Abstract

A method of reconstructing the complex amplitude of an object that is illuminated by a coherent wave from its Fresnel diffraction patterns is proposed for high-frequency wave phenomena such as x-rays and electron waves. A noniterative phase-retrieval method that uses a Gaussian filter is employed here, and it is shown that the object’s illumination with amplitude distribution in the Fraunhofer diffraction pattern of a circular aperture can be used as a substitute for the Gaussian filter. This method has an advantage over other noniterative phase-retrieval methods in that it can retrieve phase vortices.

© 2004 Optical Society of America

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    [CrossRef]
  2. Y. Takai, T. Kawasaki, Y. Kimura, T. Ikuta, R. Shimizu, “Dynamic observation of an atom-sized gold wire by phase electron microscopy,” Phys. Rev. Lett. 87, 106105 (2001).
    [CrossRef] [PubMed]
  3. M. Haider, G. Braunshausen, E. Schwan, “Correction of the spherical aberration of a 200 k TEM by means of a hexapole-corrector,” Optik (Stuttgart) 99, 167–179 (1995).
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    [CrossRef]
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    [CrossRef]
  10. 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]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  21. T. Wilson, A. R. Carlini, C. J. R. Sheppard, “Phase contrast microscopy by nearly full illumination,” Optik (Stuttgart) 70, 166–169 (1985).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  26. L. J. Allen, H. M. L. Faulkner, M. P. Oxley, D. Paganin, “Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy,” Ultramicroscopy 88, 85–97 (2001).
    [CrossRef] [PubMed]
  27. L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
    [CrossRef]
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2002

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

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

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, D. Paganin, “Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy,” Ultramicroscopy 88, 85–97 (2001).
[CrossRef] [PubMed]

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (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).
[CrossRef] [PubMed]

Y. Takai, T. Kawasaki, Y. Kimura, T. Ikuta, R. Shimizu, “Dynamic observation of an atom-sized gold wire by phase electron microscopy,” Phys. Rev. Lett. 87, 106105 (2001).
[CrossRef] [PubMed]

M. A. O’Keefe, E. C. Nelson, Y. C. Wang, A. Thust, “Sub-ångström resolution of atomistic structures below 0.8 Å,” Philos. Mag. B 81, 1861–1878 (2001).
[CrossRef]

2000

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

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

1996

W. M. J. Coene, A. Thust, M. Op de Beeck, D. Van Dyck, “Maximum-likelihood method for focus-variation image reconstruction in high resolution transmission electron microscopy,” Ultramicroscopy 64, 109–135 (1996).
[CrossRef]

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]

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]

1995

M. Haider, G. Braunshausen, E. Schwan, “Correction of the spherical aberration of a 200 k TEM by means of a hexapole-corrector,” Optik (Stuttgart) 99, 167–179 (1995).

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]

1985

T. Wilson, A. R. Carlini, C. J. R. Sheppard, “Phase contrast microscopy by nearly full illumination,” Optik (Stuttgart) 70, 166–169 (1985).

1983

1982

1976

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

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

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).

1948

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[CrossRef] [PubMed]

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]

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, D. Paganin, “Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy,” Ultramicroscopy 88, 85–97 (2001).
[CrossRef] [PubMed]

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (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]

Braunshausen, G.

M. Haider, G. Braunshausen, E. Schwan, “Correction of the spherical aberration of a 200 k TEM by means of a hexapole-corrector,” Optik (Stuttgart) 99, 167–179 (1995).

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]

Carlini, A. R.

T. Wilson, A. R. Carlini, C. J. R. Sheppard, “Phase contrast microscopy by nearly full illumination,” Optik (Stuttgart) 70, 166–169 (1985).

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]

Coene, W. M. J.

W. M. J. Coene, A. Thust, M. Op de Beeck, D. Van Dyck, “Maximum-likelihood method for focus-variation image reconstruction in high resolution transmission electron microscopy,” Ultramicroscopy 64, 109–135 (1996).
[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.

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[CrossRef]

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, D. Paganin, “Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy,” Ultramicroscopy 88, 85–97 (2001).
[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.

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[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).

Giacovazzo, C.

C. Giacovazzo, Direct Methods in Crystallography (Academic, New York, 1980).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), p. 77.

