Abstract

We discuss contrast formation in a propagating x-ray beam. We consider the validity conditions for linear relations based on the transport-of-intensity equation (TIE) and on contrast transfer functions (CTFs). From a single diffracted image, we recover the thickness of a homogeneous object which has substantial absorption and a phase-shift of -0.37radian.

© 2004 Optical Society of America

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  1. G. Schmahl, D. Rudolph, and P. Guttmann, “Phase contrast x-ray microscopy-experiments at the BESSY storage ring,” in X-ray Microscopy II, D. Sayre, M. Howells, J. Kirz, and H. Rarback, eds.,  Vol 56 in Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1988), pp. 228–232.
  2. D. Sayre and H. N. Chapman, “X-ray microscopy,” Acta. Cryst. A51, 237–252 (1995).
  3. 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]
  4. 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]
  5. U. Bonse and M. Hart, “An x-ray interferometer,” Appl. Phys. Lett. 6, 155–157 (1965).
    [Crossref]
  6. A. Momose, T. Takeda, and Y. Itai, “Phase-contrast x-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum. 66, 1434–1436 (1995).
    [Crossref]
  7. Y. Kohmura, H. Takano, Y. Suzuki, and T. Ishikawa“Shearing x-ray interferometer with an x-ray prism and its improvement,” in Proc. 7th Intern. Conf. on X-ray Microscopy, D. Joyeux and F. Polack, eds., J.de Physique IV, Vol 104J. Susini (EDP Sciences, Les Ulis, 2003), pp. 571–574.
  8. T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, M. Altissimo, J. Susini, B. Fayard, U. Neuhäusler, S. Cabrini, and F. Polack, “Differential interference contrast x-ray microscopy with twin zone plates at ESRF beamline ID21,” in Proc. 7th Intern. Conf. on X-ray microscopy, ibid.
  9. E. M. Di Fabrizio, D. Cojoc, S. Cabrini, B. Kaulich, J. Susini, P. Facci, and T. Wilhein,“Diffractive optical elements for differential interference contrast x-ray microscopy,” Opt. Express 11, 2278–2288 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-19-2278.
    [Crossref] [PubMed]
  10. E. Forster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15, 937–945 (1980).
    [Crossref]
  11. J. R. Palmer and G. R. Morrison, “Differential phase contrast imaging in the scanning transmission x-ray microscope,” in Short Wavelength Coherent Radiation, P. H. Bucksbaum and N. M. Ceglio, eds., Vol. 11 of OSA Proceedings Series (Optical Society of America, Washington, D. C., 1991), pp. 141–145.
  12. C. Jacobsen, M. Howells, J. Kirz, and S. Rothman, “X-ray holographic microscopy using photoresists,” J. Opt. Soc. Am. A 7, 1847–1861 (1990).
    [Crossref]
  13. L. J. Allen, W. McBride, and M. P. Oxley, “Exit wave reconstruction using soft x-rays,” Opt. Commun. 233, 77–82 (2004).
    [Crossref]
  14. M. R. Teague, “Deterministic phase retrieval: Greens function solution,” J. Opt. Soc. Am. A 73, 1434–41 (1983).
    [Crossref]
  15. D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
    [Crossref]
  16. T. E. Gureyev, A. Pogany, D. M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
    [Crossref]
  17. X. Wu and H. Li, “A general theoretical formalism for x-ray phase contrast imaging,” J. X-ray Sci. Tech. 11, 33–42 (2003).
  18. J.-P. Guigay, R. H. Wade, and C. Delpha, “Optical diffraction of Lorentz microscope images,” in Proceedings of the 25th meeting of the Electron Microscopy and Analysis Group, W. C. Nixon, ed. (The Institute of Physics, London, 1971), pp. 238–239.
  19. J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns,” Optik 49, 121–125 (1977).
  20. 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 (2001).
    [Crossref]
  21. P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, 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]
  22. M. H. Maleki and A. J. Devaney, “Noniterative reconstruction of complex-valued objects from two intensity measurements,” Opt. Eng. 33, 3243–3253 (1994).
    [Crossref]
  23. A. N. Tikhonov and V.Y. Arsenin, “Solutions of Ill-posed Problems” (V. H. Winston, Washington D.C., 1977).
  24. E. C. Harvey and P. T. Rumsby, “Fabrication techniques and their application to produce novel micromachined structures and devices using excimer laser projection,” in Micromachining and Microfabrication Process Technology III, S. Chang and S. W. Pang, eds., Proc. SPIE3223, 26–33 (1997).
  25. L. D. Turner, K. P. Weber, D. Paganin, and R. E. Scholten, “Off-resonant defocus-contrast imaging of cold atoms,” Opt. Lett. 29, 232–234 (2004).
    [Crossref] [PubMed]
  26. L. D. Turner, K. F. E. M. Domen, W. Rooijakkers, and R. E. Scholten, School of Physics, University of Melbourne 3010, Australia are preparing a manuscript to be called “Holographic imaging of cold atoms”.
  27. M. Centurion, Y. Pu, Z. Liu, D. Psaltis, and T.W. Hänsch, “Holographic recording of laser-induced plasma,” Opt. Lett. 29, 772–774 (2004).
    [Crossref] [PubMed]

