Quantitative phase-contrast tomography of a liquid phantom using a conventional x-ray tube source

Julia Herzen; Tilman Donath; Franz Pfeiffer; Oliver Bunk; Celestino Padeste; Felix Beckmann; Andreas Schreyer; Christian David

doi:10.1364/OE.17.010010

Quantitative phase-contrast tomography of a liquid phantom using a conventional x-ray tube source

Julia Herzen, Tilman Donath, Franz Pfeiffer, Oliver Bunk, Celestino Padeste, Felix Beckmann, Andreas Schreyer, and Christian David

Optics Express
Vol. 17,
Issue 12,
pp. 10010-10018
(2009)
•https://doi.org/10.1364/OE.17.010010

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Julia Herzen, Tilman Donath, Franz Pfeiffer, Oliver Bunk, Celestino Padeste, Felix Beckmann, Andreas Schreyer, and Christian David, "Quantitative phase-contrast tomography of a liquid phantom using a conventional x-ray tube source," Opt. Express 17, 10010-10018 (2009)

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Abstract

Over the last few years, differential phase-contrast x-ray computed tomography (PC-CT) using a hard x-ray grating interferometer and polychromatic x-ray tube sources has been developed. The method allows for simultaneous determination of the attenuation coefficient and the refractive index decrement distribution inside an object in three dimensions. Here we report experimental results of our investigation on the quantita-tiveness and accuracy of this method. For this study, a phantom consisting of several tubes filled with chemically well-defined liquids was built and measured in PC-CT. We find, that the measured attenuation coefficients and refractive index decrements closely match calculated, theoretical values. Moreover, the study demonstrates, how substances with similar attenuation coefficient or refractive index decrement, can be uniquely distinguished by the simultaneous, quantitative measurement of both quantities.

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[Crossref] [PubMed]
A. Momose, “Recent advances in X-ray phase imaging,” Jpn. J. Appl. Phys. 44, 6355–6367 (2005).
[Crossref]
F. Beckmann, K. Heise, B. Kölsch, U. Bonse, M. F. Rajewsky, M. Bartscher, and T. Biermann, “Three-Dimensional Imaging of Nerve Tissue by X-Ray Phase-Contrast Microtomography,” Biophys. J. 76, 98–102 (1999).
[Crossref] [PubMed]
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[Crossref]
T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[Crossref]
T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 12, 6296–6304 (2005).
[Crossref]
U. Bonse, F. Busch, O. Günnewig, F. Beckmann, R. Pahl, G. Delling, M. Hahn, and W. Graeff, “3D computed X-ray tomography of human cancellous bone at 8 µm,” Bone and Miner. 25, 25–38 (1994).
[Crossref]
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[Crossref]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref]
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[Crossref]
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[Crossref]
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[Crossref]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref]
M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[Crossref]
F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature Phys. 2, 258 (2006).
[Crossref]
C. Kottler, F. Pfeiffer, O. Bunk, C. Grünzweig, J. Bruder, R. Kaufmann, T. Tlustos, H. Walt, I. Briod, T. Weitkamp, and C. David, “Phase contrast X-ray imaging of large samples using an incoherent laboratory source,” Phys. Status Solidi A 204, 2728–2733 (2007).
[Crossref]
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C. David, J. Bruder, T. Rohbeck, C. Grünzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, “Fabrication of diffraction gratings for hard X-ray phase contrast imaging,” Microelectron. Eng. 84, 1172–1177 (2007).
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For the calculation of δ c the tabulated values of f 1 from L. Kissel for elastic photon-atom scattering, anomalous scattering factors were taken from the file f1f2 asf Kissel.dat of the DABAX library [28].
DABAX library, http://ftp.esrf.eu/pub/scisoft/xop2.3/DabaxFiles.
E. F. Plechaty, D. E. Cullen, and R. J. Howerton, “Tables and graphs of photon-interaction cross sections from 0.1 keV to 100 MeV derived from the LLL evaluated-nuclear-data library,” Lawrence Livermore National Laboratory (1981).
B. D. Arhatari, K. Hannah, E. Balaur, and A. G. Peele, “Phase imaging using a polychromatic X-ray laboratory source,” Opt. Express 16 (24), 19950–19956 (2008).
[Crossref]
M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microsc. 232 (1), 145–157 (2008).
[Crossref]

