Abstract

X-ray grating interferometry requires gratings with periods in the micrometer range and allows the acquisition of the dark-field contrast. The analyzer grating is designed to match the period of the interference pattern in order to translate it into a measurable intensity modulation. In this study, we explore the influence of a sample-induced mismatch between the interference pattern and the analyzer grating on the dark-field contrast. We propose a formula for the calculation of the signal due to a period mismatch and present estimations varying periods and detector pixel size. Furthermore, numerical simulations of the X-ray wave-front demonstrate that the wave-front curvature, described by the lens-term, e.g. behind a parabolic lens or edges of a sample can change the period of the interference pattern. Our results give a concrete explanation for the formation of a dark-field contrast from object edges and thus allow a better understanding of the dark-field signal obtained with a grating interferometer.

© 2015 Optical Society of America

Full Article  |  PDF Article
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References

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2015 (3)

K. Scherer, E. Braig, K. Willer, M. Willner, A. a. Fingerle, M. Chabior, J. Herzen, M. Eiber, B. Haller, M. Straub, H. Schneider, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Non-invasive Differentiation of Kidney Stone Types using X-ray Dark-Field Radiography,” Sci. Rep. 5, 9527 (2015).
[Crossref] [PubMed]

F. Prade, M. Chabior, F. Malm, C. Grosse, and F. Pfeiffer, “Observing the setting and hardening of cementitious materials by X-ray dark-field radiography,” Cem. Concr. Res. 74, 19–25 (2015).
[Crossref]

W. Yashiro and A. Momose, “Effects of unresolvable edges in grating-based X-ray differential phase imaging,” Opt. Express 23, 9233 (2015).
[Crossref] [PubMed]

2014 (1)

M. Strobl, “General solution for quantitative dark-field contrast imaging with grating interferometers,” Sci. Rep. 4, 7243 (2014).
[Crossref] [PubMed]

2013 (2)

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. O. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal x-ray dark-field scatter-contrast scanner,” Radiology 269, 427–433 (2013).
[Crossref] [PubMed]

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

2012 (3)

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

A. Malecki, G. Potdevin, and F. Pfeiffer, “Quantitative wave-optical numerical analysis of the dark-field signal in grating-based X-ray interferometry,” EPL 99, 48001 (2012).
[Crossref]

Y. Yang and X. Tang, “The second-order differential phase contrast and its retrieval for imaging with x-ray Talbot interferometry,” Med. Phys. 39, 7237–7253 (2012).
[Crossref] [PubMed]

2011 (2)

2010 (2)

W. Yashiro, Y. Terui, K. Kawabata, and A. Momose, “On the origin of visibility contrast in x-ray Talbot interferometry,” Opt. Express 18, 16890–168901 (2010).
[Crossref] [PubMed]

T. Donath, F. Pfeiffer, O. Bunk, C. Grünzweig, E. Hempel, S. Popescu, P. Vock, and C. David, “Toward clinical x-ray phase-contrast ct: Demonstration of enhanced soft-tissue contrast in human specimen,” Invest. Radiol. 45, 445–452 (2010).
[PubMed]

2008 (1)

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

2007 (1)

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. L. Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923 (2007).
[Crossref] [PubMed]

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, 258–261 (2006).
[Crossref]

2005 (2)

2004 (1)

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Gunzler, O. Kurapova, F. Zontone, A. A. Snigirev, and I. I. Snigireva, “Beryllium parabolic refractive x-ray lenses,” Proc. SPIE 5539, 1–9 (2004).
[Crossref]

2003 (1)

1997 (1)

1988 (1)

Achterhold, K.

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

Adam-Neumair, S.

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

Anton, G.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Auweter, S.

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. O. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal x-ray dark-field scatter-contrast scanner,” Radiology 269, 427–433 (2013).
[Crossref] [PubMed]

Auweter, S. D.

A. Velroyen, A. Yaroshenko, D. Hahn, A. Fehringer, A. Tapfer, M. Müller, P. B. Noël, B. Pauwels, A. Sasov, A. O. Yildirim, O. Eickelberg, K. Hellbach, S. D. Auweter, F. G. Meinel, M. F. Reiser, M. Bech, and F. Pfeiffer, “Grating-based X-ray Dark-field Computed Tomography of Living Mice,” EBioMedicine (2015), http://dx.doi.org/10.1016/j.ebiom.2015.08.014 .
[Crossref]

Auxier, J.

Bamberg, F.

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. O. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal x-ray dark-field scatter-contrast scanner,” Radiology 269, 427–433 (2013).
[Crossref] [PubMed]

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

Bayer, F.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Bech, M.

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. O. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal x-ray dark-field scatter-contrast scanner,” Radiology 269, 427–433 (2013).
[Crossref] [PubMed]

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. L. Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923 (2007).
[Crossref] [PubMed]

A. Velroyen, A. Yaroshenko, D. Hahn, A. Fehringer, A. Tapfer, M. Müller, P. B. Noël, B. Pauwels, A. Sasov, A. O. Yildirim, O. Eickelberg, K. Hellbach, S. D. Auweter, F. G. Meinel, M. F. Reiser, M. Bech, and F. Pfeiffer, “Grating-based X-ray Dark-field Computed Tomography of Living Mice,” EBioMedicine (2015), http://dx.doi.org/10.1016/j.ebiom.2015.08.014 .
[Crossref]

Beckmann, M. W.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Benner, B.