2004 (4)

L. J. Allen, W. McBride, and M. P. Oxley, “Exit wave reconstruction using soft x-rays,” Opt. Commun. 233, 77–82 (2004).
[Crossref]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[Crossref]

L. D. Turner, K. P. Weber, D. Paganin, and R. E. Scholten, “Off-resonant defocus-contrast imaging of cold atoms,” Opt. Lett. 29, 232–234 (2004).
[Crossref] [PubMed]

M. Centurion, Y. Pu, Z. Liu, D. Psaltis, and T.W. Hänsch, “Holographic recording of laser-induced plasma,” Opt. Lett. 29, 772–774 (2004).
[Crossref] [PubMed]

2003 (2)

2001 (1)

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 (2001).
[Crossref]

1999 (1)

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, 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]

1998 (1)

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

1997 (1)

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 (1)

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]

1995 (2)

A. Momose, T. Takeda, and Y. Itai, “Phase-contrast x-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum. 66, 1434–1436 (1995).
[Crossref]

D. Sayre and H. N. Chapman, “X-ray microscopy,” Acta. Cryst. A51, 237–252 (1995).

1994 (1)

M. H. Maleki and A. J. Devaney, “Noniterative reconstruction of complex-valued objects from two intensity measurements,” Opt. Eng. 33, 3243–3253 (1994).
[Crossref]

1990 (1)

1988 (1)

G. Schmahl, D. Rudolph, and P. Guttmann, “Phase contrast x-ray microscopy-experiments at the BESSY storage ring,” in X-ray Microscopy II, D. Sayre, M. Howells, J. Kirz, and H. Rarback, eds.,  Vol 56 in Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1988), pp. 228–232.

1983 (1)

M. R. Teague, “Deterministic phase retrieval: Greens function solution,” J. Opt. Soc. Am. A 73, 1434–41 (1983).
[Crossref]

1980 (1)

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

1977 (1)

J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns,” Optik 49, 121–125 (1977).

1965 (1)

U. Bonse and M. Hart, “An x-ray interferometer,” Appl. Phys. Lett. 6, 155–157 (1965).
[Crossref]

Allen, L. J.

L. J. Allen, W. McBride, and M. P. Oxley, “Exit wave reconstruction using soft x-rays,” Opt. Commun. 233, 77–82 (2004).
[Crossref]

Altissimo, M.

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, M. Altissimo, J. Susini, B. Fayard, U. Neuhäusler, S. Cabrini, and F. Polack, “Differential interference contrast x-ray microscopy with twin zone plates at ESRF beamline ID21,” in Proc. 7th Intern. Conf. on X-ray microscopy, ibid.

Arsenin, V.Y.

A. N. Tikhonov and V.Y. Arsenin, “Solutions of Ill-posed Problems” (V. H. Winston, Washington D.C., 1977).

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]

Baruchel, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, 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]

Bonse, U.

U. Bonse and M. Hart, “An x-ray interferometer,” Appl. Phys. Lett. 6, 155–157 (1965).
[Crossref]

Cabrini, S.