2008 (2)

B. D. Arhatari, K. Hannah, E. Balaur, and A. G. Peele, “Phase imaging using a polychromatic X-ray laboratory source,” Opt. Express 16 (24), 19950–19956 (2008).
[Crossref]

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microsc. 232 (1), 145–157 (2008).
[Crossref]

2007 (7)

C. Kottler, F. Pfeiffer, O. Bunk, C. Grünzweig, J. Bruder, R. Kaufmann, T. Tlustos, H. Walt, I. Briod, T. Weitkamp, and C. David, “Phase contrast X-ray imaging of large samples using an incoherent laboratory source,” Phys. Status Solidi A 204, 2728–2733 (2007).
[Crossref]

C. David, J. Bruder, T. Rohbeck, C. Grünzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, “Fabrication of diffraction gratings for hard X-ray phase contrast imaging,” Microelectron. Eng. 84, 1172–1177 (2007).
[Crossref]

T. Koyama, H. Takano, Y. Tsusaka, and Y. Kagoshima, “Tomographic quantitative phase measurement by hard X-ray micro-interferometer with 250 nm spatial resolution,” Spectrochim. Acta Part B 62, 603–607 (2007).
[Crossref]

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A (Atomic, Molecular, and Optical Physics) 76, 045804 (2007).
[Crossref]

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-Ray Phase Tomography with Low-Brilliance Sources,” Phys. Rev. Lett. 98, 108105 (2007).
[Crossref] [PubMed]

F. Pfeiffer, O. Bunk, C. Kottler, and C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Instrum. Methods Phys. Res. A 580, 925–928 (2007).
[Crossref]

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[Crossref]

2006 (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature Phys. 2, 258 (2006).
[Crossref]

2005 (2)

A. Momose, “Recent advances in X-ray phase imaging,” Jpn. J. Appl. Phys. 44, 6355–6367 (2005).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 12, 6296–6304 (2005).
[Crossref]

2003 (3)

P. J. McMahon, A. G. Peele, D. Paterson, J. J. A. Lin, T. H. K. Irving, I. McNulty, and K. A. Nugent, “Quantitative X-ray phase tomography with sub-micron resolution,” Optics Communications 217, 53–58 (2003).
[Crossref]

S.C. Mayo, T.J. Davis, T.E. Gureyev, P.R. Miller, D. Paganin, A. Pogany, A.W. Stevenson, and S.W. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289–2302 (2003).
[Crossref] [PubMed]

A. Momose, “Phase-sensitive imaging and phase tomography using X-ray interferometers,” Opt. Express 11, 2303–2314 (2003).
[Crossref] [PubMed]

2001 (1)

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, and A. W. Stevenson, “Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2001).
[Crossref]

2000 (1)

F.A. Dilmanian, Z. Zhong, B. Ren, X.Y. Wu, L.D. Chapman, I. Orion, and W.C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[Crossref] [PubMed]

1999 (3)

F. Beckmann, K. Heise, B. Kölsch, U. Bonse, M. F. Rajewsky, M. Bartscher, and T. Biermann, “Three-Dimensional Imaging of Nerve Tissue by X-Ray Phase-Contrast Microtomography,” Biophys. J. 76, 98–102 (1999).
[Crossref] [PubMed]

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]

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[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 (1)

A. Momose, “Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer,” Nucl. Instrum. Method A 352, 622–628 (1995).
[Crossref]

1994 (1)

U. Bonse, F. Busch, O. Günnewig, F. Beckmann, R. Pahl, G. Delling, M. Hahn, and W. Graeff, “3D computed X-ray tomography of human cancellous bone at 8 µm,” Bone and Miner. 25, 25–38 (1994).
[Crossref]

1980 (1)

D. R. Lide, “CRC Handbook of Chemistry and Physics (60th Edition),” Taylor and Francis, Boca Raton, FL (1980).

Als-Nielsen, J.

