Abstract

A unified theory of noninterferometric phase recovery based on the so-called ambiguity function is introduced. The theory is used to analyze previously published techniques and unify them with the methods of phase-space tomography applicable to partially coherent data. The theory is then used to propose some new approaches to noninterferometric phase recovery.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  5. P. Cloetens, W. Ludwig, J. Barucher, D. Van Dyck, J. Van Landuyr, J. P. Guigay, and M. Schlenker, "Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
    [CrossRef]
  6. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard x-rays," Nature 384, 335-338 (1996).
    [CrossRef]
  7. B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, "Imaging--phase radiography with neutrons," Nature 408, 158-159 (2000).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  12. T. E. Gureyev, A. Roberts, and K. A. Nugent, "Partially coherent fields, the transport-of-intensity equation, and phase uniqueness," J. Opt. Soc. Am. A 12, 1942-1946 (1995).
    [CrossRef]
  13. A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, and I. McNulty, "Observation of an x-ray vortex," Opt. Lett. 27, 1752-1754 (2002).
    [CrossRef]
  14. K. A. Nugent, D. Paganin, and T. E. Gureyev, "A phase odyssey," Phys. Today 54, 27-32 (2001).
    [CrossRef]
  15. L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and 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]
  16. J. P. Guigay, "Fourier-transform analysis of Fresnel diffraction patterns and in-line holograms," Optik (Jena) 49, 121-125 (1977).
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    [CrossRef]
  18. A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
    [CrossRef]
  19. L. D. Turner, B. B. Dhal, J. P. Hayes, A. P. Mancuso, K. A. Nugent, D. Paterson, R. E. Scholten, C. Q. Tran, and A. G. Peele, "X-ray phase imaging: demonstration of extended conditions with homogeneous objects," Opt. Express 12, 2960-2965 (2004).
    [CrossRef] [PubMed]
  20. K. A. Nugent and D. Paganin, "Matter wave phase measurement: a noninterferometric approach," Phys. Rev. A 61, 063614 (2000).
    [CrossRef]
  21. S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, "Quantitative phase-sensitive imaging in a transmission electron microscope," Ultramicroscopy 83, 67-73 (2000).
    [CrossRef] [PubMed]
  22. A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase microscopy," Opt. Lett. 23, 817-819 (1998).
    [CrossRef]
  23. T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, "Quantitative in-line phase-contrast imaging with multienergy x rays," Phys. Rev. Lett. 86, 5827-5830 (2001).
    [CrossRef] [PubMed]
  24. D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. 206, 33-40 (2002).
    [CrossRef] [PubMed]
  25. X. Z. Wu, H. Liu, and A. M. Yan, "X-ray phase-attenuation duality and phase retrieval," Opt. Lett. 30, 379-381 (2005).
    [CrossRef] [PubMed]
  26. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x-rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
    [CrossRef] [PubMed]
  27. W. Coene, G. Janssen, M. O. de Beek, and D. van Dyck, "Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron-microscopy," Phys. Rev. Lett. 69, 3743-3746 (1992).
    [CrossRef] [PubMed]
  28. P. Cloetens, R. Barrett., J. Baruchel, J. P. Guigay, and M. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D 29, 133-146 (1996).
    [CrossRef]
  29. R. W. Gerchberg and W. O. Saxton, "Practical algorithm for determination of phase from image and diffraction plane pictures," Optik (Jena) 35, 237-246 (1972).
  30. J. R. Fienup, "Phase retrieval algorithms--a comparison," Appl. Opt. 21, 2758-2769 (1982).
    [CrossRef] [PubMed]
  31. R. H. T. Bates, "Fourier phase problems are uniquely solvable in more than one dimension. 1. Underlying theory," Optik (Jena) 61, 247-262 (1982).
