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Transport of intensity phase reconstruction to solve the twin image problem in holographic x-ray imagingM. Krenkel, M. Bartels, and T. Salditt »View Author Affiliations
M. Krenkel,*
M. Bartels,
and T. Salditt
Institut für Röntgenphysik, Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany *Corresponding author: mkrenke@gwdg.de |
Optics Express, Vol. 21, Issue 2, pp. 2220-2235 (2013)
http://dx.doi.org/10.1364/OE.21.002220
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Abstract
We have implemented a deterministic method for solving the phase problem in hard x-ray in-line holography which overcomes the twin image problem. The phase distribution in the detector plane is retrieved by using two images with slightly different Fresnel numbers. We then use measured intensities and reconstructed phases in the detection plane to compute the exit wave in the sample plane. No further a priori information like a limited support or the assumption of pure phase objects is necessary so that it can be used for a wide range of complex samples. Using a nano-focused hard x-ray beam half period resolutions better than 30 nm are achieved.
© 2013 OSA
OCIS Codes
(340.7440) X-ray optics : X-ray imaging
(340.7460) X-ray optics : X-ray microscopy
(090.1995) Holography : Digital holography
ToC Category:
X-ray Optics
History
Original Manuscript: November 21, 2012
Revised Manuscript: December 16, 2012
Manuscript Accepted: December 16, 2012
Published: January 23, 2013
Citation
M. Krenkel, M. Bartels, and T. Salditt, "Transport of intensity phase reconstruction to solve the twin image problem in holographic x-ray imaging," Opt. Express 21, 2220-2235 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-2-2220
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References
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- A. Liu, D. Paganin, L. Bourgeois, and P. Nakashima, “Projected thickness reconstruction from a single defocused transmission electron microscope image of an amorphous object,” Ultramicroscopy111, 959–968 (2011). [CrossRef] [PubMed]
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- M. Howells, T. Beetz, H. Chapman, C. Cui, J. Holton, C. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. Shapiro, J. Spence, and D. Starodub, “An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy,” J.Electron Spectrosc.170, 4–12 (2009). [CrossRef]
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Abela, R.
Arhatari, B. D.
Barnea, Z.
Bartels, M.
Barty, A
Baruchel, J.
Baumbach, T.
Beerlink, A.
Beetz, T.
Beta, C.
Bourgeois, L.
Bronnikov, A.
Burvall, A.
Chapman, H.
Cloetens, P.
Cookson, D. F.
Cui, C.
Davis, J. C.
Davis, T.
Davis, T. J.
Dhal, B.
Diaz, A.
Dyck, D. V.
Fienup, J. R.
Fink, H.-W.
Fuhse, C.
Gabor, D.
Gao, D.
Gerchberg, R. W.
Giewekemeyer, K.
Goodman, J. W.
Groso, A.
Guigay, J. P.
Guizar-Sicairos, M.
Gullikson, E. M.
Gureyev, T.
Gureyev, T. E.
Hartmann, B.
Hayes, J.
Henke, B. L.
Hertz, H. M.
Hofmann, R.
Holton, J.
Howells, M.
Jacobsen, C.
Kalbfleisch, S.
Karvinen, P.
Kirz, J.
Krenkel, M.
Kruger, S.
Kruger, S. P.
Krüger, S. P.
Landuyt, J. V.
Larsson, D. H.
Latychevskaia, T.
Lima, E.
Liu, A.
Ludwig, W.
Lundström, U.
Mai, D. D.
Mancuso, A.
Marchesini, S.
Mayo, S.
Mayo, S. C.
Miao, H.
Miller, P.
Misell, D. L.
Mokso, R.
Moosmann, J.
Nakashima, P.
Neubauer, H.
Nugent, K.
Nugent, K. A.
Olendrowitz, C.
Osterhoff, M.
Paganin, D.
Paganin, D. M.
Parry, D.
Paterson, D.
Peele, A.
Peele, A. G.
Pogany, A.
Priebe, M.
Roggemann, M. C.
Salditt, T.
Saxton, W. O.
Sayre, D.
Schlenker, M.
Scholten, R.
Shapiro, D.
Spence, J.
Sprung, M.
Stampanoni, M.
Starodub, D.
Stevenson, A.
Stevenson, A. W.
Takman, P. A. C.
Teague, M. R.
Thurman, S. T.
Tiller, D. P. J.B
Tran, C.
Tucoulou, R.
Turner, L.
Voelz, D. G.
Wilbrandt, P. J.
Wilke, R.
Wilke, R. N.
Wilkins, S.
Wilkins, S. W.
