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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. 15, Iss. 5 — May. 1, 1998
  • pp: 1114–1120

Beam shaping in the fractional Fourier transform domain

Yan Zhang, Bi-Zhen Dong, Ben-Yuan Gu, and Guo-Zhen Yang  »View Author Affiliations


JOSA A, Vol. 15, Issue 5, pp. 1114-1120 (1998)
http://dx.doi.org/10.1364/JOSAA.15.001114


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Abstract

A new design approach for the diffractive phase elements (DPE’s) that implement beam shaping in the fractional Fourier transform (FRFT) domain is presented. The new algorithm can successfully achieve the design of DPE’s for beam shaping in both unitary and nonunitary transform systems. The unitarity transform condition of the FRFT domain is discussed. Modeling designs of the DPE’s are carried out for several fractional orders and different parameters of the beam for converting a Gaussian profile into a uniform beam. Our approach can realize beam shaping well for a nonunitary transform system in the FRFT domain.

© 1998 Optical Society of America

OCIS Codes
(070.2590) Fourier optics and signal processing : ABCD transforms
(140.3300) Lasers and laser optics : Laser beam shaping

Citation
Yan Zhang, Bi-Zhen Dong, Ben-Yuan Gu, and Guo-Zhen Yang, "Beam shaping in the fractional Fourier transform domain," J. Opt. Soc. Am. A 15, 1114-1120 (1998)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-15-5-1114


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References

  1. M. Quintanilla and A. M. de Frutos, “Holographic filter that transforms a Gaussian into a uniform beam,” Appl. Opt. 20, 879–880 (1981).
  2. Y. H. Chang, I. Yukihire, and M. Kazumi, “Reshaping collimated laser beams with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
  3. M. T. Eismann, A. M. Tai, and J. N. Cederquist, “Iterative design of a holographic beam former,” Appl. Opt. 28, 2641–2650 (1989).
  4. J. Cordingley, “Application of a binary diffractive optic for beam shaping in semiconductor processing by lasers,” Appl. Opt. 32, 2538–2549 (1993).
  5. F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. (Bellingham) 30, 529–536 (1991).
  6. D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their implementations: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
  7. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their implementations: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
  8. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
  9. H. M. Ozaktas and D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
  10. 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).
  11. S. Liu, J. Xu, Y. Zhang, L. Chen, and C. Li, “General optical implementations of fractional Fourier transforms,” Opt. Lett. 20, 1053–1055 (1995).
  12. B. Dong, Y. Zhang, B. Gu, and G. Yang, “Numerical investigation of phase retrieval in a fractional Fourier transform,” J. Opt. Soc. Am. A 14, 2709–2714 (1997).
  13. R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik (Stuttgart) 34, 275–284 (1971).
  14. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 227–246 (1972).
  15. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
  16. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. (Bellingham) 19, 297–305 (1980).
  17. Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
  18. B. Gu, G. Yang, and B. Dong, “General theory for performing an optical transform,” Appl. Opt. 25, 3197–3206 (1986).
  19. G. Yang, B. Gu, and B. Dong, “Theory of the amplitude-phase retrieval in an any linear transform system and its applications,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE 1767, 457–479 (1992).
  20. G. Yang, B. Dong, B. Gu, J. Zhuang, and O. K. Ersoy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33, 209–218 (1994).
  21. X. Tan, B. Gu, G. Yang, and B. Dong, “Diffractive phase elements for beam shaping: a new design method,” Appl. Opt. 34, 1314–1320 (1995).
  22. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 2, p. 21–25.
  23. E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1986).
  24. Xiang-Gen Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signal Process. Lett. 3, 72–74 (1996).
  25. R. Rollestion and N. George, “Image reconstruction from partial Fresnel zone information,” Appl. Opt. 25, 178–183 (1986).
  26. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
  27. D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
  28. D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics,” J. Phys. D 6, 2200–2225 (1973).
  29. R. H. Boucher, “Convergence of algorithms for phase retrieval from two intensity distributions,” in International Optical Computing Conference, W. T. Rhodes, ed., Proc. SPIE 231, 130–141 (1980).

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