Abstract

A novel algorithm that can reconstruct a symmetrical signal (both the amplitude and the phase information) with only a single Fresnel transform intensity is proposed. A new complex-convolution method is introduced, which is needed in the algorithm. The essential properties of the discrete Fresnel transform are presented as well. Numerical results show that this method can successfully rebuild the signal from one signal intensity, which is more advantageous in speed and efficiency than the conventional method that requires two intensities to accomplish this task.

© 2007 Optical Society of America

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  1. H. A. Ferwerda, "The phase reconstruction problem for wave amplitudes and coherence functions," in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978).
  2. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).
  3. R. W. Gerchberg and W. O. Saxton, "Phase determination for image and diffraction plane pictures in the electron microscope," Optik 34, 275-284 (1971).
  4. P. Van Toon and H. A. Ferwerda, "On the problem of phase retrieval in electron microscopy from image and diffraction pattern. IV. Checking of algorithm by means of simulated objects," Optik 47, 123-134 (1977).
  5. Z. Zalevsky and 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. W. Kim and M. H. Hayes, "Phase retrieval using two Fourier-transform intensities," J. Opt. Soc. Am. A 7, 441-449 (1990).
    [CrossRef]
  8. N. Nakajima, "Phase retrieval from two intensity measurements using the Fourier series expansion," J. Opt. Soc. Am. A 4, 154-158 (1987).
    [CrossRef]
  9. W. X. Cong, N. X. Chen, and B. Y. Gu, "Phase retrieval in the Fresnel transform system: a recursive algorithm," J. Opt. Soc. Am. A 16, 1827-1830 (1999).
    [CrossRef]
  10. C. Song, R. J. Damien, Y. Nishino, Y. Kohmura, T. Ishikawa, C. C. Chen, T. K. Lee, and J. Miao, "Phase retrieval from exactly oversampled diffraction intensity through deconvolution," Phys. Rev. B. 75, 012102 (2007).
    [CrossRef]
  11. M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, "Three-dimensional mapping of a deformation field inside a nanocrystal," Nature 442, 63-66 (2006).
    [CrossRef] [PubMed]
  12. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, "Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens." Nature  400, 342-345 (1999).
    [CrossRef]
  13. J. Miao, D. Sayre, and N. H. Chapman, "Phase retrieval from the magnitude of the Fourier transform of nonperiodic objects," J. Opt. Soc. Am. A. A151662-1669 (1998)
    [CrossRef]
  14. Y. M. Bruck and L. G. Sodin, ‘‘On the ambiguity of the image reconstruction problem,’’ Opt. Commun. 30, 304-308 (1979).
    [CrossRef]
  15. M. H. Hayes, ‘‘The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,’’ IEEE Trans. Acoust., Speech, Signal Process. ASSP-30, 140-154 (1982).
    [CrossRef]

2007 (1)

C. Song, R. J. Damien, Y. Nishino, Y. Kohmura, T. Ishikawa, C. C. Chen, T. K. Lee, and J. Miao, "Phase retrieval from exactly oversampled diffraction intensity through deconvolution," Phys. Rev. B. 75, 012102 (2007).
[CrossRef]

2006 (1)

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, "Three-dimensional mapping of a deformation field inside a nanocrystal," Nature 442, 63-66 (2006).
[CrossRef] [PubMed]

1999 (2)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, "Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens." Nature  400, 342-345 (1999).
[CrossRef]

W. X. Cong, N. X. Chen, and B. Y. Gu, "Phase retrieval in the Fresnel transform system: a recursive algorithm," J. Opt. Soc. Am. A 16, 1827-1830 (1999).
[CrossRef]

1998 (1)

J. Miao, D. Sayre, and N. H. Chapman, "Phase retrieval from the magnitude of the Fourier transform of nonperiodic objects," J. Opt. Soc. Am. A. A151662-1669 (1998)
[CrossRef]

1996 (1)

1990 (1)

1987 (1)

1982 (1)

M. H. Hayes, ‘‘The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,’’ IEEE Trans. Acoust., Speech, Signal Process. ASSP-30, 140-154 (1982).
[CrossRef]

1979 (1)

Y. M. Bruck and L. G. Sodin, ‘‘On the ambiguity of the image reconstruction problem,’’ Opt. Commun. 30, 304-308 (1979).
[CrossRef]

1977 (1)

P. Van Toon and H. A. Ferwerda, "On the problem of phase retrieval in electron microscopy from image and diffraction pattern. IV. Checking of algorithm by means of simulated objects," Optik 47, 123-134 (1977).

