Abstract

We demonstrate quantitative noninterferometric x-ray phase–amplitude measurement. We present results from two experimental geometries. The first geometry uses x rays diverging from a point source to produce high-resolution holograms of submicrometer-sized objects. The measured phase of the projected image agrees with the geometrically determined phase to within ±7%. The second geometry uses a direct imaging microscope setup that allows the formation of a magnified image with a zone-plate lens. Here a direct measure of the object phase is made and agrees with that of the magnified object to better than ±10%. In both cases the accuracy of the phase is limited by the pixel resolution.

© 2000 Optical Society of America

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  12. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
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  14. P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D 29, 133–146 (1996).
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  15. A. Momose, T. Takeda, Y. Itai, “Phase-contrast x-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum. 66, 1434–1436 (1995);A. Momose, T. Takeda, Y. Itai, K. Hirano, “Phase-contrast x-ray microtomography: application to human cancerous tissues,” in X-Ray Microscopy and Spectromicroscopy, J. Thieme, G. Schmahl, D. Rudolph, E. Umbach, eds. (Springer-Verlag, Berlin, 1998), pp. II-207–II-211.
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  16. P. Cloetens, M. Pateyron-Salomé, J.-Y. Buffière, G. Peix, J. Baruchel, F. Peyrin, M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
    [CrossRef]
  17. H. Rose, “Nonstandard imaging methods in electron microscopy,” Ultramicroscopy 2, 251–267 (1977).
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  20. 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] [PubMed]
  25. F. Roddier, C. Roddier, “Wave-front reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991).
    [CrossRef] [PubMed]
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    [CrossRef]
  27. P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
    [CrossRef]
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    [CrossRef]
  29. T. E. Gureyev, A. Roberts, K. A. Nugent, “Partially coherent fields, the transport of intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
    [CrossRef]
  30. K. Ichikawa, A. W. Lohmann, M. Takeda, “Phase retrieval based on the irradiance transport equation and the Fourier transform method: experiments,” Appl. Opt. 27, 3433–3436 (1988).
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  31. D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
    [CrossRef]
  32. 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);T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
    [CrossRef]
  33. A. Barty, K. A. Nugent, D. Paganin, A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
    [CrossRef]
  34. See, for example, M. Born, E. Wolf, Principles of Optics, corrected 6th ed. (Cambridge U. Press, Cambridge, UK, 1998), pp. 193–194.
  35. A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
    [CrossRef]
  36. S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
    [CrossRef] [PubMed]
  37. I. McNulty, A. Khounsary, Y. P. Feng, Y. Qian, J. Barraza, C. Benson, D. Shu, “A beamline for 1-4 keV microscopy and coherence experiments at the Advanced Photon Source,” Rev. Sci. Instrum. 67, 3372 (1996).
    [CrossRef]
  38. D. Gabor, “A new microscopic principle,” Nature (London) 161, 777–778 (1948).
    [CrossRef]
  39. C. Jacobsen, M. Howells, J. Kirz, S. Rothman, “X-ray holographic microscopy using photoresist,” J. Opt. Soc. Am. A 7, 1847–1861 (1990).
    [CrossRef]
  40. J. B. Tiller, A. Barty, D. Paganin, K. A. Nugent, “The holographic twin image problem: a deterministic phase solution,” Opt. Commun. (to be published).
  41. See, for example, M. Born, E. Wolf, Principles of Optics, corrected 6th ed. (Cambridge U. Press, Cambridge, UK, 1998), pp. 455–458.

2000

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

1999

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

1998

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

A. Barty, K. A. Nugent, D. Paganin, A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[CrossRef]

1997

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

P. Cloetens, M. Pateyron-Salomé, J.-Y. Buffière, G. Peix, J. Baruchel, F. Peyrin, M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

1996

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

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

K. A. Nugent, T. E. Gureyev, D. 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);T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[CrossRef]

I. McNulty, A. Khounsary, Y. P. Feng, Y. Qian, J. Barraza, C. Benson, D. Shu, “A beamline for 1-4 keV microscopy and coherence experiments at the Advanced Photon Source,” Rev. Sci. Instrum. 67, 3372 (1996).
[CrossRef]