J. Als-Nielsen, “Elements of Modern x-ray Physics,” John Wiley & Sons, Chichester (2008).

Arhatari, B. D.

B. D. Arhatari, K. Hannah, E. Balaur, and A. G. Peele, “Phase imaging using a polychromatic X-ray laboratory source,” Opt. Express 16 (24), 19950–19956 (2008).
[Crossref]

Balaur, E.

B. D. Arhatari, K. Hannah, E. Balaur, and A. G. Peele, “Phase imaging using a polychromatic X-ray laboratory source,” Opt. Express 16 (24), 19950–19956 (2008).
[Crossref]

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]

Bartscher, M.

Baruchel, J.

Baumann, J.

Beckmann, F.

Biermann, T.

Bonse, U.

Briod, I.

Bruder, J.

Bunk, O.

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-Ray Phase Tomography with Low-Brilliance Sources,” Phys. Rev. Lett. 98, 108105 (2007).
[Crossref] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature Phys. 2, 258 (2006).
[Crossref]

Busch, F.

Chapman, L.D.

Cloetens, P.

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 12, 6296–6304 (2005).
[Crossref]

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]

Cullen, D. E.

E. F. Plechaty, D. E. Cullen, and R. J. Howerton, “Tables and graphs of photon-interaction cross sections from 0.1 keV to 100 MeV derived from the LLL evaluated-nuclear-data library,” Lawrence Livermore National Laboratory (1981).

David, C.

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-Ray Phase Tomography with Low-Brilliance Sources,” Phys. Rev. Lett. 98, 108105 (2007).
[Crossref] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature Phys. 2, 258 (2006).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 12, 6296–6304 (2005).
[Crossref]

Davis, T. J.

Davis, T.J.

Delling, G.

Diaz, A.

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 12, 6296–6304 (2005).
[Crossref]

Dilmanian, F.A.

Engelhardt, M.

Gao, D.

Graeff, W.

Grünzweig, C.

Guigay, J. P.

Günnewig, O.

Gureyev, T. E.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[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]

Gureyev, T.E.

Hahn, M.

Hannah, K.

B. D. Arhatari, K. Hannah, E. Balaur, and A. G. Peele, “Phase imaging using a polychromatic X-ray laboratory source,” Opt. Express 16 (24), 19950–19956 (2008).
[Crossref]

Heise, K.

Howerton, R. J.

Irving, T. H. K.

James, R. W.

R. W. James, “The optical principles of the diffraction of x-rays,” (1962).

Kagoshima, Y.

Kak, A. C.

A. C. Kak and M. Slaney, “Principles of Computerized Tomography,” IEEE Press, New York (1987).

Kaufmann, R.

Kölsch, B.

Kottler, C.

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-Ray Phase Tomography with Low-Brilliance Sources,” Phys. Rev. Lett. 98, 108105 (2007).
[Crossref] [PubMed]

Koyama, T.

Lide, D. R.

D. R. Lide, “CRC Handbook of Chemistry and Physics (60th Edition),” Taylor and Francis, Boca Raton, FL (1980).

Lin, J. J. A.

Ludwig, W.

Mayo, S. C.

Mayo, S.C.

McMahon, P. J.

McNulty, I.

Miller, P. R.

Miller, P.R.

Momose, A.

A. Momose, “Recent advances in X-ray phase imaging,” Jpn. J. Appl. Phys. 44, 6355–6367 (2005).
[Crossref]

A. Momose, “Phase-sensitive imaging and phase tomography using X-ray interferometers,” Opt. Express 11, 2303–2314 (2003).
[Crossref] [PubMed]

A. Momose, “Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer,” Nucl. Instrum. Method A 352, 622–628 (1995).
[Crossref]

Myers, G. R.

Nugent, K. A.

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]

Orion, I.

Paganin, D.

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.

D. M. Paganin, “Coherent x-ray Optics,” Oxford University Press, New York (2006).

Pahl, R.

Parry, D. J.

Paterson, D.

Peele, A. G.

B. D. Arhatari, K. Hannah, E. Balaur, and A. G. Peele, “Phase imaging using a polychromatic X-ray laboratory source,” Opt. Express 16 (24), 19950–19956 (2008).
[Crossref]

Pfeiffer, F.

