Abstract

A method that effects phase reconstruction from a single image of a phase object is presented. The method, which is founded on the iterative transform algorithm, more specifically the error reduction version of this algorithm, applies this algorithm in an unusual way, in that it attempts to solve the phase problem from the measurement of a single image of the object taken by using a half-plane aperture rather than its diffraction pattern. A description of the method, an analysis of its performance, and suggestions as to its potential application are given.

© 2004 Optical Society of America

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  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).
  2. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  3. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1987).
    [CrossRef]
  4. H. Takajo, T. Takahashi, R. Ueda, M. Taninaka, “Study on the convergence property of the hybrid input–output algorithm used for phase retrieval,” J. Opt. Soc. Am. A 15, 2849–2861 (1998).
    [CrossRef]
  5. I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (four pages) (2001).
    [CrossRef] [PubMed]
  6. 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]
  7. J. Miao, T. Ishikawa, E. H. Anderson, K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (six pages) (2003).
    [CrossRef]
  8. H. He, S. Marchesini, M. Howells, U. Weierstall, G. Hembree, J. C. H. Spence, “Experimental lensless soft-x-ray imaging using iterative algorithms: phasing diffuse scattering,” Acta Crystallogr. Sect. A Cryst. Phys. Diffr. Theor. Gen. Crystallogr. 59, 143–152 (2003).
    [CrossRef]
  9. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
    [CrossRef]
  10. R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
    [CrossRef]
  11. R. P. Millane, “Phase problems for periodic images: effects of support and symmetry,” J. Opt. Soc. Am. A 10, 1037–1045 (1993).
    [CrossRef]
  12. U. Weierstall, Q. Chen, J. C. H. Spence, M. R. Howells, M. Isaacson, R. R. Panepucci, “Image reconstruction from electron and x-ray diffraction patterns using iterative algorithms: experiment and simulation,” Ultramicroscopy 90, 171–195 (2002).
    [CrossRef] [PubMed]
  13. J. C. H. Spence, M. Howells, L. D. Marks, J. Miao, “Lensless imaging: a workshop on new approaches to the phase problem for non-periodic objects,” Ultramicroscopy 90, 1–6 (2001).
    [CrossRef]
  14. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  15. D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
    [CrossRef]
  16. 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]
  17. A. Thust, W. M. J. Coene, M. Op de Beeck, D. Van Dyck, “Focal-series reconstruction in HRTEM: simulation studies on non-periodic objects,” Ultramicroscopy 64, 211–230 (1996).
    [CrossRef]
  18. L. J. Allen, W. McBride, N. L. O’Leary, M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy doi:.
    [CrossRef]
  19. D. J. Smith, “The realization of atomic resolution with the electron microscope,” Rep. Prog. Phys. 60, 1513–1580 (1997).
    [CrossRef]
  20. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).
  21. F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
    [CrossRef] [PubMed]
  22. G. S. Settles, Schlieren and Shadowgraph Techniques: Visualizing Phenomena in Transport Media (Springer-Verlag, Berlin, 2001).
  23. D. L. Misell, A. H. Greenaway, “An application of the Hilbert transform in electron microscopy: I. Bright-field microscopy,” J. Phys. D 7, 832–855 (1974).
    [CrossRef]
  24. J. C. H. Spence, “Complex image determination in electron microscopy,” Opt. Acta 21, 835–837 (1974).
    [CrossRef]
  25. J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
    [CrossRef]
  26. J. C. Bortz, B. J. Thompson, “Phase retrieval by optical differentiation,” in Wavefront Sensing, N. Bareket, C. L. Koliopoulos, eds, Proc. SPIE351, 71–79 (1983).
    [CrossRef]
  27. A. Levi, H. Stark, “Restorations from phase and magnitude by generalised projections,” in Image Recovery Theory and Application, H. Stark, ed. (Academic, London, 1987), pp. 277–320.
  28. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

2003 (3)

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]

J. Miao, T. Ishikawa, E. H. Anderson, K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (six pages) (2003).
[CrossRef]