  32. K. A. Nugent, A. G. Peele, H. N. Chapman, and H. M. Quiney, "Diffraction with wavefront curvature: a path to unique phase recovery," Acta Crystallogr., Sect. A: Found. Crystallogr. 61, 373-381 (2005).
    [CrossRef]
  33. J. W. Miao, P. Charalambous, J. Kirz, and D. Sayre, "Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens," Nature 400, 324-344 (1999).
    [CrossRef]
  34. I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, "Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction," Phys. Rev. Lett. 87, 195505 (2001).
    [CrossRef] [PubMed]
  35. S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101 (2003).
    [CrossRef]
  36. R. Neutze, R. Wouts, D. Van Der Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond x-ray pulses," Nature 406, 752-757 (2000).
    [CrossRef] [PubMed]
  37. K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, "Unique phase recovery for nonperiodic objects," Phys. Rev. Lett. 91, 203902 (2003).
    [CrossRef] [PubMed]
  38. X. H. Xiao and Q. Shen, "Wave propagation and phase retrieval in Fresnel diffraction by a distorted-object approach," Phys. Rev. B 72, 033103 (2005).
    [CrossRef]
  39. T. A. Pitts and J. F. Greenleaf, "Fresnel transform phase retrieval from magnitude," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1035-1045 (2003).
    [CrossRef] [PubMed]
  40. H. M. Quiney, K. A. Nugent, and A. G. Peele, "Iterative image reconstruction algorithms using wave-front intensity and phase variation," Opt. Lett. 30, 1638-1640 (2005).
    [CrossRef] [PubMed]
  41. H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, "Diffractive imaging of highly focused x-ray fields," Nat. Phys. 2, 101-104 (2006).
    [CrossRef]
  42. G. J. Williams, H. M. Quiney, B. B. Dhal, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, "Fresnel coherent diffractive imaging," Phys. Rev. Lett. 97, 025506 (2006).
    [CrossRef] [PubMed]
  43. W. Hoppe, "Trace structure-analysis, ptychography, phase tomography," Ultramicroscopy 10, 187-198 (1982).
    [CrossRef]
  44. P. D. Nellist and J. M. Rodenberg, "Electron ptychography. I. Experimental demonstration beyond the conventional resolution limits," Acta Crystallogr., Sect. A: Found. Crystallogr. 54, 49-60 (1998).
    [CrossRef]
  45. H. M. L. Faulkner and J. M. Rodenberg, "Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm," Phys. Rev. Lett. 93, 023903 (2004).
    [CrossRef] [PubMed]
  46. J. M. Rodenberg and H. M. L. Faulkner, "A phase retrieval algorithm for shifting illumination," Appl. Phys. Lett. 85, 4795-4797 (2004).
    [CrossRef]
  47. H. M. L. Faulkner and J. M. Rodenberg, "Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy," Ultramicroscopy 103, 153-164 (2005).
    [CrossRef]
  48. W. E. McBride, N. L. O"Leary, K. A. Nugent, and L. J. Allen, "Astigmatic electron diffraction imaging: a novel mode for structure determination," Acta Crystallogr., Sect. A: Found. Crystallogr. 61, 321-324 (2005).
    [CrossRef]
  49. K. A. Nugent, "Wave field determination using 3-dimensional intensity information," Phys. Rev. Lett. 68, 2261-2264 (1992).
    [CrossRef] [PubMed]
  50. M. G. Raymer, M. Beck, and D. F. McAlister, "Complex wave-field reconstruction using phase-space tomography," Phys. Rev. Lett. 72, 1137-1140 (1994).
    [CrossRef] [PubMed]
  51. D. M. Marks, R. A. Stack, and D. J. Brady, "Astigmatic coherence sensor for digital imaging," Opt. Lett. 25, 1726-1728 (2000).
    [CrossRef]
  52. C. Q. Tran, A. G. Peele, A. Roberts, K. A. Nugent, D. Paterson, and I. McNulty, "Synchrotron beam coherence: a spatially resolved measurement," Opt. Lett. 30, 204-206 (2005).
    [CrossRef] [PubMed]
  53. C. Q. Tran, A. P. Mancuso, B. B. Dhal, K. A. Nugent, A. G. Peele, Z. Cai, and D. Paterson, "Phase-space reconstruction of focused x-ray fields," J. Opt. Soc. Am. A 23, 1779-1786 (2006).
    [CrossRef]