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Gunzler, O. Kurapova, F. Zontone, A. A. Snigirev, and I. I. Snigireva, “Beryllium parabolic refractive x-ray lenses,” Proc. SPIE 5539, 1–9 (2004).
[Crossref]

Bennett, E. E.

Bohla, A.

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. O. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal x-ray dark-field scatter-contrast scanner,” Radiology 269, 427–433 (2013).
[Crossref] [PubMed]

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

Braig, E.

K. Scherer, E. Braig, K. Willer, M. Willner, A. a. Fingerle, M. Chabior, J. Herzen, M. Eiber, B. Haller, M. Straub, H. Schneider, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Non-invasive Differentiation of Kidney Stone Types using X-ray Dark-Field Radiography,” Sci. Rep. 5, 9527 (2015).
[Crossref] [PubMed]

Bravin, A.

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. L. Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923 (2007).
[Crossref] [PubMed]

Bronnimann, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

Bunk, O.

M. Chabior, T. Donath, C. David, O. Bunk, M. Schuster, C. Schroer, and F. Pfeiffer, “Beam hardening effects in grating-based x-ray phase-contrast imaging,” Med. Phys. 38, 1189–1195 (2011).
[Crossref] [PubMed]

T. Donath, F. Pfeiffer, O. Bunk, C. Grünzweig, E. Hempel, S. Popescu, P. Vock, and C. David, “Toward clinical x-ray phase-contrast ct: Demonstration of enhanced soft-tissue contrast in human specimen,” Invest. Radiol. 45, 445–452 (2010).
[PubMed]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. L. Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923 (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, 258–261 (2006).
[Crossref]

Chabior, M.

K. Scherer, E. Braig, K. Willer, M. Willner, A. a. Fingerle, M. Chabior, J. Herzen, M. Eiber, B. Haller, M. Straub, H. Schneider, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Non-invasive Differentiation of Kidney Stone Types using X-ray Dark-Field Radiography,” Sci. Rep. 5, 9527 (2015).
[Crossref] [PubMed]

F. Prade, M. Chabior, F. Malm, C. Grosse, and F. Pfeiffer, “Observing the setting and hardening of cementitious materials by X-ray dark-field radiography,” Cem. Concr. Res. 74, 19–25 (2015).
[Crossref]

M. Chabior, T. Donath, C. David, O. Bunk, M. Schuster, C. Schroer, and F. Pfeiffer, “Beam hardening effects in grating-based x-ray phase-contrast imaging,” Med. Phys. 38, 1189–1195 (2011).
[Crossref] [PubMed]

Cloetens, P.

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. L. Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923 (2007).
[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 13, 6296–6304 (2005).
[Crossref] [PubMed]

David, C.

M. Chabior, T. Donath, C. David, O. Bunk, M. Schuster, C. Schroer, and F. Pfeiffer, “Beam hardening effects in grating-based x-ray phase-contrast imaging,” Med. Phys. 38, 1189–1195 (2011).
[Crossref] [PubMed]

T. Donath, F. Pfeiffer, O. Bunk, C. Grünzweig, E. Hempel, S. Popescu, P. Vock, and C. David, “Toward clinical x-ray phase-contrast ct: Demonstration of enhanced soft-tissue contrast in human specimen,” Invest. Radiol. 45, 445–452 (2010).
[PubMed]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. L. Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923 (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, 258–261 (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 13, 6296–6304 (2005).
[Crossref] [PubMed]

Diaz, A.

Dommann, A.

V. Revol, B. Plank, C. Kottler, J. Kastner, R. Kaufmann, A. Dommann, and A. Neels, “Laminate fibre structure characterisation by orientation-selective X-ray grating interferometry,” Proc. 5th Conf. Ind. Comput. Tomogr.Wels, Austria1, 45–51 (2014).

Donath, T.

M. Chabior, T. Donath, C. David, O. Bunk, M. Schuster, C. Schroer, and F. Pfeiffer, “Beam hardening effects in grating-based x-ray phase-contrast imaging,” Med. Phys. 38, 1189–1195 (2011).
[Crossref] [PubMed]

T. Donath, F. Pfeiffer, O. Bunk, C. Grünzweig, E. Hempel, S. Popescu, P. Vock, and C. David, “Toward clinical x-ray phase-contrast ct: Demonstration of enhanced soft-tissue contrast in human specimen,” Invest. Radiol. 45, 445–452 (2010).
[PubMed]

Duc, G. L.

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. L. Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923 (2007).
[Crossref] [PubMed]

Durst, J.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Eiber, M.

K. Scherer, E. Braig, K. Willer, M. Willner, A. a. Fingerle, M. Chabior, J. Herzen, M. Eiber, B. Haller, M. Straub, H. Schneider, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Non-invasive Differentiation of Kidney Stone Types using X-ray Dark-Field Radiography,” Sci. Rep. 5, 9527 (2015).
[Crossref] [PubMed]

Eickelberg, O.