E. M. Di Fabrizio, D. Cojoc, S. Cabrini, B. Kaulich, J. Susini, P. Facci, and T. Wilhein,“Diffractive optical elements for differential interference contrast x-ray microscopy,” Opt. Express 11, 2278–2288 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-19-2278.
[Crossref] [PubMed]

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, M. Altissimo, J. Susini, B. Fayard, U. Neuhäusler, S. Cabrini, and F. Polack, “Differential interference contrast x-ray microscopy with twin zone plates at ESRF beamline ID21,” in Proc. 7th Intern. Conf. on X-ray microscopy, ibid.

Centurion, M.

Chapman, H. N.

D. Sayre and H. N. Chapman, “X-ray microscopy,” Acta. Cryst. A51, 237–252 (1995).

Cloetens, P.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, 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]

Cojoc, D.

Cookson, D. F.

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]

Delpha, C.

J.-P. Guigay, R. H. Wade, and C. Delpha, “Optical diffraction of Lorentz microscope images,” in Proceedings of the 25th meeting of the Electron Microscopy and Analysis Group, W. C. Nixon, ed. (The Institute of Physics, London, 1971), pp. 238–239.

Devaney, A. J.

M. H. Maleki and A. J. Devaney, “Noniterative reconstruction of complex-valued objects from two intensity measurements,” Opt. Eng. 33, 3243–3253 (1994).
[Crossref]

Di Fabrizio, E.

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, M. Altissimo, J. Susini, B. Fayard, U. Neuhäusler, S. Cabrini, and F. Polack, “Differential interference contrast x-ray microscopy with twin zone plates at ESRF beamline ID21,” in Proc. 7th Intern. Conf. on X-ray microscopy, ibid.

Di Fabrizio, E. M.

Domen, K. F. E. M.

L. D. Turner, K. F. E. M. Domen, W. Rooijakkers, and R. E. Scholten, School of Physics, University of Melbourne 3010, Australia are preparing a manuscript to be called “Holographic imaging of cold atoms”.

Facci, P.

Fayard, B.

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, M. Altissimo, J. Susini, B. Fayard, U. Neuhäusler, S. Cabrini, and F. Polack, “Differential interference contrast x-ray microscopy with twin zone plates at ESRF beamline ID21,” in Proc. 7th Intern. Conf. on X-ray microscopy, ibid.

Forster, E.

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

Gao, D.

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]

Goetz, K.

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

Guigay, J.-P.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, 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.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns,” Optik 49, 121–125 (1977).

J.-P. Guigay, R. H. Wade, and C. Delpha, “Optical diffraction of Lorentz microscope images,” in Proceedings of the 25th meeting of the Electron Microscopy and Analysis Group, W. C. Nixon, ed. (The Institute of Physics, London, 1971), pp. 238–239.

Gureyev, T. E.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[Crossref]

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 (2001).
[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]

Guttmann, P.

G. Schmahl, D. Rudolph, and P. Guttmann, “Phase contrast x-ray microscopy-experiments at the BESSY storage ring,” in X-ray Microscopy II, D. Sayre, M. Howells, J. Kirz, and H. Rarback, eds.,  Vol 56 in Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1988), pp. 228–232.

Hänsch, T.W.

Hart, M.

U. Bonse and M. Hart, “An x-ray interferometer,” Appl. Phys. Lett. 6, 155–157 (1965).
[Crossref]

Harvey, E. C.

E. C. Harvey and P. T. Rumsby, “Fabrication techniques and their application to produce novel micromachined structures and devices using excimer laser projection,” in Micromachining and Microfabrication Process Technology III, S. Chang and S. W. Pang, eds., Proc. SPIE3223, 26–33 (1997).

Howells, M.

Ishikawa, T.

Y. Kohmura, H. Takano, Y. Suzuki, and T. Ishikawa“Shearing x-ray interferometer with an x-ray prism and its improvement,” in Proc. 7th Intern. Conf. on X-ray Microscopy, D. Joyeux and F. Polack, eds., J.de Physique IV, Vol 104J. Susini (EDP Sciences, Les Ulis, 2003), pp. 571–574.

Itai, Y.

A. Momose, T. Takeda, and Y. Itai, “Phase-contrast x-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum. 66, 1434–1436 (1995).
[Crossref]

Jacobsen, C.

Kaulich, B.