1975 (1)

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

1972 (1)

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

1971 (1)

R. W. Gerchberg and W. O. Saxton, "Phase determination for image and diffraction plane pictures in the electron microscope," Optik 34, 275-284 (1971).

Bruck, Y. M.

Y. M. Bruck and L. G. Sodin, ‘‘On the ambiguity of the image reconstruction problem,’’ Opt. Commun. 30, 304-308 (1979).
[CrossRef]

Chapman, N. H.

J. Miao, D. Sayre, and N. H. Chapman, "Phase retrieval from the magnitude of the Fourier transform of nonperiodic objects," J. Opt. Soc. Am. A. A151662-1669 (1998)
[CrossRef]

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, "Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens." Nature  400, 342-345 (1999).
[CrossRef]

Chen, C. C.

C. Song, R. J. Damien, Y. Nishino, Y. Kohmura, T. Ishikawa, C. C. Chen, T. K. Lee, and J. Miao, "Phase retrieval from exactly oversampled diffraction intensity through deconvolution," Phys. Rev. B. 75, 012102 (2007).
[CrossRef]

Chen, N. X.

Cong, W. X.

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).

Damien, R. J.

C. Song, R. J. Damien, Y. Nishino, Y. Kohmura, T. Ishikawa, C. C. Chen, T. K. Lee, and J. Miao, "Phase retrieval from exactly oversampled diffraction intensity through deconvolution," Phys. Rev. B. 75, 012102 (2007).
[CrossRef]

Dorsch, R. G.

Ferwerda, H. A.

P. Van Toon and H. A. Ferwerda, "On the problem of phase retrieval in electron microscopy from image and diffraction pattern. IV. Checking of algorithm by means of simulated objects," Optik 47, 123-134 (1977).

Gerchberg, R. W.

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

R. W. Gerchberg and W. O. Saxton, "Phase determination for image and diffraction plane pictures in the electron microscope," Optik 34, 275-284 (1971).

Gu, B. Y.

Harder, R.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, "Three-dimensional mapping of a deformation field inside a nanocrystal," Nature 442, 63-66 (2006).
[CrossRef] [PubMed]

Hayes, M. H.

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

M. H. Hayes, ‘‘The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,’’ IEEE Trans. Acoust., Speech, Signal Process. ASSP-30, 140-154 (1982).
[CrossRef]

Ishikawa, T.

C. Song, R. J. Damien, Y. Nishino, Y. Kohmura, T. Ishikawa, C. C. Chen, T. K. Lee, and J. Miao, "Phase retrieval from exactly oversampled diffraction intensity through deconvolution," Phys. Rev. B. 75, 012102 (2007).
[CrossRef]

Kim, W.

Kirz, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, "Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens." Nature  400, 342-345 (1999).
[CrossRef]

Kohmura, Y.

C. Song, R. J. Damien, Y. Nishino, Y. Kohmura, T. Ishikawa, C. C. Chen, T. K. Lee, and J. Miao, "Phase retrieval from exactly oversampled diffraction intensity through deconvolution," Phys. Rev. B. 75, 012102 (2007).
[CrossRef]

Lee, T. K.

C. Song, R. J. Damien, Y. Nishino, Y. Kohmura, T. Ishikawa, C. C. Chen, T. K. Lee, and J. Miao, "Phase retrieval from exactly oversampled diffraction intensity through deconvolution," Phys. Rev. B. 75, 012102 (2007).
[CrossRef]

Miao, J.