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-Ray Phase Tomography with Low-Brilliance Sources,” Phys. Rev. Lett. 98, 108105 (2007).
[Crossref] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature Phys. 2, 258 (2006).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 12, 6296–6304 (2005).
[Crossref]

Plechaty, E. F.

Pogany, A.

Rajewsky, M. F.

Raven, C.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[Crossref]

Ren, B.

Rohbeck, T.

Schlenker, M.

Schroer, C.

Schuster, M.

Slaney, M.

A. C. Kak and M. Slaney, “Principles of Computerized Tomography,” IEEE Press, New York (1987).

Snigirev, A.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[Crossref]

Snigireva, I.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[Crossref]

Stampanoni, M.

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 12, 6296–6304 (2005).
[Crossref]

Stevenson, A. W.

Stevenson, A.W.

Takano, H.

Thomlinson, W.C.

Tlustos, T.

Tsusaka, Y.

van Dyck, D.

van Landuyt, J.

Walt, H.

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature Phys. 2, 258 (2006).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 12, 6296–6304 (2005).
[Crossref]

Wilkins, S. W.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[Crossref]

Wilkins, S.W.

Wu, X.Y.

Zhong, Z.

Ziegler, E.

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 12, 6296–6304 (2005).
[Crossref]

Appl. Phys. Lett. (2)

Biophys. J. (1)

Bone and Miner. (1)

J. Microsc. (2)

J. Phys. D (1)

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[Crossref]

Jpn. J. Appl. Phys. (1)

A. Momose, “Recent advances in X-ray phase imaging,” Jpn. J. Appl. Phys. 44, 6355–6367 (2005).
[Crossref]

Microelectron. Eng. (1)

Nature Phys. (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature Phys. 2, 258 (2006).
[Crossref]

Nucl. Instrum. Method A (1)

A. Momose, “Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer,” Nucl. Instrum. Method A 352, 622–628 (1995).
[Crossref]

Nucl. Instrum. Methods Phys. Res. A (1)

Opt. Express (4)

B. D. Arhatari, K. Hannah, E. Balaur, and A. G. Peele, “Phase imaging using a polychromatic X-ray laboratory source,” Opt. Express 16 (24), 19950–19956 (2008).
[Crossref]

A. Momose, “Phase-sensitive imaging and phase tomography using X-ray interferometers,” Opt. Express 11, 2303–2314 (2003).
[Crossref] [PubMed]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 12, 6296–6304 (2005).
[Crossref]

Optics Communications (1)

Phys. Med. Biol. (1)

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Phys. Rev. Lett. (2)

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-Ray Phase Tomography with Low-Brilliance Sources,” Phys. Rev. Lett. 98, 108105 (2007).
[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]

Phys. Status Solidi A (1)

Spectrochim. Acta Part B (1)

Other (8)

A. C. Kak and M. Slaney, “Principles of Computerized Tomography,” IEEE Press, New York (1987).

J. Als-Nielsen, “Elements of Modern x-ray Physics,” John Wiley & Sons, Chichester (2008).

D. M. Paganin, “Coherent x-ray Optics,” Oxford University Press, New York (2006).

R. W. James, “The optical principles of the diffraction of x-rays,” (1962).

D. R. Lide, “CRC Handbook of Chemistry and Physics (60th Edition),” Taylor and Francis, Boca Raton, FL (1980).

For the calculation of δ c the tabulated values of f 1 from L. Kissel for elastic photon-atom scattering, anomalous scattering factors were taken from the file f1f2 asf Kissel.dat of the DABAX library [28].

DABAX library, http://ftp.esrf.eu/pub/scisoft/xop2.3/DabaxFiles.

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Fig. 1.

X-ray grating interferometer for differential phase-contrast imaging. A phase object in the beam path causes a slight deflection of x-rays changing the locally transmitted intensity through the arrangement formed by the gratings G1 and G2. The sample is placed on a tomographic rotation stage.

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Fig. 2.