E. M. Di Fabrizio, D. Cojoc, S. Cabrini, B. Kaulich, J. Susini, P. Facci, and T. Wilhein,“Diffractive optical elements for differential interference contrast x-ray microscopy,” Opt. Express 11, 2278–2288 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-19-2278.
[Crossref] [PubMed]

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, M. Altissimo, J. Susini, B. Fayard, U. Neuhäusler, S. Cabrini, and F. Polack, “Differential interference contrast x-ray microscopy with twin zone plates at ESRF beamline ID21,” in Proc. 7th Intern. Conf. on X-ray microscopy, ibid.

Kirz, J.

Kohmura, Y.

Y. Kohmura, H. Takano, Y. Suzuki, and T. Ishikawa“Shearing x-ray interferometer with an x-ray prism and its improvement,” in Proc. 7th Intern. Conf. on X-ray Microscopy, D. Joyeux and F. Polack, eds., J.de Physique IV, Vol 104J. Susini (EDP Sciences, Les Ulis, 2003), pp. 571–574.

Li, H.

X. Wu and H. Li, “A general theoretical formalism for x-ray phase contrast imaging,” J. X-ray Sci. Tech. 11, 33–42 (2003).

Liu, Z.

Ludwig, W.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, 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]

Maleki, M. H.

M. H. Maleki and A. J. Devaney, “Noniterative reconstruction of complex-valued objects from two intensity measurements,” Opt. Eng. 33, 3243–3253 (1994).
[Crossref]

Mayo, S. C.

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 (2001).
[Crossref]

McBride, W.

L. J. Allen, W. McBride, and M. P. Oxley, “Exit wave reconstruction using soft x-rays,” Opt. Commun. 233, 77–82 (2004).
[Crossref]

Miller, P. R.

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 (2001).
[Crossref]

Momose, A.

A. Momose, T. Takeda, and Y. Itai, “Phase-contrast x-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum. 66, 1434–1436 (1995).
[Crossref]

Morrison, G. R.

J. R. Palmer and G. R. Morrison, “Differential phase contrast imaging in the scanning transmission x-ray microscope,” in Short Wavelength Coherent Radiation, P. H. Bucksbaum and N. M. Ceglio, eds., Vol. 11 of OSA Proceedings Series (Optical Society of America, Washington, D. C., 1991), pp. 141–145.

Neuhäusler, U.

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, M. Altissimo, J. Susini, B. Fayard, U. Neuhäusler, S. Cabrini, and F. Polack, “Differential interference contrast x-ray microscopy with twin zone plates at ESRF beamline ID21,” in Proc. 7th Intern. Conf. on X-ray microscopy, ibid.

Nugent, K. A.

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

Oxley, M. P.

L. J. Allen, W. McBride, and M. P. Oxley, “Exit wave reconstruction using soft x-rays,” Opt. Commun. 233, 77–82 (2004).
[Crossref]

Paganin, D.

L. D. Turner, K. P. Weber, D. Paganin, and R. E. Scholten, “Off-resonant defocus-contrast imaging of cold atoms,” Opt. Lett. 29, 232–234 (2004).
[Crossref] [PubMed]

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 (2001).
[Crossref]

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

Paganin, D. M.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[Crossref]

Palmer, J. R.

J. R. Palmer and G. R. Morrison, “Differential phase contrast imaging in the scanning transmission x-ray microscope,” in Short Wavelength Coherent Radiation, P. H. Bucksbaum and N. M. Ceglio, eds., Vol. 11 of OSA Proceedings Series (Optical Society of America, Washington, D. C., 1991), pp. 141–145.

Pogany, A.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[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]

Polack, F.

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, M. Altissimo, J. Susini, B. Fayard, U. Neuhäusler, S. Cabrini, and F. Polack, “Differential interference contrast x-ray microscopy with twin zone plates at ESRF beamline ID21,” in Proc. 7th Intern. Conf. on X-ray microscopy, ibid.

Psaltis, D.

Pu, Y.

Romanato, F.

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, M. Altissimo, J. Susini, B. Fayard, U. Neuhäusler, S. Cabrini, and F. Polack, “Differential interference contrast x-ray microscopy with twin zone plates at ESRF beamline ID21,” in Proc. 7th Intern. Conf. on X-ray microscopy, ibid.

Rooijakkers, W.

L. D. Turner, K. F. E. M. Domen, W. Rooijakkers, and R. E. Scholten, School of Physics, University of Melbourne 3010, Australia are preparing a manuscript to be called “Holographic imaging of cold atoms”.

Rothman, S.

