Abstract

We first discuss the discrete Fresnel transform and present some essential properties. We then propose a recursive algorithm to implement phase retrieval from two intensities in the Fresnel transform domain. This approach can significantly simplify computational manipulations and does not need an initial phase value compared with conventional iterative algorithms. Simulation results show that this approach can successfully recover the phase from two intensities.

© 1999 Optical Society of America

PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problem in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 13–19.
  2. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  3. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  4. G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, O. K. Ersoy, “Gerchberg–Saxton and Yang–Gu algorithm for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33, 209–218 (1994).
    [CrossRef] [PubMed]
  5. Z. Zalevsky, R. G. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
    [CrossRef] [PubMed]
  6. W. J. Dallas, “Digital computation of image complex amplitude from image and diffraction intensity: an alternative to holography,” Optik 44, 45–59 (1975).
  7. A. M. J. Huiser, P. V. Toorn, H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern. III. The development of an algorithm,” Optik 47, 1–8 (1977).
  8. W. Kim, M. H. Hayes, “Phase retrieval using two Fourier-transform intensities,” J. Opt. Soc. Am. A 7, 441–449 (1990).
    [CrossRef]
  9. H. H. Arsenault, K. Chalasinska-Macukow, “A solution to the phase-retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
    [CrossRef]
  10. N. Nakajima, “Phase retrieval from two intensity measurements using the Fourier series expansion,” J. Opt. Soc. Am. A 4, 154–158 (1987).
    [CrossRef]
  11. N. Nakajima, “Phase retrieval from Fresnel zone intensity measurement by use of Gaussian filtering,” Appl. Opt. 37, 6219–6226 (1998).
    [CrossRef]
  12. H. Takajo, T. Takahashi, H. Kawanami, R. Ueda, “Numerical investigation of iterative phase-retrieval stagnation problem: territories of convergence objects and holes in their boundaries,” J. Opt. Soc. Am. A 12, 3175–3187 (1997).
    [CrossRef]
  13. R. P. Millane, W. J. Stroud, “Reconstructing symmetric images from their undersampled Fourier intensities,” J. Opt. Soc. Am. A 3, 568–579 (1997).
    [CrossRef]
  14. P.-T. Chen, M. A. Fiddy, G. W. Liao, D. A. Pommet, “Blind deconvolution and phase retrieval from point zeros,” J. Opt. Soc. Am. A 7, 1524–1531 (1996).
    [CrossRef]
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

1998

1997

H. Takajo, T. Takahashi, H. Kawanami, R. Ueda, “Numerical investigation of iterative phase-retrieval stagnation problem: territories of convergence objects and holes in their boundaries,” J. Opt. Soc. Am. A 12, 3175–3187 (1997).
[CrossRef]

R. P. Millane, W. J. Stroud, “Reconstructing symmetric images from their undersampled Fourier intensities,” J. Opt. Soc. Am. A 3, 568–579 (1997).
[CrossRef]

1996

P.-T. Chen, M. A. Fiddy, G. W. Liao, D. A. Pommet, “Blind deconvolution and phase retrieval from point zeros,” J. Opt. Soc. Am. A 7, 1524–1531 (1996).
[CrossRef]

Z. Zalevsky, R. G. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[CrossRef] [PubMed]

1994

1990

1987

1983

H. H. Arsenault, K. Chalasinska-Macukow, “A solution to the phase-retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
[CrossRef]

1982

1977

A. M. J. Huiser, P. V. Toorn, H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern. III. The development of an algorithm,” Optik 47, 1–8 (1977).

1975

W. J. Dallas, “Digital computation of image complex amplitude from image and diffraction intensity: an alternative to holography,” Optik 44, 45–59 (1975).

1972

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Arsenault, H. H.

H. H. Arsenault, K. Chalasinska-Macukow, “A solution to the phase-retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
[CrossRef]

Chalasinska-Macukow, K.

H. H. Arsenault, K. Chalasinska-Macukow, “A solution to the phase-retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
[CrossRef]

Chen, P.-T.

P.-T. Chen, M. A. Fiddy, G. W. Liao, D. A. Pommet, “Blind deconvolution and phase retrieval from point zeros,” J. Opt. Soc. Am. A 7, 1524–1531 (1996).
[CrossRef]

Dallas, W. J.

W. J. Dallas, “Digital computation of image complex amplitude from image and diffraction intensity: an alternative to holography,” Optik 44, 45–59 (1975).

Dong, B. Z.

Dorsch, R. G.

Ersoy, O. K.

Ferwerda, H. A.

A. M. J. Huiser, P. V. Toorn, H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern. III. The development of an algorithm,” Optik 47, 1–8 (1977).