C. Song, R. J. Damien, Y. Nishino, Y. Kohmura, T. Ishikawa, C. C. Chen, T. K. Lee, and J. Miao, "Phase retrieval from exactly oversampled diffraction intensity through deconvolution," Phys. Rev. B. 75, 012102 (2007).
[CrossRef]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, "Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens." Nature  400, 342-345 (1999).
[CrossRef]

J. Miao, D. Sayre, and N. H. Chapman, "Phase retrieval from the magnitude of the Fourier transform of nonperiodic objects," J. Opt. Soc. Am. A. A151662-1669 (1998)
[CrossRef]

Nakajima, N.

Nishino, Y.

C. Song, R. J. Damien, Y. Nishino, Y. Kohmura, T. Ishikawa, C. C. Chen, T. K. Lee, and J. Miao, "Phase retrieval from exactly oversampled diffraction intensity through deconvolution," Phys. Rev. B. 75, 012102 (2007).
[CrossRef]

Pfeifer, M. A.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, "Three-dimensional mapping of a deformation field inside a nanocrystal," Nature 442, 63-66 (2006).
[CrossRef] [PubMed]

Robinson, I. K.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, "Three-dimensional mapping of a deformation field inside a nanocrystal," Nature 442, 63-66 (2006).
[CrossRef] [PubMed]

Saxton, W. O.

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

R. W. Gerchberg and W. O. Saxton, "Phase determination for image and diffraction plane pictures in the electron microscope," Optik 34, 275-284 (1971).

Sayre, D.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, "Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens." Nature  400, 342-345 (1999).
[CrossRef]

J. Miao, D. Sayre, and N. H. Chapman, "Phase retrieval from the magnitude of the Fourier transform of nonperiodic objects," J. Opt. Soc. Am. A. A151662-1669 (1998)
[CrossRef]

Sodin, L. G.

Y. M. Bruck and L. G. Sodin, ‘‘On the ambiguity of the image reconstruction problem,’’ Opt. Commun. 30, 304-308 (1979).
[CrossRef]

Song, C.

C. Song, R. J. Damien, Y. Nishino, Y. Kohmura, T. Ishikawa, C. C. Chen, T. K. Lee, and J. Miao, "Phase retrieval from exactly oversampled diffraction intensity through deconvolution," Phys. Rev. B. 75, 012102 (2007).
[CrossRef]

Van Toon, P.

P. Van Toon and H. A. Ferwerda, "On the problem of phase retrieval in electron microscopy from image and diffraction pattern. IV. Checking of algorithm by means of simulated objects," Optik 47, 123-134 (1977).

Vartanyants, I. A.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, "Three-dimensional mapping of a deformation field inside a nanocrystal," Nature 442, 63-66 (2006).
[CrossRef] [PubMed]

Williams, G. J.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, "Three-dimensional mapping of a deformation field inside a nanocrystal," Nature 442, 63-66 (2006).
[CrossRef] [PubMed]

Zalevsky, Z.

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. A. (1)

J. Miao, D. Sayre, and N. H. Chapman, "Phase retrieval from the magnitude of the Fourier transform of nonperiodic objects," J. Opt. Soc. Am. A. A151662-1669 (1998)
[CrossRef]

Nature (2)

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, "Three-dimensional mapping of a deformation field inside a nanocrystal," Nature 442, 63-66 (2006).
[CrossRef] [PubMed]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, "Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens." Nature  400, 342-345 (1999).
[CrossRef]

Opt. Commun. (1)

Y. M. Bruck and L. G. Sodin, ‘‘On the ambiguity of the image reconstruction problem,’’ Opt. Commun. 30, 304-308 (1979).
[CrossRef]

Opt. Lett. (1)

Optik (4)

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

R. W. Gerchberg and W. O. Saxton, "Phase determination for image and diffraction plane pictures in the electron microscope," Optik 34, 275-284 (1971).

P. Van Toon and H. A. Ferwerda, "On the problem of phase retrieval in electron microscopy from image and diffraction pattern. IV. Checking of algorithm by means of simulated objects," Optik 47, 123-134 (1977).