Rudolph, D.

G. Schmahl, D. Rudolph, and P. Guttmann, “Phase contrast x-ray microscopy-experiments at the BESSY storage ring,” in X-ray Microscopy II, D. Sayre, M. Howells, J. Kirz, and H. Rarback, eds.,  Vol 56 in Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1988), pp. 228–232.

Rumsby, P. T.

E. C. Harvey and P. T. Rumsby, “Fabrication techniques and their application to produce novel micromachined structures and devices using excimer laser projection,” in Micromachining and Microfabrication Process Technology III, S. Chang and S. W. Pang, eds., Proc. SPIE3223, 26–33 (1997).

Sayre, D.

D. Sayre and H. N. Chapman, “X-ray microscopy,” Acta. Cryst. A51, 237–252 (1995).

Schlenker, M.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, 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]

Schmahl, G.

G. Schmahl, D. Rudolph, and P. Guttmann, “Phase contrast x-ray microscopy-experiments at the BESSY storage ring,” in X-ray Microscopy II, D. Sayre, M. Howells, J. Kirz, and H. Rarback, eds.,  Vol 56 in Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1988), pp. 228–232.

Scholten, R. E.

L. D. Turner, K. P. Weber, D. Paganin, and R. E. Scholten, “Off-resonant defocus-contrast imaging of cold atoms,” Opt. Lett. 29, 232–234 (2004).
[Crossref] [PubMed]

L. D. Turner, K. F. E. M. Domen, W. Rooijakkers, and R. E. Scholten, School of Physics, University of Melbourne 3010, Australia are preparing a manuscript to be called “Holographic imaging of cold atoms”.

Susini, J.

E. M. Di Fabrizio, D. Cojoc, S. Cabrini, B. Kaulich, J. Susini, P. Facci, and T. Wilhein,“Diffractive optical elements for differential interference contrast x-ray microscopy,” Opt. Express 11, 2278–2288 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-19-2278.
[Crossref] [PubMed]

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, M. Altissimo, J. Susini, B. Fayard, U. Neuhäusler, S. Cabrini, and F. Polack, “Differential interference contrast x-ray microscopy with twin zone plates at ESRF beamline ID21,” in Proc. 7th Intern. Conf. on X-ray microscopy, ibid.

Suzuki, Y.

Y. Kohmura, H. Takano, Y. Suzuki, and T. Ishikawa“Shearing x-ray interferometer with an x-ray prism and its improvement,” in Proc. 7th Intern. Conf. on X-ray Microscopy, D. Joyeux and F. Polack, eds., J.de Physique IV, Vol 104J. Susini (EDP Sciences, Les Ulis, 2003), pp. 571–574.

Takano, H.

Y. Kohmura, H. Takano, Y. Suzuki, and T. Ishikawa“Shearing x-ray interferometer with an x-ray prism and its improvement,” in Proc. 7th Intern. Conf. on X-ray Microscopy, D. Joyeux and F. Polack, eds., J.de Physique IV, Vol 104J. Susini (EDP Sciences, Les Ulis, 2003), pp. 571–574.

Takeda, T.

A. Momose, T. Takeda, and Y. Itai, “Phase-contrast x-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum. 66, 1434–1436 (1995).
[Crossref]

Teague, M. R.

M. R. Teague, “Deterministic phase retrieval: Greens function solution,” J. Opt. Soc. Am. A 73, 1434–41 (1983).
[Crossref]

Tikhonov, A. N.

A. N. Tikhonov and V.Y. Arsenin, “Solutions of Ill-posed Problems” (V. H. Winston, Washington D.C., 1977).

Turner, L. D.

L. D. Turner, K. P. Weber, D. Paganin, and R. E. Scholten, “Off-resonant defocus-contrast imaging of cold atoms,” Opt. Lett. 29, 232–234 (2004).
[Crossref] [PubMed]

L. D. Turner, K. F. E. M. Domen, W. Rooijakkers, and R. E. Scholten, School of Physics, University of Melbourne 3010, Australia are preparing a manuscript to be called “Holographic imaging of cold atoms”.

Van Dyck, D.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, 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]

Van Landuyt, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, 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]

Wade, R. H.