H. He, S. Marchesini, M. Howells, U. Weierstall, G. Hembree, J. C. H. Spence, “Experimental lensless soft-x-ray imaging using iterative algorithms: phasing diffuse scattering,” Acta Crystallogr. Sect. A Cryst. Phys. Diffr. Theor. Gen. Crystallogr. 59, 143–152 (2003).
[CrossRef]

2002 (1)

U. Weierstall, Q. Chen, J. C. H. Spence, M. R. Howells, M. Isaacson, R. R. Panepucci, “Image reconstruction from electron and x-ray diffraction patterns using iterative algorithms: experiment and simulation,” Ultramicroscopy 90, 171–195 (2002).
[CrossRef] [PubMed]

2001 (2)

J. C. H. Spence, M. Howells, L. D. Marks, J. Miao, “Lensless imaging: a workshop on new approaches to the phase problem for non-periodic objects,” Ultramicroscopy 90, 1–6 (2001).
[CrossRef]

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (four pages) (2001).
[CrossRef] [PubMed]

1998 (2)

1997 (1)

D. J. Smith, “The realization of atomic resolution with the electron microscope,” Rep. Prog. Phys. 60, 1513–1580 (1997).
[CrossRef]

1996 (2)

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]

A. Thust, W. M. J. Coene, M. Op de Beeck, D. Van Dyck, “Focal-series reconstruction in HRTEM: simulation studies on non-periodic objects,” Ultramicroscopy 64, 211–230 (1996).
[CrossRef]

1993 (1)

1987 (3)

1983 (1)

1982 (1)

1981 (1)

J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
[CrossRef]

1974 (2)

D. L. Misell, A. H. Greenaway, “An application of the Hilbert transform in electron microscopy: I. Bright-field microscopy,” J. Phys. D 7, 832–855 (1974).
[CrossRef]

J. C. H. Spence, “Complex image determination in electron microscopy,” Opt. Acta 21, 835–837 (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).

1955 (1)

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[CrossRef] [PubMed]

Allen, L. J.

L. J. Allen, W. McBride, N. L. O’Leary, M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy doi:.
[CrossRef]

Anderson, E. H.

J. Miao, T. Ishikawa, E. H. Anderson, K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (six pages) (2003).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).

Bortz, J. C.

J. C. Bortz, B. J. Thompson, “Phase retrieval by optical differentiation,” in Wavefront Sensing, N. Bareket, C. L. Koliopoulos, eds, Proc. SPIE351, 71–79 (1983).
[CrossRef]

Chen, Q.

U. Weierstall, Q. Chen, J. C. H. Spence, M. R. Howells, M. Isaacson, R. R. Panepucci, “Image reconstruction from electron and x-ray diffraction patterns using iterative algorithms: experiment and simulation,” Ultramicroscopy 90, 171–195 (2002).
[CrossRef] [PubMed]

Coene, W. M. J.

A. Thust, W. M. J. Coene, M. Op de Beeck, D. Van Dyck, “Focal-series reconstruction in HRTEM: simulation studies on non-periodic objects,” Ultramicroscopy 64, 211–230 (1996).
[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]

Fienup, J. R.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

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.

D. L. Misell, A. H. Greenaway, “An application of the Hilbert transform in electron microscopy: I. Bright-field microscopy,” J. Phys. D 7, 832–855 (1974).
[CrossRef]

He, H.

H. He, S. Marchesini, M. Howells, U. Weierstall, G. Hembree, J. C. H. Spence, “Experimental lensless soft-x-ray imaging using iterative algorithms: phasing diffuse scattering,” Acta Crystallogr. Sect. A Cryst. Phys. Diffr. Theor. Gen. Crystallogr. 59, 143–152 (2003).
[CrossRef]

Hembree, G.

H. He, S. Marchesini, M. Howells, U. Weierstall, G. Hembree, J. C. H. Spence, “Experimental lensless soft-x-ray imaging using iterative algorithms: phasing diffuse scattering,” Acta Crystallogr. Sect. A Cryst. Phys. Diffr. Theor. Gen. Crystallogr. 59, 143–152 (2003).
[CrossRef]

Hodgson, K. O.

J. Miao, T. Ishikawa, E. H. Anderson, K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (six pages) (2003).
[CrossRef]

Howells, M.