H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problem in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 13–19.

Fiddy, M. A.

P.-T. Chen, M. A. Fiddy, G. W. Liao, D. A. Pommet, “Blind deconvolution and phase retrieval from point zeros,” J. Opt. Soc. Am. A 7, 1524–1531 (1996).
[CrossRef]

Fienup, J. R.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gu, B. Y.

Hayes, M. H.

Huiser, A. M. J.

A. M. J. Huiser, P. V. Toorn, H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern. III. The development of an algorithm,” Optik 47, 1–8 (1977).

Kawanami, H.

H. Takajo, T. Takahashi, H. Kawanami, R. Ueda, “Numerical investigation of iterative phase-retrieval stagnation problem: territories of convergence objects and holes in their boundaries,” J. Opt. Soc. Am. A 12, 3175–3187 (1997).
[CrossRef]

Kim, W.

Liao, G. W.

P.-T. Chen, M. A. Fiddy, G. W. Liao, D. A. Pommet, “Blind deconvolution and phase retrieval from point zeros,” J. Opt. Soc. Am. A 7, 1524–1531 (1996).
[CrossRef]

Millane, R. P.

R. P. Millane, W. J. Stroud, “Reconstructing symmetric images from their undersampled Fourier intensities,” J. Opt. Soc. Am. A 3, 568–579 (1997).
[CrossRef]

Nakajima, N.

Pommet, D. A.

P.-T. Chen, M. A. Fiddy, G. W. Liao, D. A. Pommet, “Blind deconvolution and phase retrieval from point zeros,” J. Opt. Soc. Am. A 7, 1524–1531 (1996).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Stroud, W. J.

R. P. Millane, W. J. Stroud, “Reconstructing symmetric images from their undersampled Fourier intensities,” J. Opt. Soc. Am. A 3, 568–579 (1997).
[CrossRef]

Takahashi, T.

H. Takajo, T. Takahashi, H. Kawanami, R. Ueda, “Numerical investigation of iterative phase-retrieval stagnation problem: territories of convergence objects and holes in their boundaries,” J. Opt. Soc. Am. A 12, 3175–3187 (1997).
[CrossRef]

Takajo, H.

H. Takajo, T. Takahashi, H. Kawanami, R. Ueda, “Numerical investigation of iterative phase-retrieval stagnation problem: territories of convergence objects and holes in their boundaries,” J. Opt. Soc. Am. A 12, 3175–3187 (1997).
[CrossRef]

Toorn, P. V.

A. M. J. Huiser, P. V. Toorn, H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern. III. The development of an algorithm,” Optik 47, 1–8 (1977).

Ueda, R.

H. Takajo, T. Takahashi, H. Kawanami, R. Ueda, “Numerical investigation of iterative phase-retrieval stagnation problem: territories of convergence objects and holes in their boundaries,” J. Opt. Soc. Am. A 12, 3175–3187 (1997).
[CrossRef]

Yang, G. Z.

Zalevsky, Z.

Zhuang, J. Y.

Appl. Opt.

J. Opt. Soc. Am. A

N. Nakajima, “Phase retrieval from two intensity measurements using the Fourier series expansion,” J. Opt. Soc. Am. A 4, 154–158 (1987).
[CrossRef]

W. Kim, M. H. Hayes, “Phase retrieval using two Fourier-transform intensities,” J. Opt. Soc. Am. A 7, 441–449 (1990).
[CrossRef]

H. Takajo, T. Takahashi, H. Kawanami, R. Ueda, “Numerical investigation of iterative phase-retrieval stagnation problem: territories of convergence objects and holes in their boundaries,” J. Opt. Soc. Am. A 12, 3175–3187 (1997).
[CrossRef]

R. P. Millane, W. J. Stroud, “Reconstructing symmetric images from their undersampled Fourier intensities,” J. Opt. Soc. Am. A 3, 568–579 (1997).
[CrossRef]

P.-T. Chen, M. A. Fiddy, G. W. Liao, D. A. Pommet, “Blind deconvolution and phase retrieval from point zeros,” J. Opt. Soc. Am. A 7, 1524–1531 (1996).
[CrossRef]

Opt. Commun.

H. H. Arsenault, K. Chalasinska-Macukow, “A solution to the phase-retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
[CrossRef]

Opt. Lett.

Optik

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

W. J. Dallas, “Digital computation of image complex amplitude from image and diffraction intensity: an alternative to holography,” Optik 44, 45–59 (1975).

A. M. J. Huiser, P. V. Toorn, H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern. III. The development of an algorithm,” Optik 47, 1–8 (1977).

Other

H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problem in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 13–19.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Metrics