H. N. Chapman, “Phase-retrieval x-ray microscopy by Wigner-distribution deconvolution,” Ultramicroscopy 66, 153–172 (1996).
[CrossRef]

1995

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

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995);T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
[CrossRef] [PubMed]

A. Momose, T. Takeda, Y. Itai, “Phase-contrast x-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum. 66, 1434–1436 (1995);A. Momose, T. Takeda, Y. Itai, K. Hirano, “Phase-contrast x-ray microtomography: application to human cancerous tissues,” in X-Ray Microscopy and Spectromicroscopy, J. Thieme, G. Schmahl, D. Rudolph, E. Umbach, eds. (Springer-Verlag, Berlin, 1998), pp. II-207–II-211.
[CrossRef]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, 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]

1992

W. Coene, G. Janssen, M. Op de Beeck, 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]

R. G. Lane, M. Tallon, “Wave-front reconstruction using a Shack–Hartmann sensor,” Appl. Opt. 31, 6902–6908 (1992).
[CrossRef] [PubMed]

V. Yu Ivanov, V. P. Sivokon, M. A. Vorontsov, “Phase retrieval from a set of intensity measurements,” J. Opt. Soc. Am. A 9, 1515–1524 (1992).
[CrossRef]

1991

1990

1988

1987

B. X. Yang, J. Kirz, T. K. Sham, “Oxygen K-edge extended x-ray-absorption fine-structure studies of molecules containing oxygen and carbon atoms,” Phys. Rev. A 36, 4298–4310 (1987);J. Kirz, C. Jacobsen, M. Howells, “Soft x-ray microscopes and their biological applications,” Q. Rev. Biophys. 28, 33–130 (1995).
[CrossRef] [PubMed]

J. E. Trebes, S. B. Brown, E. M. Campbell, D. L. Matthews, D. G. Nilson, G. F. Stone, D. A. Whelan, “Demonstration of x-ray holography with an x-ray laser,” Science 238, 517–519 (1987);J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. DaSilva, G. F. Stone, M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588–591 (1992);K. A. Nugent, J. E. Trebes, “Coherence measurement technique for short-wavelength light source,” Rev. Sci. Instrum. 63, 2146–2151 (1992).
[CrossRef] [PubMed]

1985

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

1983

1980

E. Forster, K. Goetz, P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser-fusion experiments,” Krist. Tech. 15, 937–945 (1980).
[CrossRef]

1979

E. M. Waddel, J. N. Chapman, “Linear imaging of strong phase objects using asymmetrical detectors in STEM,” Optik (Stuttgart) 54, 83–96 (1979).

1977

H. Rose, “Nonstandard imaging methods in electron microscopy,” Ultramicroscopy 2, 251–267 (1977).
[CrossRef] [PubMed]

1948

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

1896

W. C. Röntgen, “On a new kind of rays,” Nature 53, 274–276 (1896).
[CrossRef]

Bajt, S.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, 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. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Barraza, J.

I. McNulty, A. Khounsary, Y. P. Feng, Y. Qian, J. Barraza, C. Benson, D. Shu, “A beamline for 1-4 keV microscopy and coherence experiments at the Advanced Photon Source,” Rev. Sci. Instrum. 67, 3372 (1996).
[CrossRef]

Barrett, R.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, 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, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

A. Barty, K. A. Nugent, D. Paganin, A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[CrossRef]

J. B. Tiller, A. Barty, D. Paganin, K. A. Nugent, “The holographic twin image problem: a deterministic phase solution,” Opt. Commun. (to be published).

Baruchel, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, 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, M. Pateyron-Salomé, J.-Y. Buffière, G. Peix, J. Baruchel, F. Peyrin, M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

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

Benson, C.

I. McNulty, A. Khounsary, Y. P. Feng, Y. Qian, J. Barraza, C. Benson, D. Shu, “A beamline for 1-4 keV microscopy and coherence experiments at the Advanced Photon Source,” Rev. Sci. Instrum. 67, 3372 (1996).
[CrossRef]

Born, M.