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. O. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal x-ray dark-field scatter-contrast scanner,” Radiology 269, 427–433 (2013).
[Crossref] [PubMed]

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

A. Velroyen, A. Yaroshenko, D. Hahn, A. Fehringer, A. Tapfer, M. Müller, P. B. Noël, B. Pauwels, A. Sasov, A. O. Yildirim, O. Eickelberg, K. Hellbach, S. D. Auweter, F. G. Meinel, M. F. Reiser, M. Bech, and F. Pfeiffer, “Grating-based X-ray Dark-field Computed Tomography of Living Mice,” EBioMedicine (2015), http://dx.doi.org/10.1016/j.ebiom.2015.08.014 .
[Crossref]

Eikenberry, E. F.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

Fasching, P. a.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Fehringer, A.

A. Velroyen, A. Yaroshenko, D. Hahn, A. Fehringer, A. Tapfer, M. Müller, P. B. Noël, B. Pauwels, A. Sasov, A. O. Yildirim, O. Eickelberg, K. Hellbach, S. D. Auweter, F. G. Meinel, M. F. Reiser, M. Bech, and F. Pfeiffer, “Grating-based X-ray Dark-field Computed Tomography of Living Mice,” EBioMedicine (2015), http://dx.doi.org/10.1016/j.ebiom.2015.08.014 .
[Crossref]

Fingerle, A. a.

K. Scherer, E. Braig, K. Willer, M. Willner, A. a. Fingerle, M. Chabior, J. Herzen, M. Eiber, B. Haller, M. Straub, H. Schneider, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Non-invasive Differentiation of Kidney Stone Types using X-ray Dark-Field Radiography,” Sci. Rep. 5, 9527 (2015).
[Crossref] [PubMed]

Gifford, M.

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

Goodman, J.

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company Publishers, Greenwood Village CO, USA, 2005).

Grosse, C.

F. Prade, M. Chabior, F. Malm, C. Grosse, and F. Pfeiffer, “Observing the setting and hardening of cementitious materials by X-ray dark-field radiography,” Cem. Concr. Res. 74, 19–25 (2015).
[Crossref]

Grünzweig, C.

T. Donath, F. Pfeiffer, O. Bunk, C. Grünzweig, E. Hempel, S. Popescu, P. Vock, and C. David, “Toward clinical x-ray phase-contrast ct: Demonstration of enhanced soft-tissue contrast in human specimen,” Invest. Radiol. 45, 445–452 (2010).
[PubMed]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

Gunzler, T. F.

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Gunzler, O. Kurapova, F. Zontone, A. A. Snigirev, and I. I. Snigireva, “Beryllium parabolic refractive x-ray lenses,” Proc. SPIE 5539, 1–9 (2004).
[Crossref]

Haas, W.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Hahn, D.

A. Velroyen, A. Yaroshenko, D. Hahn, A. Fehringer, A. Tapfer, M. Müller, P. B. Noël, B. Pauwels, A. Sasov, A. O. Yildirim, O. Eickelberg, K. Hellbach, S. D. Auweter, F. G. Meinel, M. F. Reiser, M. Bech, and F. Pfeiffer, “Grating-based X-ray Dark-field Computed Tomography of Living Mice,” EBioMedicine (2015), http://dx.doi.org/10.1016/j.ebiom.2015.08.014 .
[Crossref]

Haller, B.

K. Scherer, E. Braig, K. Willer, M. Willner, A. a. Fingerle, M. Chabior, J. Herzen, M. Eiber, B. Haller, M. Straub, H. Schneider, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Non-invasive Differentiation of Kidney Stone Types using X-ray Dark-Field Radiography,” Sci. Rep. 5, 9527 (2015).
[Crossref] [PubMed]

Hartmann, A.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Hellbach, K.

A. Velroyen, A. Yaroshenko, D. Hahn, A. Fehringer, A. Tapfer, M. Müller, P. B. Noël, B. Pauwels, A. Sasov, A. O. Yildirim, O. Eickelberg, K. Hellbach, S. D. Auweter, F. G. Meinel, M. F. Reiser, M. Bech, and F. Pfeiffer, “Grating-based X-ray Dark-field Computed Tomography of Living Mice,” EBioMedicine (2015), http://dx.doi.org/10.1016/j.ebiom.2015.08.014 .
[Crossref]

Hempel, E.

T. Donath, F. Pfeiffer, O. Bunk, C. Grünzweig, E. Hempel, S. Popescu, P. Vock, and C. David, “Toward clinical x-ray phase-contrast ct: Demonstration of enhanced soft-tissue contrast in human specimen,” Invest. Radiol. 45, 445–452 (2010).
[PubMed]

Herzen, J.

K. Scherer, E. Braig, K. Willer, M. Willner, A. a. Fingerle, M. Chabior, J. Herzen, M. Eiber, B. Haller, M. Straub, H. Schneider, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Non-invasive Differentiation of Kidney Stone Types using X-ray Dark-Field Radiography,” Sci. Rep. 5, 9527 (2015).
[Crossref] [PubMed]

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

Ichikawa, K.

Kastner, J.

V. Revol, B. Plank, C. Kottler, J. Kastner, R. Kaufmann, A. Dommann, and A. Neels, “Laminate fibre structure characterisation by orientation-selective X-ray grating interferometry,” Proc. 5th Conf. Ind. Comput. Tomogr.Wels, Austria1, 45–51 (2014).

Kaufmann, R.

V. Revol, B. Plank, C. Kottler, J. Kastner, R. Kaufmann, A. Dommann, and A. Neels, “Laminate fibre structure characterisation by orientation-selective X-ray grating interferometry,” Proc. 5th Conf. Ind. Comput. Tomogr.Wels, Austria1, 45–51 (2014).