2006

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, "Diffractive imaging of highly focused x-ray fields," Nat. Phys. 2, 101-104 (2006).
[CrossRef]

G. J. Williams, H. M. Quiney, B. B. Dhal, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, "Fresnel coherent diffractive imaging," Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef] [PubMed]

C. Q. Tran, A. P. Mancuso, B. B. Dhal, K. A. Nugent, A. G. Peele, Z. Cai, and D. Paterson, "Phase-space reconstruction of focused x-ray fields," J. Opt. Soc. Am. A 23, 1779-1786 (2006).
[CrossRef]

2005

H. M. Quiney, K. A. Nugent, and A. G. Peele, "Iterative image reconstruction algorithms using wave-front intensity and phase variation," Opt. Lett. 30, 1638-1640 (2005).
[CrossRef] [PubMed]

X. H. Xiao and Q. Shen, "Wave propagation and phase retrieval in Fresnel diffraction by a distorted-object approach," Phys. Rev. B 72, 033103 (2005).
[CrossRef]

H. M. L. Faulkner and J. M. Rodenberg, "Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy," Ultramicroscopy 103, 153-164 (2005).
[CrossRef]

W. E. McBride, N. L. O"Leary, K. A. Nugent, and L. J. Allen, "Astigmatic electron diffraction imaging: a novel mode for structure determination," Acta Crystallogr., Sect. A: Found. Crystallogr. 61, 321-324 (2005).
[CrossRef]

C. Q. Tran, A. G. Peele, A. Roberts, K. A. Nugent, D. Paterson, and I. McNulty, "Synchrotron beam coherence: a spatially resolved measurement," Opt. Lett. 30, 204-206 (2005).
[CrossRef] [PubMed]

C. Q. Tran, A. G. Peele, A. Roberts, K. A. Nugent, D. Faterson, and I. Mc Nulty, "X-ray imaging: a generalized approach using phase-space tomography," J. Opt. Soc. Am. A 22, 1691-1700 (2005).
[CrossRef]

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, "Optimization of phase contrast imaging using hard x-ray," Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

X. Z. Wu, H. Liu, and A. M. Yan, "X-ray phase-attenuation duality and phase retrieval," Opt. Lett. 30, 379-381 (2005).
[CrossRef] [PubMed]

K. A. Nugent, A. G. Peele, H. N. Chapman, and H. M. Quiney, "Diffraction with wavefront curvature: a path to unique phase recovery," Acta Crystallogr., Sect. A: Found. Crystallogr. 61, 373-381 (2005).
[CrossRef]

2004

L. D. Turner, B. B. Dhal, J. P. Hayes, A. P. Mancuso, K. A. Nugent, D. Paterson, R. E. Scholten, C. Q. Tran, and A. G. Peele, "X-ray phase imaging: demonstration of extended conditions with homogeneous objects," Opt. Express 12, 2960-2965 (2004).
[CrossRef] [PubMed]

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

J. M. Rodenberg and H. M. L. Faulkner, "A phase retrieval algorithm for shifting illumination," Appl. Phys. Lett. 85, 4795-4797 (2004).
[CrossRef]

2003

T. A. Pitts and J. F. Greenleaf, "Fresnel transform phase retrieval from magnitude," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1035-1045 (2003).
[CrossRef] [PubMed]

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

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, "Unique phase recovery for nonperiodic objects," Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef] [PubMed]

2002

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. 206, 33-40 (2002).
[CrossRef] [PubMed]

A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, and I. McNulty, "Observation of an x-ray vortex," Opt. Lett. 27, 1752-1754 (2002).
[CrossRef]

2001

K. A. Nugent, D. Paganin, and T. E. Gureyev, "A phase odyssey," Phys. Today 54, 27-32 (2001).
[CrossRef]

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and 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]

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

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, "Quantitative in-line phase-contrast imaging with multienergy x rays," Phys. Rev. Lett. 86, 5827-5830 (2001).
[CrossRef] [PubMed]

2000

K. A. Nugent and D. Paganin, "Matter wave phase measurement: a noninterferometric approach," Phys. Rev. A 61, 063614 (2000).
[CrossRef]

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

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, "Imaging--phase radiography with neutrons," Nature 408, 158-159 (2000).
[CrossRef] [PubMed]

R. Neutze, R. Wouts, D. Van Der Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond x-ray pulses," Nature 406, 752-757 (2000).
[CrossRef] [PubMed]

D. M. Marks, R. A. Stack, and D. J. Brady, "Astigmatic coherence sensor for digital imaging," Opt. Lett. 25, 1726-1728 (2000).
[CrossRef]

1999

P. Cloetens, W. Ludwig, J. Barucher, D. Van Dyck, J. Van Landuyr, J. P. Guigay, and M. Schlenker, "Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

J. W. Miao, P. Charalambous, J. Kirz, and D. Sayre, "Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens," Nature 400, 324-344 (1999).
[CrossRef]

1998

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

P. D. Nellist and J. M. Rodenberg, "Electron ptychography. I. Experimental demonstration beyond the conventional resolution limits," Acta Crystallogr., Sect. A: Found. Crystallogr. 54, 49-60 (1998).
[CrossRef]

1997

A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
[CrossRef]

1996

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard x-rays," Nature 384, 335-338 (1996).
[CrossRef]

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

P. Cloetens, R. Barrett., J. Baruchel, J. P. Guigay, and M. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D 29, 133-146 (1996).
[CrossRef]

1995

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486-5492 (1995).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular-momentum to absorptive particles from a laser-beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

T. E. Gureyev, A. Roberts, and K. A. Nugent, "Partially coherent fields, the transport-of-intensity equation, and phase uniqueness," J. Opt. Soc. Am. A 12, 1942-1946 (1995).
[CrossRef]

1994

M. G. Raymer, M. Beck, and D. F. McAlister, "Complex wave-field reconstruction using phase-space tomography," Phys. Rev. Lett. 72, 1137-1140 (1994).
[CrossRef] [PubMed]

