Abstract

X-ray grating interferometry has been highlighted in the last decade as a multi-modal X-ray phase-imaging technique for providing absorption, differential phase, and visibility-contrast images. It has been mainly reported that the visibility contrast in the visibility-contrast image originates from unresolvable random microstructures. In this paper, we show that the visibility contrast is even reduced by a uniform sample with flat surfaces due to the so-called “beam-hardening effect”, which has to be taken into account when X-rays with a continuous spectrum is used. We drive a criterion for determining whether the beam-hardening effect occurs or not, and propose a method for correcting the effect of beam hardening on a visibility-contrast image.

© 2015 Optical Society of America

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References

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  1. C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
    [Crossref]
  2. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
    [Crossref]
  3. T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (2005).
    [Crossref]
  4. 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]
  5. A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
    [Crossref]
  6. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
    [Crossref]
  7. A. Momose, W. Yashiro, and Y. Takeda, “X-ray phase imaging with Talbot interferometry,” in, Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems, Y. Censor, M. Jiang, and G. Wang, eds. (Medical Physics Publishing, 2010), pp. 281–320.
  8. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
    [Crossref] [PubMed]
  9. W. Yashiro, Y. Terui, K. Kawabara, and A. Momose, “On the origin of visibility contrast in x-ray Talbot interferometry,” Opt. Express 18, 16890–16901 (2010).
    [Crossref] [PubMed]
  10. W. Yashiro and A. Momose, “Effects of unresolvable edges in grating-based X-ray differential phase imaging,” Opt. Express 23, 9233–9251 (2015).
    [Crossref] [PubMed]
  11. A. Momose, W. Yashiro, H. Kuwabara, and K. Kawabata, “Grating-based X-ray phase imaging using multiline X-ray source,” Jpn. J. Appl. Phys. 48, 076512 (2009).
    [Crossref]
  12. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).
  13. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, XXVII, E. Wolf, ed. (Elsevier, 1989) pp. 1–108.
    [Crossref]
  14. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [Crossref] [PubMed]
  15. H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley Interscience, 2007), pp. 547–666.
    [Crossref]
  16. E. Hack and J. Burke, “Invited review article: measurement uncertainty of linear phase-stepping algorithms,” Rev. Sci. Instrum. 82, 061101 (2011).
    [Crossref] [PubMed]
  17. R. A. Brooks and G. Di Chiro, “Beam hardening in X-ray reconstructive tomography,” Phys. Med. Biol. 21, 390–398 (1976).
    [Crossref] [PubMed]
  18. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1989).
  19. 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]
  20. M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
    [Crossref] [PubMed]
  21. R. H. Redus, J. A. Pantazis, T. J. Pantazis, A. C. Huber, and B. J. Cross, “Characterization of CdTe detectors for quantitative X-ray spectroscopy,” IEEE Trans. Nucl. Sci. 56, 2524–2532 (2009).
    [Crossref]

2015 (1)

2011 (2)

E. Hack and J. Burke, “Invited review article: measurement uncertainty of linear phase-stepping algorithms,” Rev. Sci. Instrum. 82, 061101 (2011).
[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]

2010 (2)

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[Crossref] [PubMed]

W. Yashiro, Y. Terui, K. Kawabara, and A. Momose, “On the origin of visibility contrast in x-ray Talbot interferometry,” Opt. Express 18, 16890–16901 (2010).
[Crossref] [PubMed]

2009 (2)

R. H. Redus, J. A. Pantazis, T. J. Pantazis, A. C. Huber, and B. J. Cross, “Characterization of CdTe detectors for quantitative X-ray spectroscopy,” IEEE Trans. Nucl. Sci. 56, 2524–2532 (2009).
[Crossref]

A. Momose, W. Yashiro, H. Kuwabara, and K. Kawabata, “Grating-based X-ray phase imaging using multiline X-ray source,” Jpn. J. Appl. Phys. 48, 076512 (2009).
[Crossref]

2008 (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

2006 (2)

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[Crossref]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[Crossref]

2005 (2)

T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (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 13, 6296–6304 (2005).
[Crossref] [PubMed]

2003 (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[Crossref]

2002 (1)

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
[Crossref]

1976 (1)

R. A. Brooks and G. Di Chiro, “Beam hardening in X-ray reconstructive tomography,” Phys. Med. Biol. 21, 390–398 (1976).
[Crossref] [PubMed]

1974 (1)

1836 (1)

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).

Bech, M.