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.

Haider, M.

M. Haider, G. Braunshausen, E. Schwan, “Correction of the spherical aberration of a 200 k TEM by means of a hexapole-corrector,” Optik (Stuttgart) 99, 167–179 (1995).

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).
[CrossRef] [PubMed]

Ikuta, T.

Y. Takai, T. Kawasaki, Y. Kimura, T. Ikuta, R. Shimizu, “Dynamic observation of an atom-sized gold wire by phase electron microscopy,” Phys. Rev. Lett. 87, 106105 (2001).
[CrossRef] [PubMed]

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]

Kawasaki, T.

Y. Takai, T. Kawasaki, Y. Kimura, T. Ikuta, R. Shimizu, “Dynamic observation of an atom-sized gold wire by phase electron microscopy,” Phys. Rev. Lett. 87, 106105 (2001).
[CrossRef] [PubMed]

Kimura, Y.

Y. Takai, T. Kawasaki, Y. Kimura, T. Ikuta, R. Shimizu, “Dynamic observation of an atom-sized gold wire by phase electron microscopy,” Phys. Rev. Lett. 87, 106105 (2001).
[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]

Miao, J.

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, 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, 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).
[CrossRef] [PubMed]

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]

Nakajima, N.

Nelson, E. C.

M. A. O’Keefe, E. C. Nelson, Y. C. Wang, A. Thust, “Sub-ångström resolution of atomistic structures below 0.8 Å,” Philos. Mag. B 81, 1861–1878 (2001).
[CrossRef]

Nugent, K. A.

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (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]

M. A. O’Keefe, E. C. Nelson, Y. C. Wang, A. Thust, “Sub-ångström resolution of atomistic structures below 0.8 Å,” Philos. Mag. B 81, 1861–1878 (2001).
[CrossRef]

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]

Op de Beeck, M.

W. M. J. Coene, A. Thust, M. Op de Beeck, D. Van Dyck, “Maximum-likelihood method for focus-variation image reconstruction in high resolution transmission electron microscopy,” Ultramicroscopy 64, 109–135 (1996).
[CrossRef]

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]

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[CrossRef]

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, D. Paganin, “Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy,” Ultramicroscopy 88, 85–97 (2001).
[CrossRef] [PubMed]

Paganin, D.

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, D. Paganin, “Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy,” Ultramicroscopy 88, 85–97 (2001).
[CrossRef] [PubMed]

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (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.

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).
[CrossRef] [PubMed]

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]

Schwan, E.

M. Haider, G. Braunshausen, E. Schwan, “Correction of the spherical aberration of a 200 k TEM by means of a hexapole-corrector,” Optik (Stuttgart) 99, 167–179 (1995).

Sheppard, C. J. R.

T. Wilson, A. R. Carlini, C. J. R. Sheppard, “Phase contrast microscopy by nearly full illumination,” Optik (Stuttgart) 70, 166–169 (1985).

Shimizu, R.

Y. Takai, T. Kawasaki, Y. Kimura, T. Ikuta, R. Shimizu, “Dynamic observation of an atom-sized gold wire by phase electron microscopy,” Phys. Rev. Lett. 87, 106105 (2001).
[CrossRef] [PubMed]

Takai, Y.

Y. Takai, T. Kawasaki, Y. Kimura, T. Ikuta, R. Shimizu, “Dynamic observation of an atom-sized gold wire by phase electron microscopy,” Phys. Rev. Lett. 87, 106105 (2001).
[CrossRef] [PubMed]

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]

Thust, A.

M. A. O’Keefe, E. C. Nelson, Y. C. Wang, A. Thust, “Sub-ångström resolution of atomistic structures below 0.8 Å,” Philos. Mag. B 81, 1861–1878 (2001).
[CrossRef]

W. M. J. Coene, A. Thust, M. Op de Beeck, D. Van Dyck, “Maximum-likelihood method for focus-variation image reconstruction in high resolution transmission electron microscopy,” Ultramicroscopy 64, 109–135 (1996).
[CrossRef]

Van Dyck, D.

W. M. J. Coene, A. Thust, M. Op de Beeck, D. Van Dyck, “Maximum-likelihood method for focus-variation image reconstruction in high resolution transmission electron microscopy,” Ultramicroscopy 64, 109–135 (1996).
[CrossRef]

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]

Wang, Y. C.