See, for example, M. Born, E. Wolf, Principles of Optics, corrected 6th ed. (Cambridge U. Press, Cambridge, UK, 1998), pp. 455–458.

See, for example, M. Born, E. Wolf, Principles of Optics, corrected 6th ed. (Cambridge U. Press, Cambridge, UK, 1998), pp. 193–194.

Brown, S. B.

J. E. Trebes, S. B. Brown, E. M. Campbell, D. L. Matthews, D. G. Nilson, G. F. Stone, D. A. Whelan, “Demonstration of x-ray holography with an x-ray laser,” Science 238, 517–519 (1987);J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. DaSilva, G. F. Stone, M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588–591 (1992);K. A. Nugent, J. E. Trebes, “Coherence measurement technique for short-wavelength light source,” Rev. Sci. Instrum. 63, 2146–2151 (1992).
[CrossRef] [PubMed]

Buffière, J.-Y.

P. Cloetens, M. Pateyron-Salomé, J.-Y. Buffière, G. Peix, J. Baruchel, F. Peyrin, M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

Burge, R. E.

G. R. Morrison, A. R. Hare, R. E. Burge, “Transmission microscopy with soft x-rays,” in Proceedings of the Institute of Physics Electron Microscopy and Analysis Group Conference (Institute of Physics, Bristol, UK, 1987), pp. 333–336.

Campbell, E. M.

J. E. Trebes, S. B. Brown, E. M. Campbell, D. L. Matthews, D. G. Nilson, G. F. Stone, D. A. Whelan, “Demonstration of x-ray holography with an x-ray laser,” Science 238, 517–519 (1987);J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. DaSilva, G. F. Stone, M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588–591 (1992);K. A. Nugent, J. E. Trebes, “Coherence measurement technique for short-wavelength light source,” Rev. Sci. Instrum. 63, 2146–2151 (1992).
[CrossRef] [PubMed]

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).

Chapman, H. N.

H. N. Chapman, “Phase-retrieval x-ray microscopy by Wigner-distribution deconvolution,” Ultramicroscopy 66, 153–172 (1996).
[CrossRef]

Chapman, J. N.

E. M. Waddel, J. N. Chapman, “Linear imaging of strong phase objects using asymmetrical detectors in STEM,” Optik (Stuttgart) 54, 83–96 (1979).

Cloetens, P.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, 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, M. Pateyron-Salomé, J.-Y. Buffière, G. Peix, J. Baruchel, F. Peyrin, M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

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

Coene, W.

W. Coene, G. Janssen, M. Op de Beeck, 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]

Cookson, D.

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

David, C.

G. Schmahl, P. Guttmann, G. Schneider, B. Niemann, C. David, T. Wilhein, J. Thieme, D. Rudolph, “Phase contrast studies of hydrated specimens with the x-ray microscope at BESSY,” in X-Ray Microscopy IV, A. Erko, V. Aristov, eds. (Bogorodski Pechatnik, Chernogolovka, Moscow Region, 1994), pp. 196–206.

Davis, T. J.

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995);T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
[CrossRef] [PubMed]

Feng, Y. P.

I. McNulty, A. Khounsary, Y. P. Feng, Y. Qian, J. Barraza, C. Benson, D. Shu, “A beamline for 1-4 keV microscopy and coherence experiments at the Advanced Photon Source,” Rev. Sci. Instrum. 67, 3372 (1996).
[CrossRef]

Forster, E.

E. Forster, K. Goetz, P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser-fusion experiments,” Krist. Tech. 15, 937–945 (1980).
[CrossRef]

Gabor, D.

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

Gao, D.

A. Pogany, D. Gao, 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, A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
[CrossRef]

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A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
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T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995);T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
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T. Wilson, A. R. Carlini, C. J. R. Sheppard, “Phase contrast microscopy by nearly full illumination,” Optik (Stuttgart) 70, 166–169 (1985).

Wolf, E.

See, for example, M. Born, E. Wolf, Principles of Optics, corrected 6th ed. (Cambridge U. Press, Cambridge, UK, 1998), pp. 193–194.

