Abstract

A noniterative method of retrieving the phase of a wave field from intensities measured during scanning of a slit aperture is proposed. In the measurements, one records the diffraction intensities of wave fields transmitted through a slit that is scanned along two directions in the Fresnel-zone plane of an object’s field. From these intensities, the phase in the Fresnel-zone plane can be retrieved by a method in which a novel phase calculation technique that uses Fourier transforms is included. Because the method does not require lens systems, it provides a potentially useful means for coherent imaging by use of x rays, electrons, or nuclear particles.

© 2005 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
  5. J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
    [CrossRef] [PubMed]
  6. S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
    [CrossRef]
  7. J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
    [CrossRef] [PubMed]
  8. J. Miao, K. O. Hodgson, D. Sayre, “An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images,” Proc. Natl. Acad. Sci. USA 98, 6641–6645 (2001).
  9. J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, M. A. O’Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
    [CrossRef] [PubMed]
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    [CrossRef]
  14. T. E. Gureyev, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
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    [CrossRef]
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    [CrossRef]
  19. S. Bajt, A. Barty, K. A. Nugent, M. MaCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
    [CrossRef] [PubMed]
  20. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  24. L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
    [CrossRef]

2004 (2)

H. M. L. Faulkner, J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

N. Nakajima, “Lensless imaging from diffraction intensity measurements by use of a noniterative phase-retrieval method,” Appl. Opt. 43, 1710–1718 (2004).
[CrossRef] [PubMed]

2003 (2)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[CrossRef] [PubMed]

2002 (3)

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, M. A. O’Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

N. Nakajima, M. Watanabe, “Phase retrieval from experimental far-field intensities by use of a Gaussian beam,” Appl. Opt. 41, 4133–4139 (2002).
[CrossRef] [PubMed]

2001 (3)

K. A. Nugent, D. Paganin, T. E. Gureyev, “A phase odyssey,” Phys. Today 54(8), 27–32 (2001).
[CrossRef]

J. Miao, K. O. Hodgson, D. Sayre, “An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images,” Proc. Natl. Acad. Sci. USA 98, 6641–6645 (2001).

L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

2000 (1)

S. Bajt, A. Barty, K. A. Nugent, M. MaCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

1999 (1)

J. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

1998 (2)

1996 (2)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

T. E. Gureyev, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
[CrossRef]

1995 (1)

1983 (1)

1982 (2)

1976 (1)

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).
[CrossRef]

1974 (1)

J. F. Nye, M. V. Berry, “Dislocation in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

1972 (1)

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).

Allen, L. J.

L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

Anderson, E. H.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Bajt, S.

S. Bajt, A. Barty, K. A. Nugent, M. MaCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Barty, A.

S. Bajt, A. Barty, K. A. Nugent, M. MaCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

Berry, M. V.

J. F. Nye, M. V. Berry, “Dislocation in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Burge, R. E.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).
[CrossRef]

Chapman, H. N.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Faulkner, H. M. L.

H. M. L. Faulkner, J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

Fiddy, M. A.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).
[CrossRef]

Fienup, J. R.

Gao, M.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[CrossRef] [PubMed]

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).

Greenaway, A. H.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).
[CrossRef]

Gureyev, T. E.

Hau-Riege, S. P.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

He, H.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Hodgson, K. O.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, M. A. O’Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

J. Miao, K. O. Hodgson, D. Sayre, “An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images,” Proc. Natl. Acad. Sci. USA 98, 6641–6645 (2001).

Howells, M. R.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Ishikawa, T.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Johnson, B.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Kirz, J.

J. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Lai, B.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

MaCartney, M.

S. Bajt, A. Barty, K. A. Nugent, M. MaCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

Marchesini, S.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Miao, J.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, M. A. O’Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

J. Miao, K. O. Hodgson, D. Sayre, “An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images,” Proc. Natl. Acad. Sci. USA 98, 6641–6645 (2001).

J. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Nagahara, L. A.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[CrossRef] [PubMed]

Nakajima, N.

Noy, A.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Nugent, K. A.

K. A. Nugent, D. Paganin, T. E. Gureyev, “A phase odyssey,” Phys. Today 54(8), 27–32 (2001).
[CrossRef]

S. Bajt, A. Barty, K. A. Nugent, M. MaCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

T. E. Gureyev, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
[CrossRef]

T. E. Gureyev, A. Roberts, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12, 1932–1941 (1995).
[CrossRef]

Nye, J. F.

J. F. Nye, M. V. Berry, “Dislocation in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

O’Keefe, M. A.

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, M. A. O’Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

Ohsuna, T.

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, M. A. O’Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

Oxley, M. P.

L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

Paganin, D.

K. A. Nugent, D. Paganin, T. E. Gureyev, “A phase odyssey,” Phys. Today 54(8), 27–32 (2001).
[CrossRef]

S. Bajt, A. Barty, K. A. Nugent, M. MaCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Roberts, A.

Rodenburg, J. M.

H. M. L. Faulkner, J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

Ross, G.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).
[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 (Stuttgart) 35, 237–246 (1972).

Sayre, D.

J. Miao, K. O. Hodgson, D. Sayre, “An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images,” Proc. Natl. Acad. Sci. USA 98, 6641–6645 (2001).

J. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Spence, J. C. H.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Teague, M. R.

Terasaki, O.

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, M. A. O’Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

Vartanyants, I.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[CrossRef] [PubMed]

Wall, M.

S. Bajt, A. Barty, K. A. Nugent, M. MaCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

Watanabe, M.

Weierstrall, U.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Zhang, R.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[CrossRef] [PubMed]

Zuo, J. M.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[CrossRef] [PubMed]

Appl. Opt. (4)

J. Opt. Soc. Am. (2)

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

Nature (1)

J. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Opt. Commun. (1)

L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

Optik (Stuttgart) (1)

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).