Kawabata, K.

Kemble, C. K.

Kottler, C.

V. Revol, B. Plank, C. Kottler, J. Kastner, R. Kaufmann, A. Dommann, and A. Neels, “Laminate fibre structure characterisation by orientation-selective X-ray grating interferometry,” Proc. 5th Conf. Ind. Comput. Tomogr.Wels, Austria1, 45–51 (2014).

Kraft, P.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

Kuhlmann, M.

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Gunzler, O. Kurapova, F. Zontone, A. A. Snigirev, and I. I. Snigireva, “Beryllium parabolic refractive x-ray lenses,” Proc. SPIE 5539, 1–9 (2004).
[Crossref]

Kurapova, O.

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Gunzler, O. Kurapova, F. Zontone, A. A. Snigirev, and I. I. Snigireva, “Beryllium parabolic refractive x-ray lenses,” Proc. SPIE 5539, 1–9 (2004).
[Crossref]

Lee, W.-K.

Lengeler, B.

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Gunzler, O. Kurapova, F. Zontone, A. A. Snigirev, and I. I. Snigireva, “Beryllium parabolic refractive x-ray lenses,” Proc. SPIE 5539, 1–9 (2004).
[Crossref]

Loewen, R.

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

Lohmann, A. W.

Lynch, S. K.

Malecki, A.

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

A. Malecki, G. Potdevin, and F. Pfeiffer, “Quantitative wave-optical numerical analysis of the dark-field signal in grating-based X-ray interferometry,” EPL 99, 48001 (2012).
[Crossref]

Malm, F.

F. Prade, M. Chabior, F. Malm, C. Grosse, and F. Pfeiffer, “Observing the setting and hardening of cementitious materials by X-ray dark-field radiography,” Cem. Concr. Res. 74, 19–25 (2015).
[Crossref]

Meinel, F. G.

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. O. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal x-ray dark-field scatter-contrast scanner,” Radiology 269, 427–433 (2013).
[Crossref] [PubMed]

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

A. Velroyen, A. Yaroshenko, D. Hahn, A. Fehringer, A. Tapfer, M. Müller, P. B. Noël, B. Pauwels, A. Sasov, A. O. Yildirim, O. Eickelberg, K. Hellbach, S. D. Auweter, F. G. Meinel, M. F. Reiser, M. Bech, and F. Pfeiffer, “Grating-based X-ray Dark-field Computed Tomography of Living Mice,” EBioMedicine (2015), http://dx.doi.org/10.1016/j.ebiom.2015.08.014 .
[Crossref]

Michel, T.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Momose, A.

Morgan, N. Y.

Müller, M.

A. Velroyen, A. Yaroshenko, D. Hahn, A. Fehringer, A. Tapfer, M. Müller, P. B. Noël, B. Pauwels, A. Sasov, A. O. Yildirim, O. Eickelberg, K. Hellbach, S. D. Auweter, F. G. Meinel, M. F. Reiser, M. Bech, and F. Pfeiffer, “Grating-based X-ray Dark-field Computed Tomography of Living Mice,” EBioMedicine (2015), http://dx.doi.org/10.1016/j.ebiom.2015.08.014 .
[Crossref]

Neels, A.

V. Revol, B. Plank, C. Kottler, J. Kastner, R. Kaufmann, A. Dommann, and A. Neels, “Laminate fibre structure characterisation by orientation-selective X-ray grating interferometry,” Proc. 5th Conf. Ind. Comput. Tomogr.Wels, Austria1, 45–51 (2014).

Nikolaou, K.

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. O. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal x-ray dark-field scatter-contrast scanner,” Radiology 269, 427–433 (2013).
[Crossref] [PubMed]

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

Noël, P. B.

K. Scherer, E. Braig, K. Willer, M. Willner, A. a. Fingerle, M. Chabior, J. Herzen, M. Eiber, B. Haller, M. Straub, H. Schneider, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Non-invasive Differentiation of Kidney Stone Types using X-ray Dark-Field Radiography,” Sci. Rep. 5, 9527 (2015).
[Crossref] [PubMed]

A. Velroyen, A. Yaroshenko, D. Hahn, A. Fehringer, A. Tapfer, M. Müller, P. B. Noël, B. Pauwels, A. Sasov, A. O. Yildirim, O. Eickelberg, K. Hellbach, S. D. Auweter, F. G. Meinel, M. F. Reiser, M. Bech, and F. Pfeiffer, “Grating-based X-ray Dark-field Computed Tomography of Living Mice,” EBioMedicine (2015), http://dx.doi.org/10.1016/j.ebiom.2015.08.014 .
[Crossref]

Pai, V.

Pauwels, B.

A. Velroyen, A. Yaroshenko, D. Hahn, A. Fehringer, A. Tapfer, M. Müller, P. B. Noël, B. Pauwels, A. Sasov, A. O. Yildirim, O. Eickelberg, K. Hellbach, S. D. Auweter, F. G. Meinel, M. F. Reiser, M. Bech, and F. Pfeiffer, “Grating-based X-ray Dark-field Computed Tomography of Living Mice,” EBioMedicine (2015), http://dx.doi.org/10.1016/j.ebiom.2015.08.014 .
[Crossref]

Pelzer, G.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Pfeiffer, F.