J.-P. Guigay, R. H. Wade, and C. Delpha, “Optical diffraction of Lorentz microscope images,” in Proceedings of the 25th meeting of the Electron Microscopy and Analysis Group, W. C. Nixon, ed. (The Institute of Physics, London, 1971), pp. 238–239.

Weber, K. P.

Wilhein, T.

E. M. Di Fabrizio, D. Cojoc, S. Cabrini, B. Kaulich, J. Susini, P. Facci, and T. Wilhein,“Diffractive optical elements for differential interference contrast x-ray microscopy,” Opt. Express 11, 2278–2288 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-19-2278.
[Crossref] [PubMed]

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, M. Altissimo, J. Susini, B. Fayard, U. Neuhäusler, S. Cabrini, and F. Polack, “Differential interference contrast x-ray microscopy with twin zone plates at ESRF beamline ID21,” in Proc. 7th Intern. Conf. on X-ray microscopy, ibid.

Wilkins, S.W.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[Crossref]

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 (2001).
[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]

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X. Wu and H. Li, “A general theoretical formalism for x-ray phase contrast imaging,” J. X-ray Sci. Tech. 11, 33–42 (2003).

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P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, 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).
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G. Schmahl, D. Rudolph, and P. Guttmann, “Phase contrast x-ray microscopy-experiments at the BESSY storage ring,” in X-ray Microscopy II, D. Sayre, M. Howells, J. Kirz, and H. Rarback, eds.,  Vol 56 in Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1988), pp. 228–232.

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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 (2001).
[Crossref]

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C. Jacobsen, M. Howells, J. Kirz, and S. Rothman, “X-ray holographic microscopy using photoresists,” J. Opt. Soc. Am. A 7, 1847–1861 (1990).
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M. R. Teague, “Deterministic phase retrieval: Greens function solution,” J. Opt. Soc. Am. A 73, 1434–41 (1983).
[Crossref]

J. X-ray Sci. Tech. (1)

X. Wu and H. Li, “A general theoretical formalism for x-ray phase contrast imaging,” J. X-ray Sci. Tech. 11, 33–42 (2003).

Krist. Tech. (1)

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

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L. J. Allen, W. McBride, and M. P. Oxley, “Exit wave reconstruction using soft x-rays,” Opt. Commun. 233, 77–82 (2004).
[Crossref]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[Crossref]

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J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns,” Optik 49, 121–125 (1977).

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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).
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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. Momose, T. Takeda, and Y. Itai, “Phase-contrast x-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum. 66, 1434–1436 (1995).
[Crossref]

Other (7)

Y. Kohmura, H. Takano, Y. Suzuki, and T. Ishikawa“Shearing x-ray interferometer with an x-ray prism and its improvement,” in Proc. 7th Intern. Conf. on X-ray Microscopy, D. Joyeux and F. Polack, eds., J.de Physique IV, Vol 104J. Susini (EDP Sciences, Les Ulis, 2003), pp. 571–574.

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, M. Altissimo, J. Susini, B. Fayard, U. Neuhäusler, S. Cabrini, and F. Polack, “Differential interference contrast x-ray microscopy with twin zone plates at ESRF beamline ID21,” in Proc. 7th Intern. Conf. on X-ray microscopy, ibid.

J.-P. Guigay, R. H. Wade, and C. Delpha, “Optical diffraction of Lorentz microscope images,” in Proceedings of the 25th meeting of the Electron Microscopy and Analysis Group, W. C. Nixon, ed. (The Institute of Physics, London, 1971), pp. 238–239.

J. R. Palmer and G. R. Morrison, “Differential phase contrast imaging in the scanning transmission x-ray microscope,” in Short Wavelength Coherent Radiation, P. H. Bucksbaum and N. M. Ceglio, eds., Vol. 11 of OSA Proceedings Series (Optical Society of America, Washington, D. C., 1991), pp. 141–145.

L. D. Turner, K. F. E. M. Domen, W. Rooijakkers, and R. E. Scholten, School of Physics, University of Melbourne 3010, Australia are preparing a manuscript to be called “Holographic imaging of cold atoms”.

A. N. Tikhonov and V.Y. Arsenin, “Solutions of Ill-posed Problems” (V. H. Winston, Washington D.C., 1977).

E. C. Harvey and P. T. Rumsby, “Fabrication techniques and their application to produce novel micromachined structures and devices using excimer laser projection,” in Micromachining and Microfabrication Process Technology III, S. Chang and S. W. Pang, eds., Proc. SPIE3223, 26–33 (1997).