H. He, S. Marchesini, M. Howells, U. Weierstall, G. Hembree, J. C. H. Spence, “Experimental lensless soft-x-ray imaging using iterative algorithms: phasing diffuse scattering,” Acta Crystallogr. Sect. A Cryst. Phys. Diffr. Theor. Gen. Crystallogr. 59, 143–152 (2003).
[CrossRef]

J. C. H. Spence, M. Howells, L. D. Marks, J. Miao, “Lensless imaging: a workshop on new approaches to the phase problem for non-periodic objects,” Ultramicroscopy 90, 1–6 (2001).
[CrossRef]

Howells, M. R.

U. Weierstall, Q. Chen, J. C. H. Spence, M. R. Howells, M. Isaacson, R. R. Panepucci, “Image reconstruction from electron and x-ray diffraction patterns using iterative algorithms: experiment and simulation,” Ultramicroscopy 90, 171–195 (2002).
[CrossRef] [PubMed]

Isaacson, M.

U. Weierstall, Q. Chen, J. C. H. Spence, M. R. Howells, M. Isaacson, R. R. Panepucci, “Image reconstruction from electron and x-ray diffraction patterns using iterative algorithms: experiment and simulation,” Ultramicroscopy 90, 171–195 (2002).
[CrossRef] [PubMed]

Ishikawa, T.

J. Miao, T. Ishikawa, E. H. Anderson, K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (six pages) (2003).
[CrossRef]

Lane, R. G.

R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
[CrossRef]

Levi, A.

A. Levi, H. Stark, “Restorations from phase and magnitude by generalised projections,” in Image Recovery Theory and Application, H. Stark, ed. (Academic, London, 1987), pp. 277–320.

Marchesini, S.

H. He, S. Marchesini, M. Howells, U. Weierstall, G. Hembree, J. C. H. Spence, “Experimental lensless soft-x-ray imaging using iterative algorithms: phasing diffuse scattering,” Acta Crystallogr. Sect. A Cryst. Phys. Diffr. Theor. Gen. Crystallogr. 59, 143–152 (2003).
[CrossRef]

Marks, L. D.

J. C. H. Spence, M. Howells, L. D. Marks, J. Miao, “Lensless imaging: a workshop on new approaches to the phase problem for non-periodic objects,” Ultramicroscopy 90, 1–6 (2001).
[CrossRef]

McBride, W.

L. J. Allen, W. McBride, N. L. O’Leary, M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy doi:.
[CrossRef]

Miao, J.

J. Miao, T. Ishikawa, E. H. Anderson, K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (six pages) (2003).
[CrossRef]

J. C. H. Spence, M. Howells, L. D. Marks, J. Miao, “Lensless imaging: a workshop on new approaches to the phase problem for non-periodic objects,” Ultramicroscopy 90, 1–6 (2001).
[CrossRef]

Millane, R. P.

Misell, D. L.

D. L. Misell, A. H. Greenaway, “An application of the Hilbert transform in electron microscopy: I. Bright-field microscopy,” J. Phys. D 7, 832–855 (1974).
[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]

Nugent, K. A.

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

O’Leary, N. L.

L. J. Allen, W. McBride, N. L. O’Leary, M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy doi:.
[CrossRef]

Op de Beeck, M.

A. Thust, W. M. J. Coene, M. Op de Beeck, D. Van Dyck, “Focal-series reconstruction in HRTEM: simulation studies on non-periodic objects,” Ultramicroscopy 64, 211–230 (1996).
[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]

Oxley, M. P.

L. J. Allen, W. McBride, N. L. O’Leary, M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy doi:.
[CrossRef]

Paganin, D.

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

Panepucci, R. R.

U. Weierstall, Q. Chen, J. C. H. Spence, M. R. Howells, M. Isaacson, R. R. Panepucci, “Image reconstruction from electron and x-ray diffraction patterns using iterative algorithms: experiment and simulation,” Ultramicroscopy 90, 171–195 (2002).
[CrossRef] [PubMed]

Pfeifer, M. A.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (four pages) (2001).
[CrossRef] [PubMed]

Pitney, J. A.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (four pages) (2001).
[CrossRef] [PubMed]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Robinson, I. K.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (four pages) (2001).
[CrossRef] [PubMed]

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).

