Reed and B. Simon, Analysis of Operators, Vol. IV of Methods of Modern Mathematical Physics (Academic, London, 1978).
  19. J. Zak, “Finite translations in solid-state physics,” Phys. Rev. Lett. 19, 1385–1387 (1967).
  20. J. Zak, “Dynamics of electronics in solids in external fields,” Phys. Rev. 168, 686–695 (1968).
  21. M. Boon and J. Zak, “Amplitudes on von Neumann lattices,” J. Math. Phys. 22, 1090–1099 (1981).
  22. M. Boon, J. Zak, and I. J. Zucker, “Rational von Neumann lattices,” J. Math. Phys. 24, 316–323 (1983).
  23. A. J. E. M. Janssen, “Bargmann transform, Zak transform, and coherent states,” J. Math. Phys. 23, 720–731 (1982).
  24. A. J. E. M. Janssen, “The Zak transform: a signal transform for sampled time-continuous signals,” Philips J. Res. 43, 23–69 (1988).
  25. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products (Academic, London, 2000).
  26. D. Mumford, Tata Lectures on Theta (Birkhauser, Boston, Mass., 1983), Vols. 1 and 2.
  27. N. J. Vilenkin and A. V. Klimyk, Representation of Lie Groups and Special Functions (Kluwer Academic, Dordrecht, The Netherlands, 1972), Vol. 3.
  28. A. M. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1986).
  29. Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, Singapore, 1991).
  30. C. Bendjaballah, Introduction to Photon Communication (Springer-Verlag, Heidelberg, 1995).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited