Abstract

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

© 1998 Optical Society of America

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References

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  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.
    [CrossRef]
  2. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 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 algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33, 209–218 (1994).
    [CrossRef] [PubMed]
  5. W. J. Dallas, “Digital computation of image complex amplitude from image and diffraction intensity: an alternative to holography,” Optik (Stuttgart) 44, 45–59 (1975).
  6. A. M. J. Huiser, P. V. Toorn, H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction patterns III. The development of an algorithm,” Optik (Stuttgart) 47, 1–8 (1977).
  7. W. Kim, M. H. Hayes, “Phase retrieval using two Fourier-transform intensities,” J. Opt. Soc. Am. A 7, 441–449 (1990).
    [CrossRef]
  8. H. H. Arsenault, K. Chalasinska-Macukow, “A solution to the phase-retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
    [CrossRef]
  9. N. Nakajima, “Phase retrieval from two intensity measurements using the Fourier series expansion,” J. Opt. Soc. Am. A 4, 154–158 (1987).
    [CrossRef]
  10. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  11. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation. I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  12. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  13. 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]
  14. H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
    [CrossRef]
  15. X. G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signal Process. Lett. 3, 72–74 (1996).
    [CrossRef]

1996

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

X. G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signal Process. Lett. 3, 72–74 (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

1993

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 patterns III. The development of an algorithm,” Optik (Stuttgart) 47, 1–8 (1977).

1975

W. J. Dallas, “Digital computation of image complex amplitude from image and diffraction intensity: an alternative to holography,” Optik (Stuttgart) 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 (Stuttgart) 35, 237–246 (1972).

Arikan, O.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

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]

Bozdagi, G.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[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]

Dallas, W. J.

W. J. Dallas, “Digital computation of image complex amplitude from image and diffraction intensity: an alternative to holography,” Optik (Stuttgart) 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 patterns III. The development of an algorithm,” Optik (Stuttgart) 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.
[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 (Stuttgart) 35, 237–246 (1972).

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 patterns III. The development of an algorithm,” Optik (Stuttgart) 47, 1–8 (1977).

Kim, W.

Kutay, M. A.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

Lohmann, A. W.

Mendlovic, D.

Nakajima, N.

Ozaktas, H. M.

Saxton, W. O.

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

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 patterns III. The development of an algorithm,” Optik (Stuttgart) 47, 1–8 (1977).

Xia, X. G.

X. G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signal Process. Lett. 3, 72–74 (1996).
[CrossRef]

Yang, G. Z.

Zalevsky, Z.

Zhuang, J. Y.

Appl. Opt.

IEEE Signal Process. Lett.

X. G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signal Process. Lett. 3, 72–74 (1996).
[CrossRef]

IEEE Trans. Signal Process.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

J. Opt. Soc. Am. A

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 (Stuttgart)

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

W. J. Dallas, “Digital computation of image complex amplitude from image and diffraction intensity: an alternative to holography,” Optik (Stuttgart) 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 patterns III. The development of an algorithm,” Optik (Stuttgart) 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.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Schematic of the relation of the fractional Fourier transform between the object and the image planes.

Fig. 2
Fig. 2

Comparison of the recovered phase obtained with (a) the recursive algorithm and (b) the fractional G-S algorithm for the image function f 1(x) = η(x)exp[i sin(πx)]. The solid curves correspond to the ideal phase; the circles correspond to the calculated phase.

Fig. 3
Fig. 3

Comparison of the recovered phase obtained with (a) the recursive algorithm and (b) the fractional G-S algorithm for the image function f 2(x) =exp(-0.2x 2)exp{i[sin(5πx) +cos(8πx)]}. The solid curves correspond to the ideal phase; the circles correspond to the calculated phase. The dashed curve in (b) corresponds to the other calculated phase obtained with the fractional G-S algorithm by use of a different initial phase in the iteration.

Equations (20)

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p g x = -   B p t ,   x g t d t , B p t ,   x = A ϕ exp i π x 2 cot   ϕ - 2 xt   csc   ϕ + t 2 cot   ϕ , A ϕ = exp - i π   sgn sin   ϕ / 4 + i ϕ / 2 | sin   ϕ | 1 / 2 .
p 1 p 2 = p 1 + p 2 .
F p m Δ z = A ϕ Δ t   k = - N 2 N 2 - 1   g k Δ t exp i π   cot   ϕ m Δ z 2 + k Δ t 2 - i 2 π   km N ,
g k Δ t = A ϕ * Δ z   m = - N 2 N 2 - 1   F p m Δ z exp - i π   cot   ϕ m Δ z 2 + k Δ t 2 + i 2 π   km N ,
F p 1 F p 2 = F p 1 + p 2 ,
l = - N 2 N 2 - 1   g * l Δ t g l + k Δ t exp i 2 π   cot   ϕ   kl Δ t 2 = sin   ϕ N Δ t 2 exp - i π   cot   ϕ k Δ t 2 × m = - N 2 N 2 - 1   | F p m Δ z | 2 exp i 2 π   km N .
f 2 n = A ϕ Δ t   k = - N 2 N 2 - 1   g k exp i π u 2 n Δ x 2 + k Δ t 2 - i 2 π   kn N ,
g k = 0 ,   k = - N , ,   - N / 2 - 1 ,   N / 2 , ,   N - 1 ,
f n = A ϕ Δ t   k = - N N - 1   g k exp i π u n Δ x 2 + k Δ t 2 - i 2 π   kn 2 N .
l = - N 2 N 2 - k - 1   g * l g l + k exp i 2 π ukl Δ t 2 = sin π 2   α 2 N Δ t 2 exp - i π u k Δ t 2 n = - N N - 1   | f n | 2 exp i 2 π   kn 2 N ,
l = - N 2 N 2 - k - 1   g * l g l + k exp i 2 π wkl Δ t 2 = sin π 2   γ 2 N Δ t 2 exp - i π w k Δ t 2 m = - N N - 1   | F β m | 2 exp i 2 π   km 2 N ,
g * - N 2 g N 2 - 1 = R N - 1 exp i π uN N - 1 Δ t 2 .
g * - N 2 g N 2 - 2 = R N - 2 s 2 - T N - 2 q 2 D 2 ,
g * - N 2 + 1 g N 2 - 1 = T N - 2 p 2 - R N - 2 r 2 D 2 .
g * - N 2 g N 2 - m = a m s m - b m q m D m ,
g * - N 2 + m - 1 g N 2 - 1 = b m p m - a m r m D m .
p m = exp - i π u N - m N Δ t 2 , q m = exp - i π u N - m N - 2 m + 2 Δ t 2 , r m = exp - i π w N - m N Δ t 2 , s m = exp - i π w N - m N - 2 m + 2 Δ t 2 , a m = R N - m - j = 1 m - 2   g * - N 2 + j g N 2 - m + j × exp - i π u N - m N - 2 j Δ t 2 , b m = T N - m - j = 1 m - 2   g * - N 2 + j g N 2 - m + j × exp - i π w N - m N - 2 j Δ t 2 , D m = p m s m - q m r m .
D m = exp - i π Δ t 2 N - m N u + w × exp i 2 π Δ t 2 N - m m - 1 w × 1 - exp i 2 π Δ t 2 N - m m - 1 u - w .
g N 2 - m 0 - 1 = g - N 2 + m 0 - 1 ,
g * - N / 2 g N / 2 - m 0 - 1 g * - N / 2 + m 0 - 1 g N / 2 - 1 R N - 1 exp i π uN N - 1 Δ t 2 = | g 0 | 2 .

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