1993

J. M. Beckers, "Adaptive optics for astronomy--principles, performance, and applications," Annu. Rev. Astron. Astrophys. 31, 13-62 (1993).
[CrossRef]

1992

W. Coene, G. Janssen, M. O. de Beek, and D. van Dyck, "Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron-microscopy," Phys. Rev. Lett. 69, 3743-3746 (1992).
[CrossRef] [PubMed]

K. A. Nugent, "Wave field determination using 3-dimensional intensity information," Phys. Rev. Lett. 68, 2261-2264 (1992).
[CrossRef] [PubMed]

1988

1986

1983

1982

J. R. Fienup, "Phase retrieval algorithms--a comparison," Appl. Opt. 21, 2758-2769 (1982).
[CrossRef] [PubMed]

R. H. T. Bates, "Fourier phase problems are uniquely solvable in more than one dimension. 1. Underlying theory," Optik (Jena) 61, 247-262 (1982).

W. Hoppe, "Trace structure-analysis, ptychography, phase tomography," Ultramicroscopy 10, 187-198 (1982).
[CrossRef]

1978

J. P. Guigay, "The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams," Opt. Commun. 26, 136-138 (1978).
[CrossRef]

1977

J. P. Guigay, "Fourier-transform analysis of Fresnel diffraction patterns and in-line holograms," Optik (Jena) 49, 121-125 (1977).

1972

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

Allen, L. J.

W. E. McBride, N. L. O"Leary, K. A. Nugent, and L. J. Allen, "Astigmatic electron diffraction imaging: a novel mode for structure determination," Acta Crystallogr., Sect. A: Found. Crystallogr. 61, 321-324 (2005).
[CrossRef]

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and 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]

Allman, B. E.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, "Imaging--phase radiography with neutrons," Nature 408, 158-159 (2000).
[CrossRef] [PubMed]

Arif, M.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, "Imaging--phase radiography with neutrons," Nature 408, 158-159 (2000).
[CrossRef] [PubMed]

Bajt, S.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and 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, and Z. Barnea, "Quantitative phase imaging using hard x-rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Barrett., R.

P. Cloetens, R. Barrett., J. Baruchel, J. P. Guigay, and M. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D 29, 133-146 (1996).
[CrossRef]

Barty, A.

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

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

Baruchel, J.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, "Optimization of phase contrast imaging using hard x-ray," Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

P. Cloetens, R. Barrett., J. Baruchel, J. P. Guigay, and M. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D 29, 133-146 (1996).
[CrossRef]

Barucher, J.

P. Cloetens, W. Ludwig, J. Barucher, D. Van Dyck, J. Van Landuyr, J. P. Guigay, and M. Schlenker, "Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Bastiaans, M. J.

Beck, M.

M. G. Raymer, M. Beck, and D. F. McAlister, "Complex wave-field reconstruction using phase-space tomography," Phys. Rev. Lett. 72, 1137-1140 (1994).
[CrossRef] [PubMed]

Beckers, J. M.

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Peele, A. G.

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[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

Pogany, A.

A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard x-rays," Nature 384, 335-338 (1996).
[CrossRef]

Quiney, H. M.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, "Diffractive imaging of highly focused x-ray fields," Nat. Phys. 2, 101-104 (2006).
[CrossRef]

G. J. Williams, H. M. Quiney, B. B. Dhal, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, "Fresnel coherent diffractive imaging," Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef] [PubMed]

H. M. Quiney, K. A. Nugent, and A. G. Peele, "Iterative image reconstruction algorithms using wave-front intensity and phase variation," Opt. Lett. 30, 1638-1640 (2005).
[CrossRef] [PubMed]

K. A. Nugent, A. G. Peele, H. N. Chapman, and H. M. Quiney, "Diffraction with wavefront curvature: a path to unique phase recovery," Acta Crystallogr., Sect. A: Found. Crystallogr. 61, 373-381 (2005).
[CrossRef]

Raymer, M. G.

M. G. Raymer, M. Beck, and D. F. McAlister, "Complex wave-field reconstruction using phase-space tomography," Phys. Rev. Lett. 72, 1137-1140 (1994).
[CrossRef] [PubMed]

Roberts, A.

Robinson, I. K.

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

Roddier, F.

Rodenberg, J. M.

H. M. L. Faulkner and J. M. Rodenberg, "Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy," Ultramicroscopy 103, 153-164 (2005).
[CrossRef]

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

J. M. Rodenberg and H. M. L. Faulkner, "A phase retrieval algorithm for shifting illumination," Appl. Phys. Lett. 85, 4795-4797 (2004).
[CrossRef]

P. D. Nellist and J. M. Rodenberg, "Electron ptychography. I. Experimental demonstration beyond the conventional resolution limits," Acta Crystallogr., Sect. A: Found. Crystallogr. 54, 49-60 (1998).
[CrossRef]

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular-momentum to absorptive particles from a laser-beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Saxton, W. O.