K. Scherer, E. Braig, K. Willer, M. Willner, A. a. Fingerle, M. Chabior, J. Herzen, M. Eiber, B. Haller, M. Straub, H. Schneider, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Non-invasive Differentiation of Kidney Stone Types using X-ray Dark-Field Radiography,” Sci. Rep. 5, 9527 (2015).
[Crossref] [PubMed]

F. Prade, M. Chabior, F. Malm, C. Grosse, and F. Pfeiffer, “Observing the setting and hardening of cementitious materials by X-ray dark-field radiography,” Cem. Concr. Res. 74, 19–25 (2015).
[Crossref]

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. O. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal x-ray dark-field scatter-contrast scanner,” Radiology 269, 427–433 (2013).
[Crossref] [PubMed]

A. Malecki, G. Potdevin, and F. Pfeiffer, “Quantitative wave-optical numerical analysis of the dark-field signal in grating-based X-ray interferometry,” EPL 99, 48001 (2012).
[Crossref]

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

M. Chabior, T. Donath, C. David, O. Bunk, M. Schuster, C. Schroer, and F. Pfeiffer, “Beam hardening effects in grating-based x-ray phase-contrast imaging,” Med. Phys. 38, 1189–1195 (2011).
[Crossref] [PubMed]

T. Donath, F. Pfeiffer, O. Bunk, C. Grünzweig, E. Hempel, S. Popescu, P. Vock, and C. David, “Toward clinical x-ray phase-contrast ct: Demonstration of enhanced soft-tissue contrast in human specimen,” Invest. Radiol. 45, 445–452 (2010).
[PubMed]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. L. Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923 (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, 258–261 (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 13, 6296–6304 (2005).
[Crossref] [PubMed]

A. Velroyen, A. Yaroshenko, D. Hahn, A. Fehringer, A. Tapfer, M. Müller, P. B. Noël, B. Pauwels, A. Sasov, A. O. Yildirim, O. Eickelberg, K. Hellbach, S. D. Auweter, F. G. Meinel, M. F. Reiser, M. Bech, and F. Pfeiffer, “Grating-based X-ray Dark-field Computed Tomography of Living Mice,” EBioMedicine (2015), http://dx.doi.org/10.1016/j.ebiom.2015.08.014 .
[Crossref]

Plank, B.

V. Revol, B. Plank, C. Kottler, J. Kastner, R. Kaufmann, A. Dommann, and A. Neels, “Laminate fibre structure characterisation by orientation-selective X-ray grating interferometry,” Proc. 5th Conf. Ind. Comput. Tomogr.Wels, Austria1, 45–51 (2014).

Popescu, S.

T. Donath, F. Pfeiffer, O. Bunk, C. Grünzweig, E. Hempel, S. Popescu, P. Vock, and C. David, “Toward clinical x-ray phase-contrast ct: Demonstration of enhanced soft-tissue contrast in human specimen,” Invest. Radiol. 45, 445–452 (2010).
[PubMed]

Potdevin, G.

A. Malecki, G. Potdevin, and F. Pfeiffer, “Quantitative wave-optical numerical analysis of the dark-field signal in grating-based X-ray interferometry,” EPL 99, 48001 (2012).
[Crossref]

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

Prade, F.

F. Prade, M. Chabior, F. Malm, C. Grosse, and F. Pfeiffer, “Observing the setting and hardening of cementitious materials by X-ray dark-field radiography,” Cem. Concr. Res. 74, 19–25 (2015).
[Crossref]

Radicke, M.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Rauh, C.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Reiser, M.

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

Reiser, M. F.

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. O. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal x-ray dark-field scatter-contrast scanner,” Radiology 269, 427–433 (2013).
[Crossref] [PubMed]

A. Velroyen, A. Yaroshenko, D. Hahn, A. Fehringer, A. Tapfer, M. Müller, P. B. Noël, B. Pauwels, A. Sasov, A. O. Yildirim, O. Eickelberg, K. Hellbach, S. D. Auweter, F. G. Meinel, M. F. Reiser, M. Bech, and F. Pfeiffer, “Grating-based X-ray Dark-field Computed Tomography of Living Mice,” EBioMedicine (2015), http://dx.doi.org/10.1016/j.ebiom.2015.08.014 .
[Crossref]

Revol, V.

V. Revol, B. Plank, C. Kottler, J. Kastner, R. Kaufmann, A. Dommann, and A. Neels, “Laminate fibre structure characterisation by orientation-selective X-ray grating interferometry,” Proc. 5th Conf. Ind. Comput. Tomogr.Wels, Austria1, 45–51 (2014).

Rieger, J.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Ritter, A.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Rummeny, E. J.

K. Scherer, E. Braig, K. Willer, M. Willner, A. a. Fingerle, M. Chabior, J. Herzen, M. Eiber, B. Haller, M. Straub, H. Schneider, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Non-invasive Differentiation of Kidney Stone Types using X-ray Dark-Field Radiography,” Sci. Rep. 5, 9527 (2015).
[Crossref] [PubMed]

Ruth, R.

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

Sasov, A.