Phys. Rev. B (1)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Phys. Rev. Lett. (4)

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, M. A. O’Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

H. M. L. Faulkner, J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Phys. Today (1)

K. A. Nugent, D. Paganin, T. E. Gureyev, “A phase odyssey,” Phys. Today 54(8), 27–32 (2001).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

J. Miao, K. O. Hodgson, D. Sayre, “An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images,” Proc. Natl. Acad. Sci. USA 98, 6641–6645 (2001).

Proc. R. Soc. London Ser. A (2)

J. F. Nye, M. V. Berry, “Dislocation in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).
[CrossRef]

Science (1)

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003).
[CrossRef] [PubMed]

Ultramicroscopy (1)

S. Bajt, A. Barty, K. A. Nugent, M. MaCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

Other (1)

N. Nakajima, “Phase retrieval using the properties of entire functions,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, 1995), Vol. 93, pp. 109–171.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of the object-reconstruction system that uses phase retrieval. The object reconstruction is based on the measurement of two series of intensities recorded along lines parallel to the ξ axis at two coordinates, s and s + τ, in the detector plane as a function of slit position s.

Fig. 2
Fig. 2

Reconstruction of a complex-valued object with phase vortices in the noise-free case: (a) modulus and (c) phase of an original object; (b) modulus and (d) phase of a reconstructed object.

Fig. 3
Fig. 3

Reconstruction of the object shown in Fig. 2 from noisy intensities: (a) modulus and (c) phase of the reconstructed object; (b), (d) cross-sectional profiles of the figures in (a) and (c), respectively, taken along a horizontal line passing through the center of each figure. Dotted and solid curves represent the original and the reconstructed objects, respectively.

Equations (19)

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F ( x , y ) = σ f ( u , v ) exp { i π λ z [ ( x - u ) 2 + ( y - v ) 2 ] } d u d v ,
H ( ξ , η ) = q - F ( x , y ) R ( x - s ) × exp { i π λ l [ ( x - s ) 2 + y 2 ] } × exp { - i 2 π λ l [ x ( ξ - s ) + y η ] } d x d y ,
H ( s , η ) 2 = | - K ( x , η ) R ( x - s ) d x | 2 ,
H ( s , τ , η ) 2 = | - K ( x , η ) R ( x - s ) × exp ( - i 2 π x τ / λ l ) d x | 2 ,
K ( x , η ) = σ f ( u , v ) exp [ i π λ z ( x - u ) 2 ] × exp [ i π λ ( z + l ) v 2 ] × exp [ - i 2 π λ ( z + l ) η v ] d u d v ,
- R ( x ) exp ( i 2 π u x / λ z ) d x = 2 a sinc ( 2 π a u / λ z ) = 2 a [ 1 - ( 2 π a u / λ z ) 2 6 + ] 2 a exp [ - ( 2 π a u / λ z ) 2 / 6 ] ,
H ( s + τ , η ) 2 = exp ( - 3 c 2 / 2 a 2 ) H ( s - i c , η ) 2 ,
ln [ H ( s + τ , η ) M ( s - i c , η ) ] + 3 c 2 / 4 a 2 = - ϕ I ( s , η ) ,
M ( s - i c , η ) = - [ - M ( s , η ) exp ( i 2 π u s ) d s ] × exp ( - 2 π c u ) exp ( - i 2 π s u ) d u .
D ( s , η ) = - ϕ I ( s , η ) , = - 1 2 i [ ϕ ( s - i c , η ) - ϕ * ( s + i c , η ) ] ,
F [ D ( s , η ) ] = - 1 2 i - [ ϕ ( s - i c , η ) - ϕ * ( s + i c , η ) ] × exp ( - i 2 π α s ) d s ,
ϕ ( s - i c , η ) = - Φ ( α , η ) exp [ i 2 π α ( s - i c ) ] d α ,
- ϕ ( s - i c , η ) exp ( - i 2 π α s ) d s = Φ ( α , η ) exp ( 2 π α c ) = exp ( 2 π α c ) - ϕ ( s , η ) exp ( - i 2 π α s ) d s .
- ϕ * ( s - i c , η ) exp ( - i 2 π α s ) d s = exp ( - 2 π α c ) × - ϕ * ( s , η ) exp ( - i 2 π α s ) d s .
F [ D ( s , η ) ] = - 1 2 i - [ ϕ ( s , η ) exp ( 2 π α c ) - ϕ * ( s , η ) exp ( - 2 π α c ) ] exp ( - i 2 π α s ) d s , = i sinh ( 2 π α c ) - ϕ ( s , η ) × exp ( - i 2 π α s ) d s ,
ϕ ( s , η ) = F - 1 [ F [ D ( s , η ) ] i sinh ( 2 π α c ) ] ,
- K ( x , η ) exp [ i π λ l ( x - ξ ) 2 ] d x = i λ z l z + l σ f ( u , v ) exp [ i π λ ( z + l ) ( ξ - u ) 2 ] × exp [ i π λ ( z + l ) v 2 ] exp [ - i 2 π λ ( z + l ) η v ] d u d v ,
- exp [ i π λ l ( x - ξ ) 2 + i π λ z ( x - u ) 2 ] d x = i λ z l z + l exp [ i π λ ( z + l ) ( ξ - u ) 2 ] .
K ξ ( ξ , η ) = σ f ( u , v ) exp [ i π λ ( z + l ) ( u 2 + v 2 ) ] × exp [ - i 2 π λ ( z + l ) ( ξ u + η v ) ] d u d v .

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