Settles, G. S.

G. S. Settles, Schlieren and Shadowgraph Techniques: Visualizing Phenomena in Transport Media (Springer-Verlag, Berlin, 2001).

Smith, D. J.

D. J. Smith, “The realization of atomic resolution with the electron microscope,” Rep. Prog. Phys. 60, 1513–1580 (1997).
[CrossRef]

Spence, J. C. H.

H. He, S. Marchesini, M. Howells, U. Weierstall, G. Hembree, J. C. H. Spence, “Experimental lensless soft-x-ray imaging using iterative algorithms: phasing diffuse scattering,” Acta Crystallogr. Sect. A Cryst. Phys. Diffr. Theor. Gen. Crystallogr. 59, 143–152 (2003).
[CrossRef]

U. Weierstall, Q. Chen, J. C. H. Spence, M. R. Howells, M. Isaacson, R. R. Panepucci, “Image reconstruction from electron and x-ray diffraction patterns using iterative algorithms: experiment and simulation,” Ultramicroscopy 90, 171–195 (2002).
[CrossRef] [PubMed]

J. C. H. Spence, M. Howells, L. D. Marks, J. Miao, “Lensless imaging: a workshop on new approaches to the phase problem for non-periodic objects,” Ultramicroscopy 90, 1–6 (2001).
[CrossRef]

J. C. H. Spence, “Complex image determination in electron microscopy,” Opt. Acta 21, 835–837 (1974).
[CrossRef]

Stark, H.

A. Levi, H. Stark, “Restorations from phase and magnitude by generalised projections,” in Image Recovery Theory and Application, H. Stark, ed. (Academic, London, 1987), pp. 277–320.

Takahashi, T.

Takajo, H.

Taninaka, M.

Teague, M. R.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Thompson, B. J.

J. C. Bortz, B. J. Thompson, “Phase retrieval by optical differentiation,” in Wavefront Sensing, N. Bareket, C. L. Koliopoulos, eds, Proc. SPIE351, 71–79 (1983).
[CrossRef]

Thust, A.

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]

A. Thust, W. M. J. Coene, M. Op de Beeck, D. Van Dyck, “Focal-series reconstruction in HRTEM: simulation studies on non-periodic objects,” Ultramicroscopy 64, 211–230 (1996).
[CrossRef]

Ueda, R.

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]

A. Thust, W. M. J. Coene, M. Op de Beeck, D. Van Dyck, “Focal-series reconstruction in HRTEM: simulation studies on non-periodic objects,” Ultramicroscopy 64, 211–230 (1996).
[CrossRef]

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]

Vartanyants, I. A.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (four pages) (2001).
[CrossRef] [PubMed]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Wackerman, C. C.

Walker, J. G.

J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
[CrossRef]

Weierstall, U.

H. He, S. Marchesini, M. Howells, U. Weierstall, G. Hembree, J. C. H. Spence, “Experimental lensless soft-x-ray imaging using iterative algorithms: phasing diffuse scattering,” Acta Crystallogr. Sect. A Cryst. Phys. Diffr. Theor. Gen. Crystallogr. 59, 143–152 (2003).
[CrossRef]

U. Weierstall, Q. Chen, J. C. H. Spence, M. R. Howells, M. Isaacson, R. R. Panepucci, “Image reconstruction from electron and x-ray diffraction patterns using iterative algorithms: experiment and simulation,” Ultramicroscopy 90, 171–195 (2002).
[CrossRef] [PubMed]

Williams, G. J.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (four pages) (2001).
[CrossRef] [PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).

Zernike, F.