A. Velroyen, A. Yaroshenko, D. Hahn, A. Fehringer, A. Tapfer, M. Müller, P. B. Noël, B. Pauwels, A. Sasov, A. O. Yildirim, O. Eickelberg, K. Hellbach, S. D. Auweter, F. G. Meinel, M. F. Reiser, M. Bech, and F. Pfeiffer, “Grating-based X-ray Dark-field Computed Tomography of Living Mice,” EBioMedicine (2015), http://dx.doi.org/10.1016/j.ebiom.2015.08.014 .
[Crossref]

Scherer, K.

K. Scherer, E. Braig, K. Willer, M. Willner, A. a. Fingerle, M. Chabior, J. Herzen, M. Eiber, B. Haller, M. Straub, H. Schneider, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Non-invasive Differentiation of Kidney Stone Types using X-ray Dark-Field Radiography,” Sci. Rep. 5, 9527 (2015).
[Crossref] [PubMed]

Schleede, S.

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. O. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal x-ray dark-field scatter-contrast scanner,” Radiology 269, 427–433 (2013).
[Crossref] [PubMed]

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

Schneider, H.

K. Scherer, E. Braig, K. Willer, M. Willner, A. a. Fingerle, M. Chabior, J. Herzen, M. Eiber, B. Haller, M. Straub, H. Schneider, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Non-invasive Differentiation of Kidney Stone Types using X-ray Dark-Field Radiography,” Sci. Rep. 5, 9527 (2015).
[Crossref] [PubMed]

Schroer, C.

M. Chabior, T. Donath, C. David, O. Bunk, M. Schuster, C. Schroer, and F. Pfeiffer, “Beam hardening effects in grating-based x-ray phase-contrast imaging,” Med. Phys. 38, 1189–1195 (2011).
[Crossref] [PubMed]

Schroer, C. G.

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Gunzler, O. Kurapova, F. Zontone, A. A. Snigirev, and I. I. Snigireva, “Beryllium parabolic refractive x-ray lenses,” Proc. SPIE 5539, 1–9 (2004).
[Crossref]

Schulz-Wendtland, R.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Schuster, M.

M. Chabior, T. Donath, C. David, O. Bunk, M. Schuster, C. Schroer, and F. Pfeiffer, “Beam hardening effects in grating-based x-ray phase-contrast imaging,” Med. Phys. 38, 1189–1195 (2011).
[Crossref] [PubMed]

Sievers, P.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Snigirev, A. A.

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Gunzler, O. Kurapova, F. Zontone, A. A. Snigirev, and I. I. Snigireva, “Beryllium parabolic refractive x-ray lenses,” Proc. SPIE 5539, 1–9 (2004).
[Crossref]

Snigireva, I. I.

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Gunzler, O. Kurapova, F. Zontone, A. A. Snigirev, and I. I. Snigireva, “Beryllium parabolic refractive x-ray lenses,” Proc. SPIE 5539, 1–9 (2004).
[Crossref]

Stampanoni, M.

Stein, A. F.

Straub, M.

K. Scherer, E. Braig, K. Willer, M. Willner, A. a. Fingerle, M. Chabior, J. Herzen, M. Eiber, B. Haller, M. Straub, H. Schneider, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Non-invasive Differentiation of Kidney Stone Types using X-ray Dark-Field Radiography,” Sci. Rep. 5, 9527 (2015).
[Crossref] [PubMed]

Strobl, M.

M. Strobl, “General solution for quantitative dark-field contrast imaging with grating interferometers,” Sci. Rep. 4, 7243 (2014).
[Crossref] [PubMed]

Suleski, T. J.

Takeda, M.

Tang, X.

Y. Yang and X. Tang, “The second-order differential phase contrast and its retrieval for imaging with x-ray Talbot interferometry,” Med. Phys. 39, 7237–7253 (2012).
[Crossref] [PubMed]

Tapfer, A.

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. O. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal x-ray dark-field scatter-contrast scanner,” Radiology 269, 427–433 (2013).
[Crossref] [PubMed]

A. Velroyen, A. Yaroshenko, D. Hahn, A. Fehringer, A. Tapfer, M. Müller, P. B. Noël, B. Pauwels, A. Sasov, A. O. Yildirim, O. Eickelberg, K. Hellbach, S. D. Auweter, F. G. Meinel, M. F. Reiser, M. Bech, and F. Pfeiffer, “Grating-based X-ray Dark-field Computed Tomography of Living Mice,” EBioMedicine (2015), http://dx.doi.org/10.1016/j.ebiom.2015.08.014 .
[Crossref]

Terui, Y.

Thieme, S. F.

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

Uder, M.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Velroyen, A.

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. O. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal x-ray dark-field scatter-contrast scanner,” Radiology 269, 427–433 (2013).
[Crossref] [PubMed]

A. Velroyen, A. Yaroshenko, D. Hahn, A. Fehringer, A. Tapfer, M. Müller, P. B. Noël, B. Pauwels, A. Sasov, A. O. Yildirim, O. Eickelberg, K. Hellbach, S. D. Auweter, F. G. Meinel, M. F. Reiser, M. Bech, and F. Pfeiffer, “Grating-based X-ray Dark-field Computed Tomography of Living Mice,” EBioMedicine (2015), http://dx.doi.org/10.1016/j.ebiom.2015.08.014 .
[Crossref]

Vock, P.