- 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]
- M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, and T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy1, 10 (2012). [CrossRef]
- C. Olendrowitz, M. Bartels, M. Krenkel, A. Beerlink, R. Mokso, M. Sprung, and T. Salditt, “Phase-contrast x-ray imaging and tomography of the nematode Caenorhabditis elegans,” Phys. Med. Biol.57, 5309–5323 (2012). [CrossRef] [PubMed]
- S. P. Krüger, H. Neubauer, M. Bartels, S. Kalbfleisch, K. Giewekemeyer, P. J. Wilbrandt, M. Sprung, and T. Salditt, “Sub-10 nm beam confinement by X-ray waveguides: design, fabrication and characterization of optical properties,” J. Synchrotron Radiat.19, 227–236 (2012). [CrossRef] [PubMed]
- R. N. Wilke, M. Priebe, M. Bartels, K. Giewekemeyer, A. Diaz, P. Karvinen, and T. Salditt, “Hard x-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction,” Opt. Express20, 19232–19254 (2012). [CrossRef] [PubMed]
- S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, and T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIPConf.Proc.1365, 96–99 (2011).
- T. Salditt, S. Kalbfleisch, M. Osterhoff, S. P. Krüger, M. Bartels, K. Giewekemeyer, H. Neubauer, and M. Sprung, “Partially coherent nano-focused x-ray radiation characterized by Talbot interferometry,” Opt. Express19, 9656–9675 (2011). [CrossRef] [PubMed]
- K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A83, 023804 (2011). [CrossRef]
- S. P. Krüger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, “Sub-15 nm beam confinement by two crossed x-ray waveguides,” Opt. Express18, 13492–13501 (2010). [CrossRef] [PubMed]
- D. P. J.B Tiller, A Barty, and K. Nugent, “The holographic twin image problem: a deterministic phase solution,” Opt. Comm.183, 7–14 (2000). [CrossRef]
- P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. 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]
- C. Olendrowitz, M. Bartels, M. Krenkel, A. Beerlink, R. Mokso, M. Sprung, and T. Salditt, “Phase-contrast x-ray imaging and tomography of the nematode Caenorhabditis elegans,” Phys. Med. Biol.57, 5309–5323 (2012). [CrossRef] [PubMed]
- M. Howells, T. Beetz, H. Chapman, C. Cui, J. Holton, C. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. Shapiro, J. Spence, and D. Starodub, “An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy,” J.Electron Spectrosc.170, 4–12 (2009). [CrossRef]
- K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A83, 023804 (2011). [CrossRef]
- A. Liu, D. Paganin, L. Bourgeois, and P. Nakashima, “Projected thickness reconstruction from a single defocused transmission electron microscope image of an amorphous object,” Ultramicroscopy111, 959–968 (2011). [CrossRef] [PubMed]
- M. Howells, T. Beetz, H. Chapman, C. Cui, J. Holton, C. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. Shapiro, J. Spence, and D. Starodub, “An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy,” J.Electron Spectrosc.170, 4–12 (2009). [CrossRef]
- P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. 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]
- T. Salditt, K. Giewekemeyer, C. Fuhse, S. P. Kruger, R. Tucoulou, and P. Cloetens, “Projection phase contrast microscopy with a hard x-ray nanofocused beam: Defocus and contrast transfer,” Phys. Rev. B79, 184112(2009).
- 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]
- M. Howells, T. Beetz, H. Chapman, C. Cui, J. Holton, C. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. Shapiro, J. Spence, and D. Starodub, “An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy,” J.Electron Spectrosc.170, 4–12 (2009). [CrossRef]
- B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, Z = 1–92,” Atomic Data and Nuclear Data Tables54, 181–342 (1993). [CrossRef]
- S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging.” J. Microsc.207, 79–96 (2002). [CrossRef] [PubMed]
- P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. 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]
- T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett.98, 233901 (2007). [CrossRef] [PubMed]
- T. Salditt, K. Giewekemeyer, C. Fuhse, S. P. Kruger, R. Tucoulou, and P. Cloetens, “Projection phase contrast microscopy with a hard x-ray nanofocused beam: Defocus and contrast transfer,” Phys. Rev. B79, 184112(2009).
- D. Gabor, “A new microscopic principle,” Nature161, 777–778 (1948). [CrossRef] [PubMed]
- S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging.” J. Microsc.207, 79–96 (2002). [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,” Nature384, 335–338 (1996). [CrossRef]
- R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena)35, 237–246 (1972).
- M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, and T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy1, 10 (2012). [CrossRef]
- S. P. Krüger, H. Neubauer, M. Bartels, S. Kalbfleisch, K. Giewekemeyer, P. J. Wilbrandt, M. Sprung, and T. Salditt, “Sub-10 nm beam confinement by X-ray waveguides: design, fabrication and characterization of optical properties,” J. Synchrotron Radiat.19, 227–236 (2012). [CrossRef] [PubMed]
- R. N. Wilke, M. Priebe, M. Bartels, K. Giewekemeyer, A. Diaz, P. Karvinen, and T. Salditt, “Hard x-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction,” Opt. Express20, 19232–19254 (2012). [CrossRef] [PubMed]
- T. Salditt, S. Kalbfleisch, M. Osterhoff, S. P. Krüger, M. Bartels, K. Giewekemeyer, H. Neubauer, and M. Sprung, “Partially coherent nano-focused x-ray radiation characterized by Talbot interferometry,” Opt. Express19, 9656–9675 (2011). [CrossRef] [PubMed]
- S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, and T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIPConf.Proc.1365, 96–99 (2011).