Supplementary Material (1)

» Media 1: MPG (1894 KB)     

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Figures (2)

Fig. 1.
Fig. 1.

Inverse of the contrast transfer functions for the TIE (blue dotted line) and CTF (red line) forms calculated for an infinite grid with spatial feature size of 1.9µm and the experimental conditions. The green dashed line is at the experimental distance. The embedded movie shows the inverse CTFs as shown here with a slider indicating the z position corresponding to diffraction distance. Also shown in the movie is a plot of an input amplitude (green) and the amplitude retrieved using either the CTF (red) or TIE (blue) methods for the indicated propagation distance. The CTF method correctly accounts for the contrast reversals that arise on propagation. The TIE method should only be applied for z closer than the first contrast reversal; it may retrieve inverted amplitudes if applied at greater z. [Media 1]

Fig. 2.
Fig. 2.

(a) CTF-retrieved thickness map for the square with grid lines.(b) Column-average of retrieved thickness for the grid pattern in the region shown in (a) for the TIE solution (blue) and the CTF solution (red). The AFM result (green) shows excellent agreement. AFM measurements also confirm the presence of grid lines outside the square. These are not a retrieval artefact, unlike the circular fringes around the contaminant at centre right. The contaminating material presumably violates the assumption of an homogeneous object.

Equations (20)

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𝓕 [ I ( r , z ) ] = + f * ( r + λ z u 2 ) f ( r λ z u 2 ) exp ( 2 π i r · u ) d r .
f ( r + λ z u 2 ) = f ( r ) + 1 2 λ z u · f ( r ) ,
· ( I ( r , z ) ϕ ( r , z ) ) = 2 π λ z I ( r , z ) ,
j = 2 1 j ! ( 1 2 λ z u · ) j f ( r ) 1 .
f ( r ) = f 0 exp ( μ ( r ) + i ϕ ( r ) ) ,
f ( r ) = f 0 ( 1 μ ( r ) + i ϕ ( r ) ) .
𝓕 [ I ( r , z ) ] = I 0 ( δ ( u ) 2 cos ( π λ z u 2 ) 𝓕 [ μ ( r ) ] + 2 sin ( π λ z u 2 ) 𝓕 [ ϕ ( r ) ] )
ϕ ( r + λ z u 2 ) ϕ ( r λ z u 2 ) 1 .
μ ( r ) = k β T ( r ) and ϕ ( r ) = k δ T ( r ) ,
𝓕 [ I ( r , z ) ] = I 0 + exp ( 2 k β T ( r ) ) exp ( i k δ λ z u · T ( r ) ) exp ( 2 π i r · u ) d r .
𝓕 [ I ( r , z ) ] = I 0 + exp ( 2 k β T ( r ) ) ( 1 i k δ λ z u · T ( r ) ) exp ( 2 π i r · u ) d r .
[ I ( r , z ) ] = I 0 [ exp ( 2 k β T ( r ) ) ] ( 1 + δ β λ z u 2 ) .
T ( r ) = 1 2 k β ln 𝓕 1 [ β β + δ π λ z u 2 𝓕 [ I ( r , z ) I 0 ] ] .
𝓕 [ I ( r , z ) ] = I 0 + exp ( μ ( r + λ z u 2 ) μ ( r λ z u 2 ) + i ( ϕ ( r λ z u 2 ) ϕ ( r + λ z u 2 ) ) )
× exp ( 2 π i r · u ) d r .
𝓕 [ I ( r , z ) ] I 0 = δ ( u ) 𝓕 [ μ ( r + λ z u 2 ) + μ ( r λ z u 2 ) ]
+ i 𝓕 [ ϕ ( r λ z u 2 ) ϕ ( r + λ z u 2 ) ] .
T ( r ) = 𝓕 1 [ 1 2 k ( δ sin ( π λ z u 2 ) + β cos ( π λ z u 2 ) ) 𝓕 [ I ( r , z ) I 0 1 ] ] .
2 μ ( r ) 1 and ϕ ( r + λ z u 2 ) ϕ ( r λ z u 2 ) 1 .
I R 1 ( r , R 2 ) = 1 M 2 I ( r M , R 2 M ) ,

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