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

Sayre, D.

J. W. Miao, P. Charalambous, J. Kirz, and D. Sayre, "Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens," Nature 400, 324-344 (1999).
[CrossRef]

Schelokov, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486-5492 (1995).
[CrossRef]

Schlenker, M.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, "Optimization of phase contrast imaging using hard x-ray," Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

P. Cloetens, W. Ludwig, J. Barucher, D. Van Dyck, J. Van Landuyr, J. P. Guigay, and M. Schlenker, "Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

P. Cloetens, R. Barrett., J. Baruchel, J. P. Guigay, and M. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D 29, 133-146 (1996).
[CrossRef]

Scholten, R. E.

Shen, Q.

X. H. Xiao and Q. Shen, "Wave propagation and phase retrieval in Fresnel diffraction by a distorted-object approach," Phys. Rev. B 72, 033103 (2005).
[CrossRef]

Snigirev, A.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486-5492 (1995).
[CrossRef]

Snigireva, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486-5492 (1995).
[CrossRef]

Stack, R. A.

Stevenson, A. W.

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, "Quantitative in-line phase-contrast imaging with multienergy x rays," Phys. Rev. Lett. 86, 5827-5830 (2001).
[CrossRef] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard x-rays," Nature 384, 335-338 (1996).
[CrossRef]

T. Bates, R. H.

R. H. T. Bates, "Fourier phase problems are uniquely solvable in more than one dimension. 1. Underlying theory," Optik (Jena) 61, 247-262 (1982).

Teague, M. R.

Tran, C. Q.

Turner, L. D.

Van Der Spoel, D.

R. Neutze, R. Wouts, D. Van Der Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond x-ray pulses," Nature 406, 752-757 (2000).
[CrossRef] [PubMed]

Van Dyck, D.

P. Cloetens, W. Ludwig, J. Barucher, D. Van Dyck, J. Van Landuyr, J. P. Guigay, and M. Schlenker, "Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

W. Coene, G. Janssen, M. O. de Beek, and D. van Dyck, "Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron-microscopy," Phys. Rev. Lett. 69, 3743-3746 (1992).
[CrossRef] [PubMed]

Van Landuyr, J.

P. Cloetens, W. Ludwig, J. Barucher, D. Van Dyck, J. Van Landuyr, J. P. Guigay, and M. Schlenker, "Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Vartanyants, I. A.

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

Wall, M.

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

Weckert, E.

R. Neutze, R. Wouts, D. Van Der Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond x-ray pulses," Nature 406, 752-757 (2000).
[CrossRef] [PubMed]

Weierstall, U.

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

Werner, S. A.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, "Imaging--phase radiography with neutrons," Nature 408, 158-159 (2000).
[CrossRef] [PubMed]

Wilkins, S. W.

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. 206, 33-40 (2002).
[CrossRef] [PubMed]

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, "Quantitative in-line phase-contrast imaging with multienergy x rays," Phys. Rev. Lett. 86, 5827-5830 (2001).
[CrossRef] [PubMed]

A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard x-rays," Nature 384, 335-338 (1996).
[CrossRef]

Williams, G. J.

G. J. Williams, H. M. Quiney, B. B. Dhal, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, "Fresnel coherent diffractive imaging," Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef] [PubMed]

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

Wouts, R.

R. Neutze, R. Wouts, D. Van Der Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond x-ray pulses," Nature 406, 752-757 (2000).
[CrossRef] [PubMed]

Wu, X. Z.

Xiao, X. H.

X. H. Xiao and Q. Shen, "Wave propagation and phase retrieval in Fresnel diffraction by a distorted-object approach," Phys. Rev. B 72, 033103 (2005).
[CrossRef]

Yan, A. M.

Zabler, S.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, "Optimization of phase contrast imaging using hard x-ray," Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

Acta Crystallogr., Sect. A: Found. Crystallogr.

K. A. Nugent, A. G. Peele, H. N. Chapman, and H. M. Quiney, "Diffraction with wavefront curvature: a path to unique phase recovery," Acta Crystallogr., Sect. A: Found. Crystallogr. 61, 373-381 (2005).
[CrossRef]

P. D. Nellist and J. M. Rodenberg, "Electron ptychography. I. Experimental demonstration beyond the conventional resolution limits," Acta Crystallogr., Sect. A: Found. Crystallogr. 54, 49-60 (1998).
[CrossRef]

W. E. McBride, N. L. O"Leary, K. A. Nugent, and L. J. Allen, "Astigmatic electron diffraction imaging: a novel mode for structure determination," Acta Crystallogr., Sect. A: Found. Crystallogr. 61, 321-324 (2005).
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Appl. Phys. Lett.