See, for example, M. Born, E. Wolf, Principles of Optics, corrected 6th ed. (Cambridge U. Press, Cambridge, UK, 1998), pp. 455–458.

Yang, B. X.

B. X. Yang, J. Kirz, T. K. Sham, “Oxygen K-edge extended x-ray-absorption fine-structure studies of molecules containing oxygen and carbon atoms,” Phys. Rev. A 36, 4298–4310 (1987);J. Kirz, C. Jacobsen, M. Howells, “Soft x-ray microscopes and their biological applications,” Q. Rev. Biophys. 28, 33–130 (1995).
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P. Cloetens, M. Pateyron-Salomé, J.-Y. Buffière, G. Peix, J. Baruchel, F. Peyrin, M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
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[CrossRef]

Krist. Tech.

E. Forster, K. Goetz, P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser-fusion experiments,” Krist. Tech. 15, 937–945 (1980).
[CrossRef]

Nature

W. C. Röntgen, “On a new kind of rays,” Nature 53, 274–276 (1896).
[CrossRef]

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

Nature (London)

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

Opt. Lett.

Optik (Stuttgart)

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

E. M. Waddel, J. N. Chapman, “Linear imaging of strong phase objects using asymmetrical detectors in STEM,” Optik (Stuttgart) 54, 83–96 (1979).

Phys. Rev. A

B. X. Yang, J. Kirz, T. K. Sham, “Oxygen K-edge extended x-ray-absorption fine-structure studies of molecules containing oxygen and carbon atoms,” Phys. Rev. A 36, 4298–4310 (1987);J. Kirz, C. Jacobsen, M. Howells, “Soft x-ray microscopes and their biological applications,” Q. Rev. Biophys. 28, 33–130 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett.

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995);T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
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A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
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A. Momose, T. Takeda, Y. Itai, “Phase-contrast x-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum. 66, 1434–1436 (1995);A. Momose, T. Takeda, Y. Itai, K. Hirano, “Phase-contrast x-ray microtomography: application to human cancerous tissues,” in X-Ray Microscopy and Spectromicroscopy, J. Thieme, G. Schmahl, D. Rudolph, E. Umbach, eds. (Springer-Verlag, Berlin, 1998), pp. II-207–II-211.
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H. N. Chapman, “Phase-retrieval x-ray microscopy by Wigner-distribution deconvolution,” Ultramicroscopy 66, 153–172 (1996).
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Other

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See, for example, Advanced Photon Source, http://aps.anl.gov ; European Synchrotron Radiation Facility, http://www.esrf.fr ; Super Photon Ring, http://www.spring8.or.jp .

G. Schmahl, D. Rudolf, P. Guttmann, “Phase contrast x-ray microscopy experiments at the BESSY storage ring,” in X-Ray Microscopy II, D. Sayre, M. Howells, J. Kirz, H. Rarback, eds., Vol. 56 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1988), pp. 228–232; G. Schmahl, D. Rudolph, G. Schneider, P. Guttman, B. Niemann, “Phase contrast x-ray microscopy studies,” Optik (Stuttgart) , 97, 181–182 (1994).

G. Schmahl, P. Guttmann, G. Schneider, B. Niemann, C. David, T. Wilhein, J. Thieme, D. Rudolph, “Phase contrast studies of hydrated specimens with the x-ray microscope at BESSY,” in X-Ray Microscopy IV, A. Erko, V. Aristov, eds. (Bogorodski Pechatnik, Chernogolovka, Moscow Region, 1994), pp. 196–206.

See, for example, M. Born, E. Wolf, Principles of Optics, corrected 6th ed. (Cambridge U. Press, Cambridge, UK, 1998), pp. 193–194.

P. Schiske, “Image processing using additional statistical information about the object,” in Image Processing and Computer-Aided Design in Electron Optics, P. W. Hawkes, ed. (Academic, New York, 1973), p. 82.

J. B. Tiller, A. Barty, D. Paganin, K. A. Nugent, “The holographic twin image problem: a deterministic phase solution,” Opt. Commun. (to be published).

See, for example, M. Born, E. Wolf, Principles of Optics, corrected 6th ed. (Cambridge U. Press, Cambridge, UK, 1998), pp. 455–458.