T. Donath, F. Pfeiffer, O. Bunk, C. Grünzweig, E. Hempel, S. Popescu, P. Vock, and C. David, “Toward clinical x-ray phase-contrast ct: Demonstration of enhanced soft-tissue contrast in human specimen,” Invest. Radiol. 45, 445–452 (2010).
[PubMed]

Wachter, D. L.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Weber, T.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

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, 258–261 (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 13, 6296–6304 (2005).
[Crossref] [PubMed]

Wen, H. H.

Wenkel, E.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Willer, K.

K. Scherer, E. Braig, K. Willer, M. Willner, A. a. Fingerle, M. Chabior, J. Herzen, M. Eiber, B. Haller, M. Straub, H. Schneider, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Non-invasive Differentiation of Kidney Stone Types using X-ray Dark-Field Radiography,” Sci. Rep. 5, 9527 (2015).
[Crossref] [PubMed]

Willner, M.

K. Scherer, E. Braig, K. Willer, M. Willner, A. a. Fingerle, M. Chabior, J. Herzen, M. Eiber, B. Haller, M. Straub, H. Schneider, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Non-invasive Differentiation of Kidney Stone Types using X-ray Dark-Field Radiography,” Sci. Rep. 5, 9527 (2015).
[Crossref] [PubMed]

Xiao, X.

Yang, Y.

Y. Yang and X. Tang, “The second-order differential phase contrast and its retrieval for imaging with x-ray Talbot interferometry,” Med. Phys. 39, 7237–7253 (2012).
[Crossref] [PubMed]

Yaroshenko, A.

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. O. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal x-ray dark-field scatter-contrast scanner,” Radiology 269, 427–433 (2013).
[Crossref] [PubMed]

A. Velroyen, A. Yaroshenko, D. Hahn, A. Fehringer, A. Tapfer, M. Müller, P. B. Noël, B. Pauwels, A. Sasov, A. O. Yildirim, O. Eickelberg, K. Hellbach, S. D. Auweter, F. G. Meinel, M. F. Reiser, M. Bech, and F. Pfeiffer, “Grating-based X-ray Dark-field Computed Tomography of Living Mice,” EBioMedicine (2015), http://dx.doi.org/10.1016/j.ebiom.2015.08.014 .
[Crossref]

Yashiro, W.

Yildirim, A.

S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U.S.A. 109, 17880–17885 (2012).
[Crossref] [PubMed]

Yildirim, A. O.

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. O. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal x-ray dark-field scatter-contrast scanner,” Radiology 269, 427–433 (2013).
[Crossref] [PubMed]

A. Velroyen, A. Yaroshenko, D. Hahn, A. Fehringer, A. Tapfer, M. Müller, P. B. Noël, B. Pauwels, A. Sasov, A. O. Yildirim, O. Eickelberg, K. Hellbach, S. D. Auweter, F. G. Meinel, M. F. Reiser, M. Bech, and F. Pfeiffer, “Grating-based X-ray Dark-field Computed Tomography of Living Mice,” EBioMedicine (2015), http://dx.doi.org/10.1016/j.ebiom.2015.08.014 .
[Crossref]

Zanette, I.

I. Zanette, “Interférométrie Xà réseaux pour l imagerie et l’analyse de front d’ondes au synchrotron,” Ph.D. thesis (2011).

Zang, A.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. a. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Ziegler, E.

Zontone, F.

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Gunzler, O. Kurapova, F. Zontone, A. A. Snigirev, and I. I. Snigireva, “Beryllium parabolic refractive x-ray lenses,” Proc. SPIE 5539, 1–9 (2004).
[Crossref]

Appl. Opt. (3)

Cem. Concr. Res. (1)

F. Prade, M. Chabior, F. Malm, C. Grosse, and F. Pfeiffer, “Observing the setting and hardening of cementitious materials by X-ray dark-field radiography,” Cem. Concr. Res. 74, 19–25 (2015).
[Crossref]

EPL (1)

A. Malecki, G. Potdevin, and F. Pfeiffer, “Quantitative wave-optical numerical analysis of the dark-field signal in grating-based X-ray interferometry,” EPL 99, 48001 (2012).
[Crossref]

Invest. Radiol. (1)

T. Donath, F. Pfeiffer, O. Bunk, C. Grünzweig, E. Hempel, S. Popescu, P. Vock, and C. David, “Toward clinical x-ray phase-contrast ct: Demonstration of enhanced soft-tissue contrast in human specimen,” Invest. Radiol. 45, 445–452 (2010).
[PubMed]

Jpn. J. Appl. Phys. (1)

A. Momose, “Recent Advances in X-ray Phase Imaging,” Jpn. J. Appl. Phys. 44, 6355–6367 (2005).
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Med. Phys. (2)

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F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
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Nature Phys. (1)

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Figures (6)