P. Cloetens, W. Ludwig, J. Barucher, D. Van Dyck, J. Van Landuyr, J. P. Guigay, and M. Schlenker, "Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

J. M. Rodenberg and H. M. L. Faulkner, "A phase retrieval algorithm for shifting illumination," Appl. Phys. Lett. 85, 4795-4797 (2004).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control

T. A. Pitts and J. F. Greenleaf, "Fresnel transform phase retrieval from magnitude," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1035-1045 (2003).
[CrossRef] [PubMed]

J. Microsc.

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. 206, 33-40 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D

P. Cloetens, R. Barrett., J. Baruchel, J. P. Guigay, and M. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D 29, 133-146 (1996).
[CrossRef]

Nat. Phys.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, "Diffractive imaging of highly focused x-ray fields," Nat. Phys. 2, 101-104 (2006).
[CrossRef]

Nature

J. W. Miao, P. Charalambous, J. Kirz, and D. Sayre, "Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens," Nature 400, 324-344 (1999).
[CrossRef]

R. Neutze, R. Wouts, D. Van Der Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond x-ray pulses," Nature 406, 752-757 (2000).
[CrossRef] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard x-rays," Nature 384, 335-338 (1996).
[CrossRef]

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, "Imaging--phase radiography with neutrons," Nature 408, 158-159 (2000).
[CrossRef] [PubMed]

Opt. Commun.

J. P. Guigay, "The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams," Opt. Commun. 26, 136-138 (1978).
[CrossRef]

Opt. Express

Opt. Lett.

Optik (Jena)

J. P. Guigay, "Fourier-transform analysis of Fresnel diffraction patterns and in-line holograms," Optik (Jena) 49, 121-125 (1977).

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

R. H. T. Bates, "Fourier phase problems are uniquely solvable in more than one dimension. 1. Underlying theory," Optik (Jena) 61, 247-262 (1982).

Phys. Rev. A

K. A. Nugent and D. Paganin, "Matter wave phase measurement: a noninterferometric approach," Phys. Rev. A 61, 063614 (2000).
[CrossRef]

Phys. Rev. B

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

X. H. Xiao and Q. Shen, "Wave propagation and phase retrieval in Fresnel diffraction by a distorted-object approach," Phys. Rev. B 72, 033103 (2005).
[CrossRef]

Phys. Rev. Lett.

G. J. Williams, H. M. Quiney, B. B. Dhal, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, "Fresnel coherent diffractive imaging," Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef] [PubMed]

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

K. A. Nugent, "Wave field determination using 3-dimensional intensity information," Phys. Rev. Lett. 68, 2261-2264 (1992).
[CrossRef] [PubMed]

M. G. Raymer, M. Beck, and D. F. McAlister, "Complex wave-field reconstruction using phase-space tomography," Phys. Rev. Lett. 72, 1137-1140 (1994).
[CrossRef] [PubMed]

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, "Quantitative in-line phase-contrast imaging with multienergy x rays," Phys. Rev. Lett. 86, 5827-5830 (2001).
[CrossRef] [PubMed]

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

W. Coene, G. Janssen, M. O. de Beek, and D. van Dyck, "Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron-microscopy," Phys. Rev. Lett. 69, 3743-3746 (1992).
[CrossRef] [PubMed]

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, "Unique phase recovery for nonperiodic objects," Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef] [PubMed]

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

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular-momentum to absorptive particles from a laser-beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Phys. Today

K. A. Nugent, D. Paganin, and T. E. Gureyev, "A phase odyssey," Phys. Today 54, 27-32 (2001).
[CrossRef]

Rev. Sci. Instrum.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, "Optimization of phase contrast imaging using hard x-ray," Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
[CrossRef]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486-5492 (1995).
[CrossRef]

Ultramicroscopy

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and 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]

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

H. M. L. Faulkner and J. M. Rodenberg, "Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy," Ultramicroscopy 103, 153-164 (2005).
[CrossRef]

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[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Fourier transform of the measured intensity corresponds to the plane through the four-dimensional ambiguity function. The sloped line is described by the equation x = z 0 q , where the symbols are defined in the text. The slice measured obviously depends on the location at which the intensity distribution is measured, and the projections for contact images ( z = 0 ) and the Fraunhofer diffraction ( z ) are also shown.

Fig. 2
Fig. 2

Method of Guigay uses two quite widely spaced intensity measurements and so uses projections through the ambiguity function that has a relatively wide angular separation.