- K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A83, 023804 (2011). [CrossRef]
- S. P. Krüger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, “Sub-15 nm beam confinement by two crossed x-ray waveguides,” Opt. Express18, 13492–13501 (2010). [CrossRef] [PubMed]
- T. Salditt, K. Giewekemeyer, C. Fuhse, S. P. Kruger, R. Tucoulou, and P. Cloetens, “Projection phase contrast microscopy with a hard x-ray nanofocused beam: Defocus and contrast transfer,” Phys. Rev. B79, 184112(2009).
- K. Giewekemeyer, “A study on new approaches in coherent x-ray microscopy of biological specimens,” Ph.D. thesis, Universität Göttingen (2011).
- J. W. Goodman, Introduction to Fourier optics (Roberts & Company: Englewood, Colorado, 2005).
- P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. 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]
- B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, Z = 1–92,” Atomic Data and Nuclear Data Tables54, 181–342 (1993). [CrossRef]
- D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc.206, 33–40 (2002). [CrossRef] [PubMed]
- S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging.” J. Microsc.207, 79–96 (2002). [CrossRef] [PubMed]
- T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first born- and rytov-type approximations,” Appl. Opt.43, 2418–2430 (2004). [CrossRef] [PubMed]
- T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun.220, 49 – 58 (2003). [CrossRef]
- S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384, 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]
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Atomic Data and Nuclear Data Tables
- B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, Z = 1–92,” Atomic Data and Nuclear Data Tables54, 181–342 (1993). [CrossRef]
J. Microsc.
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- D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc.206, 33–40 (2002). [CrossRef] [PubMed]
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J. Phys. D: Appl. Phys
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J. Synchrotron Radiat.
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J.Electron Spectrosc.
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Nature
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Opt. Comm.
- D. P. J.B Tiller, A Barty, and K. Nugent, “The holographic twin image problem: a deterministic phase solution,” Opt. Comm.183, 7–14 (2000). [CrossRef]
Opt. Commun.
- T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun.220, 49 – 58 (2003). [CrossRef]
Opt. Express
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Opt. Lett.
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Opt. Nanoscopy
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Optik (Jena)
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Phys. Med. Biol.
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Phys. Rev. A
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Phys. Rev. B
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Phys. Rev. Lett.
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2012, Bartels, Opt. Nanoscopy
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- C. Olendrowitz, M. Bartels, M. Krenkel, A. Beerlink, R. Mokso, M. Sprung, and T. Salditt, “Phase-contrast x-ray imaging and tomography of the nematode Caenorhabditis elegans,” Phys. Med. Biol.57, 5309–5323 (2012). [CrossRef] [PubMed]
- S. P. Krüger, H. Neubauer, M. Bartels, S. Kalbfleisch, K. Giewekemeyer, P. J. Wilbrandt, M. Sprung, and T. Salditt, “Sub-10 nm beam confinement by X-ray waveguides: design, fabrication and characterization of optical properties,” J. Synchrotron Radiat.19, 227–236 (2012). [CrossRef] [PubMed]
- A. Liu, D. Paganin, L. Bourgeois, and P. Nakashima, “Projected thickness reconstruction from a single defocused transmission electron microscope image of an amorphous object,” Ultramicroscopy111, 959–968 (2011). [CrossRef] [PubMed]
- S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, and T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIPConf.Proc.1365, 96–99 (2011).
- K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A83, 023804 (2011). [CrossRef]
- K. A. Nugent, “Coherent methods in the x-ray sciences,” Adv. Phys.59, 1–99 (2010). [CrossRef]
- M. Howells, T. Beetz, H. Chapman, C. Cui, J. Holton, C. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. Shapiro, J. Spence, and D. Starodub, “An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy,” J.Electron Spectrosc.170, 4–12 (2009). [CrossRef]
- T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett.98, 233901 (2007). [CrossRef] [PubMed]
- T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun.220, 49 – 58 (2003). [CrossRef]
- D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc.206, 33–40 (2002). [CrossRef] [PubMed]
- S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging.” J. Microsc.207, 79–96 (2002). [CrossRef] [PubMed]
- D. P. J.B Tiller, A Barty, and K. Nugent, “The holographic twin image problem: a deterministic phase solution,” Opt. Comm.183, 7–14 (2000). [CrossRef]
- P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. 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]
- D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett.80, 2586–2589 (1998). [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]
- 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]
- S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384, 335–338 (1996). [CrossRef]
- B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, Z = 1–92,” Atomic Data and Nuclear Data Tables54, 181–342 (1993). [CrossRef]
- D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: Ii. sources of error,” J. Phys. D: Appl. Phys6, 2217 (1973). [CrossRef]
- R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena)35, 237–246 (1972).
- D. Gabor, “A new microscopic principle,” Nature161, 777–778 (1948). [CrossRef] [PubMed]
- T. Salditt, K. Giewekemeyer, C. Fuhse, S. P. Kruger, R. Tucoulou, and P. Cloetens, “Projection phase contrast microscopy with a hard x-ray nanofocused beam: Defocus and contrast transfer,” Phys. Rev. B79, 184112(2009).
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