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

Phys. Rev. B. (1)

C. Song, R. J. Damien, Y. Nishino, Y. Kohmura, T. Ishikawa, C. C. Chen, T. K. Lee, and J. Miao, "Phase retrieval from exactly oversampled diffraction intensity through deconvolution," Phys. Rev. B. 75, 012102 (2007).
[CrossRef]

Signal Process. (1)

M. H. Hayes, ‘‘The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,’’ IEEE Trans. Acoust., Speech, Signal Process. ASSP-30, 140-154 (1982).
[CrossRef]

Other (1)

H. A. Ferwerda, "The phase reconstruction problem for wave amplitudes and coherence functions," in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978).

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Figures (5)

Fig. 1.
Fig. 1.

The configuration to illustrate the Fresnel transform.

Fig. 2.
Fig. 2.

Signal reconstruction algorithm based on a single Fresnel transform intensity

Fig. 3.
Fig. 3.

Numerical results of the proposed recursive algorithm. (a) The magnitude of the recovered signal (the open circles) and the magnitude of original signal f(xo ) (the solid curve); (b) The phase of the recovered signal (the open circles) and the phase of original signal f(xo ) (the solid curve).

Fig. 4.
Fig. 4.

Numerical results of the proposed recursive algorithm for a two dimensional signal f(xo , yo ) = exp[-10(x 2 o +3y 2 o )+j0.2sin(48πx 2 o )cos(32πy 2 o )]. (a) The magnitude of the original signal (b) The phase of the original signal (c) The magnitude of the recovered signal (d) The phase of the recovered signal.

Fig. 5.
Fig. 5.

Numerical results of the proposed recursive algorithm with different extent of random noise. (a) and (b): The magnitude and phase of the original signal (the solid curve) and the recovered signal (the open circles) with S/N = 10 (c) and (d): The magnitude and phase of the original signal (the solid curve) and the recovered signal (the open circles) with S/N = 3.

Equations (20)