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[CrossRef] [PubMed]

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]

Acta Crystallogr. Sect. A Cryst. Phys. Diffr. Theor. Gen. Crystallogr. (1)

H. He, S. Marchesini, M. Howells, U. Weierstall, G. Hembree, J. C. H. Spence, “Experimental lensless soft-x-ray imaging using iterative algorithms: phasing diffuse scattering,” Acta Crystallogr. Sect. A Cryst. Phys. Diffr. Theor. Gen. Crystallogr. 59, 143–152 (2003).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

J. Phys. D (1)

D. L. Misell, A. H. Greenaway, “An application of the Hilbert transform in electron microscopy: I. Bright-field microscopy,” J. Phys. D 7, 832–855 (1974).
[CrossRef]

Opt. Acta (2)

J. C. H. Spence, “Complex image determination in electron microscopy,” Opt. Acta 21, 835–837 (1974).
[CrossRef]

J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
[CrossRef]

Opt. Commun. (1)

R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
[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)

J. Miao, T. Ishikawa, E. H. Anderson, K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (six pages) (2003).
[CrossRef]

Phys. Rev. Lett. (2)

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (four pages) (2001).
[CrossRef] [PubMed]

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

Rep. Prog. Phys. (1)

D. J. Smith, “The realization of atomic resolution with the electron microscope,” Rep. Prog. Phys. 60, 1513–1580 (1997).
[CrossRef]

Science (2)

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[CrossRef] [PubMed]

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Figures (9)

Fig. 1
Fig. 1

Proposed optical configuration: a half-plane aperture is placed in the back focal plane of the lens. This allows phase reconstruction from a single image of a phase object by use of an ER and HIO-based phase reconstruction method.

Fig. 2
Fig. 2

Diagram showing how the algorithm can be implemented: FT is Fourier transform; iFT is inverse Fourier transform; δ is a constant used to avoid division by zero. The diagram shows how the algorithm is a two-step process. In the first step half of the diffraction pattern is reconstructed and stored in ψs(q). In the second step, since the object is a phase object, its image is assumed to have unit amplitude that along with ψs(q) is used as a constraint to complete the reconstruction of the wave function.

Fig. 3
Fig. 3

a, Phase map of the phase object constructed to test the algorithm; its phase range in radians is displayed at the base of the figure. b, Image of the phase object taken with a half-plane aperture; the maximum and minimum intensity values are also shown at the base of the figure. The proposed phase reconstruction algorithm uses b to reconstruct a.

Fig. 4
Fig. 4

a, Original phase map (used in construction of the phase object). b, Phase map reconstructed from Fig. 3b. Their phase ranges in radians are displayed at the base of each figure. Although the phase maps are very similar, they are not identical. This difference can be identified by inspection of the Fourier transforms of a and b shown in c and d and noticing that the algorithm has not reconstructed the frequencies along the line qx=0 correctly.

Fig. 5
Fig. 5

By use of a modified aperture [see Eq. (7)], the simulation shown in Fig. 4 has been repeated. It is clear, through comparison of the phase ranges (displayed in radians at the base of a and b) and inspection of the Fourier transforms of a and b shown in c and d, that the phase has been reconstructed correctly.

Fig. 6
Fig. 6

Obtained by subtracting Fig. 4a from Fig. 4b. The difference between the two is obviously quite small as demonstrated by the maximum and minimum values for the phase difference in radians shown at the base of the figure. The horizontal lines are expected given that the phase along the line qx=0 cannot be reconstructed correctly.

Fig. 7
Fig. 7

(a) Convergence curves for (a) step one and (b) step two of the method when applied to Fig. 3b plotted on a log scale.

Fig. 8
Fig. 8

Noise has been added to the image formed with the half-plane aperture at levels of 5% and 10% of the maximum intensity value of that image; the images and their maximum and minimum intensity values are shown. Adjacent to each image is the phase map reconstructed by use of our ER and HIO style algorithm; the algorithm is robust in the presence of noisy data.

Fig. 9
Fig. 9

a, General phase object whose phase range is in radians; b, Fourier transform of a plotted on a log scale; c, image formed when a is imaged with a half-plane aperture; the intensity range is displayed at the base of the figure; d, phase map reconstructed from c.

Equations (7)

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ψ(r)=A(r)exp[iθ(r)],
ψ(z)=exp(i2πkz),
ψt(r)=exp[iϕ(r)],
ψt(q)=A(q)F[ψt(r)],
A(q)=1ifqx00ifqx<0,
ψi(r)=F-1[ψt(q)]=a(r)ψt(r),
A(q)=1ifqx0andqy01ifqx>0andqy00otherwise.

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