Every Fourier optical system is equivalent to consecutive fractional-Fourier-domain filtering

Applied Optics

Volumn 35, Issue 17, 1996, Pages 3167-3170

  Ozaktas, Haldun M ^a   Mendlovic, David ^b  

^a BILKENT UNIVERSITY   (Turkey)

^b TEL AVIV UNIVERSITY   (Israel)

Author keywords

Fourier optics; Fractional Fourier transform; Optical information processing

Indexed keywords

EID: 0038584838     PISSN: 00036935     EISSN: 15394522     Source Type: Journal    
DOI: 10.1364/AO.35.003167     Document Type: Article

References (24)

1. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
2. H. M. Ozaktas and D. Mendlovic, "Fractional Fourier Optics," J. Opt. Soc. Am. A 12, 743-751 (1995).
3. M. J. Bastiaans, "Wigner distribution function and its application to first-order optics," J. Opt. Soc. Am. A 69, 1710-1716 (1979).
4. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).
5. Quadratic graded-index media also belong to this class, and our results remain valid also for systems containing arbitrary sections of such media.
6. H. M. Ozaktas, B. Barshan, D. Mendlovic, and L. Onural, "Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms," J. Opt. Soc. Am. A 11, 547-559 (1994).
7. A. C. McBride and F. H. Kerr, "On Namia's fractional Fourier transform," IMA J. Appl. Math. 39, 159-175 (1987).
8. L. B. Almeida, "The fractional Fourier transform and time-frequency representations," IEEE Trans. Acoust. Speech Signal Process. 42, 3084-3091 (1994).
9. H. M. Ozaktas and D. Mendlovic, "Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators," Opt. Lett. 19, 1678-1680 (1994).
10. 11
11. S. Abe and J. T. Sheridan, "Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation," Opt. Lett. 19, 1801-1803 (1994).
12. M. Nazarathy and J. Shamir, "First-order optics - a canonical operator representation: lossless systems," J. Opt. Soc. Am. 72, 356-364 (1982).
13. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).
14. A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
15. We can find similar results by setting a to any other value as well. The conventional Fourier transform is no more privileged than fractional transforms of other orders.
16. A. W. Lohmann, "Image rotation, Wigner rotation, and the fractional Fourier transform," J. Opt. Soc. Am. A 10, 2181-2186 (1993).
17. L. M. Bernardo and O. D. D. Scares, "Fractional Fourier transforms and imaging," J. Opt. Soc. Am. A 11, 2622-2626 (1994).
18. P. Pellat-Finet and G. Bonnet, "Fractional order Fourier transform and Fourier optics," Opt. Commun. 111, 141-154 (1994).
19. H. M. Ozaktas and D. Mendlovic, "Fractional Fourier transforms and their optical implementations. II," J. Opt. Soc. Am. A 10, 2522-2531 (1993).
20. H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, "Digital computation of the fractional Fourier transform," IEEE Trans. Signal Process, (to be published).
21. M. A. Kutay, H. M. Ozaktas, O. Arikan, and L. Onural, "Optimal filtering in fractional Fourier domains," IEEE Trans. Acoust. Speech Signal Process, (to be published).
22. Convincing examples of the utility of the fractional Fourier transform in designing filtering systems may be found in Refs. 6, 21, and 23, and in D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, "Fractional correlation," Appl. Opt. 34, 303-309 (1995).
23. H. M. Ozaktas, B. Barshan, and D. Mendlovic, "Convolution and filtering in fractional Fourier domains," Opt. Rev. 1, 15-16 (1994); R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, and H. M. Ozaktas, "Chirp filtering in the fractional Fourier domain," Appl. Opt. 33, 7599-7602 (1994).
24. H. M. Ozaktas, B. Barshan, and D. Mendlovic, "Convolution and filtering in fractional Fourier domains," Opt. Rev. 1, 15-16 (1994); R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, and H. M. Ozaktas, "Chirp filtering in the fractional Fourier domain," Appl. Opt. 33, 7599-7602 (1994).