Fig. 3
Fig. 3

Transport of intensity approach uses two intensity measurements that are separated by a differential distance, and so the projections through the ambiguity function are also separated by a differential angle. This diagram also applies to the multiple wavelength approach, where the difference in slope arises from a change in the wavelength rather than a change in the measurement plane.

Fig. 4
Fig. 4

Transport of the intensity equation measures the phase and intensity of the propagating field and so may be applied at any point in the wave field. That is, the projections need not take place around the plane of the object at z = 0 .

Fig. 5
Fig. 5

Homogeneous object approach makes a priori assumptions about the composition of the sample, and this assumption eliminates the need for a second measurement plane. The additional assumption that the object only weakly interacts with the sample allows for the measurement plane to be a nondifferential distance from the sample, as indicated by the large angle between the projection and the q axis.

Fig. 6
Fig. 6

Phase-only approach makes an assumption about the wave field—that there is a negligible amplitude modulation by the sample. Thus, only one measurement plane is required. The phase modulation can be quite high, in which case the measurement plane should be a differential distance from the sample.

Fig. 7
Fig. 7

Multiple plane method obtains many data sets at different distances from the sample and then undergoes a fitting procedure to find the wave field consistent with all of the data.

Fig. 8
Fig. 8

Gerchberg–Saxton approach assumes that an intensity measurement of the sample is obtained (corresponding to a projection along the q axis), and a far-field diffraction measurement is obtained (corresponding to a projection along the x axis). An iterative method is then used to find a consistent complex amplitude.

Fig. 9
Fig. 9

Gerchberg–Saxton–Fienup approach uses a single set of far-field data and planar sample illumination and knowledge about the support (size and shape) of the object. The data are therefore described using the Fraunhofer formalism. The a priori size and shape knowledge serves to eliminate the need for a second data set. An iterative method is then used to find a consistent solution.

Fig. 10
Fig. 10

Gerchberg–Saxton–Fienup approach in the Fresnel region allows for some phase curvature on the sample or at the detector and the data are described by the Fresnel formalism. This curvature introduces a correlation between the position in the measurement plane and position in the sample. As a result, the convergence is much better than with the Fraunhofer diffracted data, but the method is now sensitive to sample movement.

Fig. 11
Fig. 11

Astigmatic diffraction case illuminates the sample with waves that have orthogonal cylindrical curvature (the lighter curve in the figure indicates a wave that is curved out of the plane sketched in this two-dimensional representation). Although experimentally challenging, the convergence is excellent, and the solution is truly unique. Indeed, the formalism has a close relationship to that of the transport of the intensity equation, and it should be possible to perform a direct reconstruction, though this is probably unnecessary.

Fig. 12
Fig. 12

Figures 10, 11 indicate methods that use modifications to the sample illumination. In principle, these approaches can be generalized to allow for arbitrary phase curved illumination, as indicated. It is conjectured that this opens up new approaches to phase recovery that may have significant experimental advantages in some circumstances.

Tables (1)

Tables Icon

Table 1 Simple Classification System Proposed for the Approaches to Phase Recovery Discussed in This Paper a

Equations (57)

Equations on this page are rendered with MathJax. Learn more.