Fig. 1
Fig. 1 Sketch of an X-ray grating interferometry (XGI) imaging setup with a three-grating Talbot-Lau interferometer. While G0 enables the use of a standard X-ray source by increasing the spatial coherence, a defined phase-shift introduced by G1 creates a periodic interference pattern at certain distances downstream. Generally, the period of this interference pattern is much smaller than the size of the detector pixels and therefore cannot be resolved directly. The analyzer grating G2 has typically a period matching the interference pattern. It is moved in a number of equidistant steps over one period of G2 perpendicular to the grating bars. For each step the intensity at the detector is recorded and, thus, the pattern is translated into a intensity curve (”stepping curve”) in each detector pixel.
Fig. 2
Fig. 2 Effect of period mismatch between interference pattern and G2 on the stepping curve. In XGI the use of an analyzer grating relies on the fact that the grating period (black bars) matches the period of the interference pattern (light gray bars). Then in an optimal case (a), there is a step of maximum intensity (step 0) and a step of minimum intensity (step 4, remaining intensity depends on blocking capacity and height of grating material). This leads to a maximum visibility value (blue stepping curve plotted over two grating periods (c)). If the periods of the grating and the interference pattern do not match (b), no perfect overlap is possible and the visibility is reduced (red curve). (Note: For cases with detector pixels covering only very few periods even an increased visibility is possible.)
Fig. 3
Fig. 3 Definition of relevant parameters used for Eqs. (6)(11) and Eqs. (14)(22) presented in the appendix. p1 (here period of the intensity pattern) and p2 (here period of G2) denote the smaller and the larger period, respectively. n specifies the half-periods of p1. Even numbers stand for minima of the interference pattern (as in this example) or grating bars. Odd numbers define maxima or grating spaces. nmax is the highest number that still lies within the detector pixel (dashed line). The intensity is calculated based on the portions of the inference maxima that are not blocked by the grating (sn). In addition to the periods, the distance to the pixel border (lpix) and the relative displacement of the periods (ldiff) are input parameters. dn and c are values necessary to calculate sn.
Fig. 4
Fig. 4 Effect of period mismatch between interference pattern and G2 on the dark-field contrast (DFC). (a) shows calculations based on Eq. (6), where the period of G2 was set to 2.4 μm and the period of the interference pattern was varied. Both, smaller and larger periods of the interference pattern lead to decreased visibility and consequently a DFC signal, relative to the case of matching periods. The influence of the detector pixel size is shown in (b) for the specific case with periods of 2.37 and 2.4 μm. Both, in (a) and (b), points of zero visibility occur when the resulting moiré pattern period has the same size as the detector pixel or a multiple thereof. The error bars indicate the range of possible DFC values for variation of the positions of interference pattern and grating.
Fig. 5
Fig. 5 Simulations for a parabolic beryllium X-ray lens with a radius of 50 μm [25]. Panel (a) presents a line plot of the wave-front intensity at the detector. The wave-front shows the characteristic interference pattern of high and low intensity (b), which is visible at fractional Talbot distances. In the simulated configuration the interference has a period of 2.4 μm outside of the lens. The lens causes strong distortions at its edges (b). The parabolic shape of the lens leads to a constant compression of the interference pattern. The distances between the maxima (marked by red triangles in (a)) behind the lens therefore measure only 2.37 μm, as shown in (c). Binning of the wave-front into detector pixels of 50 μm size (indicated by dashed lines) leads to an almost constant DFC signal behind the lens (d).
Fig. 6
Fig. 6 DFC signal from the edge of an epoxy cylinder. The cylinder covers only the left part of the image. Experimental measurement (a) and simulation (b) demonstrate that a DFC signal can be detected over several pixels at the edge of the cylinder. As reported in [20] the signal decreases slower towards the center of the object than at the other side of the edge (c, arrows). The signal larger than unity in the area behind the cylinder in the simulation with polychromatic radiation is most likely caused by beam hardening, as simulations with monochromatic radiation and a purely phase-shifting cylinder do not show this behavior. Simulations of the wave-front reveal that two effects are responsible for the signal: the strong disturbance of the pattern directly behind the edge (d), and the increased period of the interference pattern due to the curvature of the cylinder acting as a diverging lens for the X-rays (e). The point spread function smears the signal out over a few pixels.

Equations (22)

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x y I x y Φ + I x y 2 Φ + I z = 0 .
T = a 0 s a 0 r ,
P = arg ( a 1 s a 1 r ) ,
D = V s V r = | a 1 s | | a 0 r | | a 0 s | | a 1 r | ,
p moire = p 1 p 2 p 2 p 1 .
I pixel = i factor n = 1 n max s n with n odd ,
n max = 2 ( w pix l pix ) p 1 ,
s n = | p 1 2 d n | p 2 p 1 2 m with m = { 0 if d n p 1 2 1 if d n > p 1 2 ,
d n = [ ( n c ) p 2 p 1 2 + l offset ] mod p 2 ,
l offset = { ( i step p 2 N steps + l diff ) mod p 2 initially , s n ^ p 1 2 for n > n ^ if s n ^ > p 1 2 s n ^ p 1 2 ,
c = { 0 initially , n ^ + 1 for n > n ^ if s n ^ > p 1 2 s n ^ p 1 2 .
f = R 2 δ ,
p = p 1 2 f d f ,
s 1 * = { l pix , if d 1 p 1 2 and s 1 > l pix , s 1 ( p 1 2 l pix ) , if d 1 > p 1 2 .
r n max = ( ( w pix l pix ) mod p 1 2 ) / p 1 2
s n max * = { s n max p 1 2 ( 1 r max ) , if d n max p 1 2 , p 1 2 r n max , if d n max > p 1 2 and s n max > p 1 2 r n max .
l pix , i = l pix + i step p 1 N steps .
l pix , i * = l pix , i p 1 ,
l pix * = l pix p 1 ,
l diff * = l diff ( p 2 p 1 ) .
l diff , i = ( l diff i step p 1 N steps ) mod p 2 ,
l offset , i = l diff , i .

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