Tomographic reconstructions from the same slice of the phantom showing (a) the attenuation coefficient Δµ(x, y) and (b) the refractive index decrement Δδ(x, y). The color-bar (from black to white) ranges from -2.0 ·10⁻² to 3.0 ·10⁻² mm⁻¹ in (a) and from -7.5·10⁻⁸ to 11.3·10⁻⁸ in (b). The fluids in the 13 polyethylene tubes of the phantom have been labeled with their identifiers according to the fluid description in Table 1. A centered, circular averaging region, as shown for one tube in (b), was defined for each tube to obtain the experimental values given in Table 2.

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Fig. 3.

Scatter plot of measured and calculated attenuation coefficients Δµ, Δµ_c and refractive index decrements Δδ, Δδ_c . The plot shows the data given in Table 2.

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Tables (2)

Table 1. Density and elemental composion of the fluids. These values were used in the calculation of Δµc and Δδc listed in Table 2.

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Table 2 Measured and calculated attenuation coefficients Δµ and refractive index decrements Δδ for all fluids in the phantom (relative to water). The standard deviations σ_µ and σ_δ within each averaging region are given. The calculated Δµ_c and Δδ_c values were determined for effective photon energies of E_µ =30.1 keV and E_δ =28.3 keV, respectively, as discussed in the text. The data is plotted in Fig. 3.

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Equations (7)

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T^{Θ} (x') = \exp [- \int_{0}^{d} \frac{4 π}{λ} β (x', y') d y'] = \exp [- \int_{0}^{d} μ (x', y') d y'],

{(\frac{μ}{ρ})}_{subst} = \sum_{i = 1}^{I} {(\frac{μ}{ρ})}_{i} W_{i},

α^{Θ} (x') = \frac{λ}{2 π} \frac{\partial Φ^{Θ} (x')}{\partial x'} = \int_{0}^{d} \frac{\partial δ (x', y')}{\partial x'} d y',

δ_{subst} = \frac{r_{e} λ^{2}}{2 π} \sum_{i = 1}^{I} N_{i} f_{i}^{1},

μ (x, y) = - \int_{0}^{π} ℱ^{-1} {ℱ [l n T^{Θ} (ω)] ∙ ℱ [k (ω)]} d Θ,

δ (x, y) = \int_{0}^{π} ℱ^{- 1} {ℱ [α^{Θ} (ω)] ∙ ℱ [h (ω)]} d Θ,

CNR = \frac{∣ S_{a} - S_{b} ∣}{σ_{S}},

			Elemental weight fraction W_i
Solution [wt.%]	Identifier	Density [g/cm³]	H [wt.%]	O [wt.%]	C [wt.%]	Na [wt.%]	Cl [wt.%]	I [wt.%]
H₂O (demineralized)	H₂O	0.9982	11.19	88.81	–	–	–	–
glycerol (99.5%)	Gly	1.26	8.76	52.12	39.13	–	–	–
ethanol (p.a. 99.9%)	EtOH	0.7894	13.13	34.73	52.14	–	–	–
H₂O+NaCl 1.25%	H2O-NaCl1.25	1.0072	11.05	87.70	–	0.49	0.76	–
H₂O+NaCl 2.5%	H2O-NaCl2.5	1.0160	10.91	86.59	–	0.98	1.52	–
H₂O+NaCl 5%	H2O-NaCl5	1.0340	10.63	84.37	–	1.97	3.03	–
H₂O+NaCl 10%	H₂O-NaCl10	1.0707	10.07	79.93	–	3.93	6.07	–
ethanol+NaI 1.25%	EtOH-NaI1.25	–	12.96	34.29	51.49	0.19	–	1.06
ethanol+NaI 2.5%	EtOH-NaI2.5	–	12.80	33.86	50.84	0.38	–	2.12
ethanol+NaI 5%	EtOH-NaI5	–	12.47	32.99	49.54	0.77	–	4.23
ethanol (75%) + glycerol (25%)	EtOH75-Gly25	–	–	–	–	–	–	–
ethanol (50%) + glycerol (50%)	EtOH50-Gly50	–	–	–	–	–	–	–
ethanol (25%) + glycerol (75%)	EtOH25-Gly75	–	–	–	–	–	–	–