M. A. O’Keefe, E. C. Nelson, Y. C. Wang, A. Thust, “Sub-ångström resolution of atomistic structures below 0.8 Å,” Philos. Mag. B 81, 1861–1878 (2001).
[CrossRef]

Watanabe, M.

Wilson, T.

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Appl. Opt.

J. Opt. Soc. Am.

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[CrossRef]

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[CrossRef] [PubMed]

Opt. Commun.

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[CrossRef]

Optik (Stuttgart)

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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).

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Philos. Mag. B

M. A. O’Keefe, E. C. Nelson, Y. C. Wang, A. Thust, “Sub-ångström resolution of atomistic structures below 0.8 Å,” Philos. Mag. B 81, 1861–1878 (2001).
[CrossRef]

Phys. Rev. E

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[CrossRef]

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[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of the object-reconstruction system that uses phase retrieval. The object function is reconstructed from Fresnel-zone intensities of the object unshifted and shifted in each direction of the u and v axes in the illumination of the Fraunhofer diffraction pattern of a circular aperture.

Fig. 2
Fig. 2

Difference between the functions 2J 1(r)/r and exp(-r 2/8) as a function of r.

Fig. 3
Fig. 3

Original object function used in the computer simulation: (a) modulus and (b) phase of an object with a circular extent of radius ∼14 μm.

Fig. 4
Fig. 4

Reconstruction of the object shown in Fig. 3 from noisy Fresnel-zone intensities when, in the system of Fig. 1, the distances z 1 = 600 mm and z 2 = 80 mm, the aperture radius is 5 μm, and the coherent illumination has a wavelength λ = 0.682 nm. (a) Modulus and (b) phase of the reconstructed object, and (c) and (d) are cross-sectional profiles of the functions in (a) and (b), respectively, taken along horizontal lines passing through the center. The dotted and solid curves represent the original and the reconstructed objects, respectively.

Fig. 5
Fig. 5

Original object function used in the computer simulation: (a) modulus and (b) phase of an object with two first-order vortices of Laguerre–Gaussian mode.

Fig. 6
Fig. 6

Reconstruction of the object shown in Fig. 5 from noisy Fresnel-zone intensities when the same parameters used in Fig. 4 are used in the system of Fig. 1. (a) Modulus and (b) phase of the reconstructed object, and (c) and (d) are cross-sectional profiles of the functions in (a) and (b), respectively, taken along horizontal lines passing through the center. The dotted and solid curves represent the original and the reconstructed objects, respectively.

Equations (19)

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bu, v=πw2 expikz1iλz1expi k2z1u2+v2×2J1kwu2+v2/z1kwu2+v2/z1,
F1x, y=expikz2iλz2expi k2z2x2+y2U1x, y,
U1x, y=σ fu, vbu, vexpi k2z2u2+v2×exp-i 2πλz2xu+yvdudv,
F2x, y=A σ fu-τ, vbu, vexpi k2z2u2+v2exp-i 2πλz2xu+yvdudv=A σ fu, vbu+τ, v×expi k2z2u2+v2×exp-i 2πλz2x-τu+yvdudv,
2J1rr=1-r28+r4192+,
r=kwu2+v2z1.
exp-r28=1-r28+r4128+.
bu, vπw2 expikz1iλz1expi k2z1u2+v2×exp-k2w2u2+v28z12,
bu+τ, v=bu, vexp-πwλz12τu×expi kτuz1×expi kτ22z1-12πwτλz12.
|F2x, y|=exp-12πwτλz12|U1x-s-ic, y|,
s=1+z2z1τ,
c=λz22ππwλz12τ.
U1x, y=Mx, yexpiϕx, y.
|U1x-ic, y|=|Mx-ic, y|×exp-Im ϕx-ic, y,
ln|F2x+s, y||Mx-ic, y|+12πwτλz12=-Im ϕx-ic, y.
|F3x, y|=exp-12πwνλz12|U1x, y-s-ic|,
s=1+z2z1ν,
c=λz22ππwλz12ν.
ER=u,v |fu, v-fru, v|2u,v |fu, v|21/2,

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