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Figures (13)

Fig. 1
Fig. 1

Schematic of the point-source-projection, holographic geometry, noninterferometric phase-imaging experiment. The zone plate forms a first-order focus approximately 6 mm downstream. A 5-μm order-sorting aperture placed just before the focus masks other focal orders. The focus acts as a real point source and illuminates the sample placed less than 2 mm away. The projected image is detected by the CCD 530 mm further downstream. Images in closely spaced planes are taken by translating the CCD along the beam axis.

Fig. 2
Fig. 2

Holographic geometry images of the aligned flat-field (sample-free) intensity distribution: (a) under-focused and (b) (magnified) over-focused intensity distribution, (c) intensity difference, (d) retrieved phase, and (e) rendered surface plot of the retrieved phase.

Fig. 3
Fig. 3

Profile through the spherical phase distribution (points), with circular fit (solid curve).

Fig. 4
Fig. 4

Holographic geometry images of the scaled and aligned flat-field (sample-free) intensity distribution: (a) (scaled) under-focused and (b) over-focused intensity distribution, (c) intensity difference, (d) retrieved phase, and (e) rendered surface plot of the retrieved phase.

Fig. 5
Fig. 5

Holographic geometry images of the scaled and misaligned flat-field (sample-free) intensity distribution: (a) (vertically shifted) under-focused and (b) over-focused intensity distribution, (c) intensity difference, (d) retrieved phase, and (e) rendered surface plot of the retrieved phase.

Fig. 6
Fig. 6

Holographic geometry images of aluminum microspheres, scaled and aligned: (a) in-focus with microspheres and (b) in-focus flat-field intensity distribution, (c) intensity distribution flat-field subtracted, (d) intensity difference between defocused images, (e) retrieved phase, and (f) rendered surface plot of the retrieved phase.

Fig. 7
Fig. 7

(a) Horizontal and (b) vertical profiles through the phase images of the aluminum balls. The solid line and curve represent a linear fit to the phase tilt and a circular fit to the ball and indicate highly spherical phase objects. The phase effects of the fold in the substrate can be seen in the horizontal profile.

Fig. 8
Fig. 8

Schematic of the imaging geometry, noninterferometric phase-retrieval experiment. A 20-μm aperture (defining the beam dimensions) and the sample are placed upstream of the zone plate and off center to its axis. An in-focus image of the sample that is clear of the zeroth-order beam is formed at the CCD 980 mm further downstream. Images in closely spaced planes are achieved by translating the zone plate along the beam axis.

Fig. 9
Fig. 9

Flat-field intensity distribution in imaging geometry. (a) The in-focus first-order image is seen to be well clear of the zeroth-order direct beam. (b) A logarithmic scale image is displayed that shows the central zeroth-order beam and beam stop, the in-focus +1 and out-of-focus +2 orders (upper left), and the out-of-focus -1 and -2 orders (lower right).

Fig. 10
Fig. 10

Imaging geometry phase retrieval. (a) Under-focus, (b) in-focus, and (c) over-focus intensity images for the aluminum microspheres sample. (d) The intensity difference is used to retrieve (e) the phase and (f) rendered view. (g) A high-pass filter highlights the spheres while removing the low-frequency background signal.

Fig. 11
Fig. 11

Vertical profile through the phase image of the middle aluminum sphere of the three. The solid line and curve represent a linear fit to the phase tilt and a circular fit to the sphere and indicate a highly spherical phase object.

Fig. 12
Fig. 12

(a) Under-focus, (b) in-focus, and (c) over-focus intensity images for a length of polycarbonate optical fiber that has been folded back on itself. (d) The intensity difference is used to retrieve (e) the phase. (f) The rendered view gives the best visual impression of the large phase excursion from the projection through the fiber overlaying itself.

Fig. 13
Fig. 13

(a) Under-focus, (b) in-focus, and (c) over-focus intensity images for a spider silk sample. (d) The intensity difference is used to retrieve (e) the phase and (f) the rendered view.

Equations (2)

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[I(r)ϕ(r)]=0,
kI(r)z=-[I(r)ϕ(r)],

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