J z ( r 1 , r 2 ) = E z ( r 1 ) E z * ( r 2 ) ,
I z ( r ) = J z ( r , r ) .
B z ( r , u ) ( k 2 π ) 2 J z ( r + x 2 , r x 2 ) exp [ i k x u ] d x .
B z + Δ z ( r , u ) = B z ( r Δ z u , u ) ,
I z ( r ) = B z ( r , u ) d u .
A z ( q , x ) = J z ( r + x 2 , r x 2 ) exp [ i k q r ] d r .
A ( q , x ) = E ( r + x 2 ) E * ( r x 2 ) exp [ i k q r ] d r .
I ̂ ( q ) = A ( q , z q ) .
I ̂ z ( q ) = J z ( r + z q 2 , r z q 2 ) exp [ i k q r ] d r .
I ̂ z ( q ) = E ( r + z q 2 ) E * ( r z q 2 ) exp [ i k q r ] d r .
E z ( r , θ ) ( k 2 π ) 2 exp [ i k r 2 2 z ] f ( ρ , φ ) exp [ i k ρ 2 2 z ] exp [ i k r ρ z cos φ ] ρ d ρ d φ .
E 0 ( ρ , φ ) = A ( ρ ) exp [ i m φ ] ,
E z ( r , θ ) ( k 2 π ) 2 C ( r ) A ( ρ ) exp [ i k ρ 2 2 z ] { exp [ i ( m φ k r ρ z cos φ ) ] d φ } ρ d ρ ,
E z ( r , θ ) ( k 2 π ) 2 C ( r ) exp [ i m π 2 ] A ( ρ ) J m ( k r ρ z ) exp [ i k ρ 2 2 z ] ρ d ρ ,
I ̂ z 1 ( q ) = A ( q , z 1 q ) ,
I ̂ z 2 ( q ) = A ( q , z 2 q ) .
E ( r ) = E 0 exp [ μ ( r ) + i ϕ ( r ) ] E 0 ( 1 μ ( r ) + i ϕ ( r ) ) .
I ̂ z ( q ) = I 0 ( δ ( q ) 2 cos ( π λ z q 2 ) F [ μ ( r ) ] + 2 sin ( π λ z q 2 ) F [ ϕ ( r ) ] ) ,
I ̂ z ( q ) = A ( q , z q ) ,
I ̂ z + δ z ( q ) = A ( q , ( z + δ z ) q ) .
A ( q , ( z + δ z ) q ) A ( q , z q ) + δ z A ( q , z q ) z .
z A ( q ) = q 2 q E ( r + z q 2 ) E * ( r z q 2 ) exp [ i k q r ] d r .
j ( r ) = i k lim x 0 x E ( r + x 2 ) E * ( r x 2 ) .
z A ( q , 0 ) = i k q j ( r ) exp [ i k q r ] d r .
I ( r ) z = 1 k j ( r ) ,
j ( r ) = 1 k I ( r ) Φ ( r ) ,
I ( r ) z = 1 k [ I ( r ) Φ ( r ) ] .
I ̂ z , λ ( q ) = A ( q , z q ) ,
I ̂ z , k + δ k ( q ) = A ( q , z ( 1 + δ k k ) q ) ,
δ I ( r ) = z δ k k 2 [ I ( r ) Φ ( r ) ] .
I ̂ z ( q ) = A ( q , z q ) ,
E ( r ) = E 0 exp [ i k 0 ( δ i β ) t ( r ) ] ,
I ̂ z ( q ) = I 0 exp [ k 0 β ( t ( r + z q 2 ) + t ( r z q 2 ) ) i k 0 δ ( t ( r + z q 2 ) t ( r z q 2 ) ) ] exp [ i k q r ] d r .
t ( r ± z q 2 ) t ( r ) ± z q 2 t ( r ) ,
I ̂ z ( q ) = I 0 exp [ 2 k 0 β t ( r ) i k 0 z q t ( r ) ] exp [ i k 0 q r ] d r .
F [ t ( r ) ] = 2 π i q F [ t ( r ) ] ,
[ exp ( k 0 t ( r ) ) ] = k 0 t ( r ) exp ( k 0 t ( r ) ) ,
I ̂ z ( r ) = I 0 F [ exp ( 2 k 0 β t ( r ) ) ] ( 1 + δ β λ z q 2 ) .
t ( r ) = 1 2 k 0 β ln F 1 [ β β + λ z δ q 2 I ̂ z ( q ) I 0 ] .
I ̂ z ( q ) = A ( q , δ z q ) .
A ( q , 0 ) z = 1 δ z [ I ̂ δ z ( q ) I 0 δ ( q ) ] ,
A ( q , 0 ) z = 1 k 0 q 2 F [ ϕ ( r ) ] ,
ϕ ( r ) = k 0 δ z F 1 [ 1 q 2 [ I ̂ δ z ( q ) I 0 δ ( q ) ] ] .
I ̂ z n ( q ) = A ( q , z n q ) , n = 1 , 2 , , N .
I ̂ 0 ( q ) = A ( q , 0 ) ,
I ̂ ( q ) = A ( 0 , q ) .
I ̂ ( q ) = A ( 0 , q ) ,
I ̂ α ( q ) = A ( q , α q ) ,
I ̂ y ( q x , q y ) = A ( q x , 0 , β q x , q y ) ,
I ̂ x ( q x , q y ) = A ( 0 , q y , q x , α q y ) .
A 2 ( q , x ) = E ( r + x 2 ) E * ( r x 2 ) exp [ i k ( χ ( r + x 2 ) χ ( r x 2 ) ) ] exp [ i k q r ] d r .
χ ( r + x 2 ) χ ( r x 2 ) = γ ( r , x ) ,
A ( q , x ) = E ( r + x 2 ) E * ( r x 2 ) exp [ i k ( q r γ ( r , x ) ) ] d r .
I ̂ ( q ) = E ( r + z q 2 ) E * ( r z q 2 ) exp [ i k ( q r γ ( r , z q ) ) ] d r .
γ ( r , z q ) = r β ( z q ) ,
I ̂ Φ ( q ) = A ( q , z ( q β ( z q ) ) ) .
I ̂ Φ ( q ) I ̂ 0 ( q ) + i k γ ( r , z q ) E ( r + z q 2 ) E * ( r z q 2 ) exp [ i k q r ] d r .

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