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Fr T Z [ f ( x o ) ] = F Z ( x p ) = exp ( i 2 πz λ ) iλz f ( x o ) exp [ λz ( x o x p ) 2 ] d x o ,
DFr T z [ f ( x o ) ] = F z ( x p ) = δ x o m = N 2 N 2 1 κ m n z f ( x o ) ,
κ m n z = exp ( i 2 πz λ ) iλz exp [ λz ( x o x p ) 2 ] .
f ( x o ) = δ x p n = −N 2 N 2 1 κ * m n z F z ( x p ) ,
m = N 2 N 2 1 f * ( m ) f ( m + k ) exp [ i 2 πmk ( δ x o ) 2 λz ] = δ x p δ x o exp [ ( x o ) 2 λz ] n = N 2 N 2 1 F z ( n ) 2 exp ( i 2 πnk N ) ,
f ( k ) = 0 for k = N , , N 2 1 , N 2 , , N 1 ,
m = N 2 N 2 k 1 f * ( m ) f ( m + k ) exp [ i 2 πmk ( δ x o ) 2 λz ] = δ x p δ x o exp [ ( x o ) 2 λz ] n = N N 1 F z ( n ) 2 exp ( i 2 πnk 2 N ) ,
R ( k ) = δ x p δ x o exp [ ( x o ) 2 λz ] n = N N 1 F z ( n ) 2 exp ( i 2 πnk 2 N ) .
f * ( N 2 ) f ( N 2 1 ) = R ( N 1 ) exp [ iπN ( N 1 ) ( δ x o ) 2 λz ] .
f * ( N 2 ) f ( N 2 2 ) exp [ iπN ( N 2 ) ( δ x o ) 2 λz ] + f * ( N 2 + 1 ) f ( N 2 1 ) exp [ ( N 2 ) ( N 2 ) ( δ x o ) 2 λz ] = R ( N 2 ) .
f * ( N 2 ) f ( N 2 m ) exp [ iπN ( N m ) ( δ x o ) 2 λz ] + j = 1 m 2 f * ( N 2 + j ) f ( N 2 m + j ) exp [ ( N m ) ( N 2 j ) ( δ x o ) 2 λz ] + f * ( N 2 + m 1 ) f ( N 2 1 ) exp [ ( N m ) ( N 2 m + 2 ) ( δ x o ) 2 λz ] = R ( N m ) .
Part A k = N 1 f * ( N 2 ) f ( N 2 1 ) A ( 1 , 1 ) k = N− 2 f * ( N 2 ) f ( N 2 2 ) , f * ( N 2 + 1 ) f ( N 2 1 ) A ( 2 , 1 ) , A ( 2 , 2 ) f * ( N 2 ) f ( N 2 3 ) , f * ( N 2 + 1 ) f ( N 2 2 ) , f * ( N 2 + 2 ) f ( N 2 1 ) ................................................................................. f * ( N 2 ) f ( 1 ) , ................. ( k = N 2 + 1 ) ................. , f * ( 2 ) f ( N 2 1 ) f * ( N 2 ) f ( 0 ) , f * ( N 2 + 1 ) f ( 1 ) , ................. ( k = N 2 ) ..............., f * ( 2 ) f ( N 2 2 ) , f * ( 1 ) f ( N 2 1 ) f * ( N 2 ) f ( 1 ) , f * ( N 2 + 1 ) f ( 0 ) , ........................ ( k = N 2 1 ) ..................... , f * ( 1 ) f ( N 2 2 ) , f * ( 0 ) f ( N 2 1 )
m = N 2 N 2 + k f * ( m ) f ( k m ) exp [ i 2 πmk ( δ x o ) 2 λz ] = δ x p δ x o exp [ ( x o ) 2 λz ] n = N N 1 F z ( n ) 2 exp ( i 2 πnk 2 N ) .
R ( k ) = δ x p δ x o exp [ ( x o ) 2 λz ] n = N N 1 F z ( n ) 2 exp ( i 2 πnk 2 N ) .
f * ( N 2 ) f ( N 2 ) = R ( N ) exp [ ( x o ) 2 λz ] = f ( N 2 ) 2 .
f * ( N 2 ) f ( N 2 + 1 ) exp [ iπN ( N + 1 ) ( δ x o ) 2 λz ] + f * ( N 2 + 1 ) f ( N 2 ) exp [ ( N 2 ) ( N + 1 ) ( δ x o ) 2 λz ] = R ( N + 1 ) .
f * ( N 2 ) f ( N 2 + m ) exp [ iπN ( N m ) ( δx o ) 2 λz + j = 1 m = 1 f * ( N 2 + j ) f ( N 2 + m j ) exp [ ( N m ) ( N 2 j ) ( δx o ) 2 λz ] + f * ( N 2 + m ) f ( N 2 ) exp [ ( N m ) ( N 2 m ) ( δx o ) 2 λz ] = R ' ( N + m ) .
Part B k = N f * ( N 2 ) f ( N 2 ) B ( 1 , 1 ) k=−N + 1 f * ( N 2 ) f ( N 2 + 1 ) , f * ( N 2 + 1 ) f ( N 2 ) B ( 2 , 1 ) , B ( 2 , 2 ) f * ( N 2 ) f ( N 2 + 2 ) , f * ( N 2 + 1 ) f ( N 2 + 1 ) , f * ( N 2 + 2 ) f ( N 2 ) f * ( N 2 ) f ( N 2 + 3 ) , f * ( N 2 + 1 ) f ( N 2 + 2 ) , f * ( N 2 + 2 ) f ( N 2 + 1 ) , f * ( N 2 + 3 ) f ( N 2 ) ........................................................................................
f * ( N 2 + 1 ) f ( N 2 + 1 ) = f * ( N 2 ) f ( N 2 + 1 ) × f * ( N 2 + 1 ) f ( N 2 ) f * ( N 2 ) f ( N 2 ) .
[ f * ( N 2 ) f ( 0 ) ] × [ f * ( 0 ) f ( N 2 1 ) ] = f ( 0 ) 2 R ( N 1 ) exp [ iπN ( N 1 ) ( δx o ) 2 λz ] .

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