Identifier	Δ _µ ±σ _µ [10⁻²/mm]	Δµ_c (E_µ ) [10⁻²/mm]	Δδ±σ_δ [10⁻⁸]	Δδ_c (E_δ ) [10⁻⁸]
H₂O	0.023±0.019	0	−0.015±0.049	0
Gly	0.358±0.022	0.382	5.831±0.077	6.736
EtOH	−1.220±0.054	−1.219	−5.632±0.076	−5.634
H₂O-NaCl1.25	0.247±0.028	0.201	0.122±0.053	0.208
H₂O-NaCl2.5	0.467±0.022	0.406	0.486±0.022	0.413
H₂O-NaCl5	0.878±0.028	0.827	0.662±0.028	0.829
H₂O-NaCl10	1.796±0.018	1.707	1.388±0.028	1.670
EtOH-NaI2.5	1.272±0.028	–	−5.013±0.045	–
EtOH-NaI5	3.539±0.038	–	−4.782±0.038	–
EtOH75-Gly25	−0.906±0.027	–	−3.361±0.066	–
EtOH50-Gly50	−0.540±0.025	–	−0.340±0.042	–
EtOH25-Gly75	−0.115±0.017	–	2.767±0.044	–

			Elemental weight fraction W_i
Solution [wt.%]	Identifier	Density [g/cm³]	H [wt.%]	O [wt.%]	C [wt.%]	Na [wt.%]	Cl [wt.%]	I [wt.%]
H₂O (demineralized)	H₂O	0.9982	11.19	88.81	–	–	–	–
glycerol (99.5%)	Gly	1.26	8.76	52.12	39.13	–	–	–
ethanol (p.a. 99.9%)	EtOH	0.7894	13.13	34.73	52.14	–	–	–
H₂O+NaCl 1.25%	H2O-NaCl1.25	1.0072	11.05	87.70	–	0.49	0.76	–
H₂O+NaCl 2.5%	H2O-NaCl2.5	1.0160	10.91	86.59	–	0.98	1.52	–
H₂O+NaCl 5%	H2O-NaCl5	1.0340	10.63	84.37	–	1.97	3.03	–
H₂O+NaCl 10%	H₂O-NaCl10	1.0707	10.07	79.93	–	3.93	6.07	–
ethanol+NaI 1.25%	EtOH-NaI1.25	–	12.96	34.29	51.49	0.19	–	1.06
ethanol+NaI 2.5%	EtOH-NaI2.5	–	12.80	33.86	50.84	0.38	–	2.12
ethanol+NaI 5%	EtOH-NaI5	–	12.47	32.99	49.54	0.77	–	4.23
ethanol (75%) + glycerol (25%)	EtOH75-Gly25	–	–	–	–	–	–	–
ethanol (50%) + glycerol (50%)	EtOH50-Gly50	–	–	–	–	–	–	–
ethanol (25%) + glycerol (75%)	EtOH25-Gly75	–	–	–	–	–	–	–

Identifier	Δ _µ ±σ _µ [10⁻²/mm]	Δµ_c (E_µ ) [10⁻²/mm]	Δδ±σ_δ [10⁻⁸]	Δδ_c (E_δ ) [10⁻⁸]
H₂O	0.023±0.019	0	−0.015±0.049	0
Gly	0.358±0.022	0.382	5.831±0.077	6.736
EtOH	−1.220±0.054	−1.219	−5.632±0.076	−5.634
H₂O-NaCl1.25	0.247±0.028	0.201	0.122±0.053	0.208
H₂O-NaCl2.5	0.467±0.022	0.406	0.486±0.022	0.413
H₂O-NaCl5	0.878±0.028	0.827	0.662±0.028	0.829
H₂O-NaCl10	1.796±0.018	1.707	1.388±0.028	1.670
EtOH-NaI2.5	1.272±0.028	–	−5.013±0.045	–
EtOH-NaI5	3.539±0.038	–	−4.782±0.038	–
EtOH75-Gly25	−0.906±0.027	–	−3.361±0.066	–
EtOH50-Gly50	−0.540±0.025	–	−0.340±0.042	–
EtOH25-Gly75	−0.115±0.017	–	2.767±0.044	–