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[Crossref] [PubMed]

Brangaccio, D. J.

Brooks, R. A.

R. A. Brooks and G. Di Chiro, “Beam hardening in X-ray reconstructive tomography,” Phys. Med. Biol. 21, 390–398 (1976).
[Crossref] [PubMed]

Bruning, J. H.

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]

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[Crossref] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[Crossref]

Burke, J.

E. Hack and J. Burke, “Invited review article: measurement uncertainty of linear phase-stepping algorithms,” Rev. Sci. Instrum. 82, 061101 (2011).
[Crossref] [PubMed]

Chabior, 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]

Cloetens, P.

Cross, B. J.

R. H. Redus, J. A. Pantazis, T. J. Pantazis, A. C. Huber, and B. J. Cross, “Characterization of CdTe detectors for quantitative X-ray spectroscopy,” IEEE Trans. Nucl. Sci. 56, 2524–2532 (2009).
[Crossref]

David, C

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[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]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[Crossref]

T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (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 13, 6296–6304 (2005).
[Crossref] [PubMed]

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
[Crossref]

Di Chiro, G.

R. A. Brooks and G. Di Chiro, “Beam hardening in X-ray reconstructive tomography,” Phys. Med. Biol. 21, 390–398 (1976).
[Crossref] [PubMed]

Diaz, A.

T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (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 13, 6296–6304 (2005).
[Crossref] [PubMed]

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]

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[Crossref] [PubMed]

Feidenhans’l, R.

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[Crossref] [PubMed]

Gallagher, J. E.

Hack, E.

E. Hack and J. Burke, “Invited review article: measurement uncertainty of linear phase-stepping algorithms,” Rev. Sci. Instrum. 82, 061101 (2011).
[Crossref] [PubMed]

Hamaishi, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[Crossref]

Hattori, T.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[Crossref]

Herriott, D. R.

Huber, A. C.

R. H. Redus, J. A. Pantazis, T. J. Pantazis, A. C. Huber, and B. J. Cross, “Characterization of CdTe detectors for quantitative X-ray spectroscopy,” IEEE Trans. Nucl. Sci. 56, 2524–2532 (2009).
[Crossref]

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1989).

Kawabara, K.

Kawabata, K.

A. Momose, W. Yashiro, H. Kuwabara, and K. Kawabata, “Grating-based X-ray phase imaging using multiline X-ray source,” Jpn. J. Appl. Phys. 48, 076512 (2009).
[Crossref]

Kawamoto, S.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[Crossref]

Koyama, I.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[Crossref]

Kuwabara, H.

A. Momose, W. Yashiro, H. Kuwabara, and K. Kawabata, “Grating-based X-ray phase imaging using multiline X-ray source,” Jpn. J. Appl. Phys. 48, 076512 (2009).
[Crossref]

Momose, A.

W. Yashiro and A. Momose, “Effects of unresolvable edges in grating-based X-ray differential phase imaging,” Opt. Express 23, 9233–9251 (2015).
[Crossref] [PubMed]

W. Yashiro, Y. Terui, K. Kawabara, and A. Momose, “On the origin of visibility contrast in x-ray Talbot interferometry,” Opt. Express 18, 16890–16901 (2010).
[Crossref] [PubMed]

A. Momose, W. Yashiro, H. Kuwabara, and K. Kawabata, “Grating-based X-ray phase imaging using multiline X-ray source,” Jpn. J. Appl. Phys. 48, 076512 (2009).
[Crossref]

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[Crossref]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[Crossref]

A. Momose, W. Yashiro, and Y. Takeda, “X-ray phase imaging with Talbot interferometry,” in, Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems, Y. Censor, M. Jiang, and G. Wang, eds. (Medical Physics Publishing, 2010), pp. 281–320.

Nöhammer, B.

T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (2005).
[Crossref]

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
[Crossref]

Pantazis, J. A.

R. H. Redus, J. A. Pantazis, T. J. Pantazis, A. C. Huber, and B. J. Cross, “Characterization of CdTe detectors for quantitative X-ray spectroscopy,” IEEE Trans. Nucl. Sci. 56, 2524–2532 (2009).
[Crossref]

Pantazis, T. J.

R. H. Redus, J. A. Pantazis, T. J. Pantazis, A. C. Huber, and B. J. Cross, “Characterization of CdTe detectors for quantitative X-ray spectroscopy,” IEEE Trans. Nucl. Sci. 56, 2524–2532 (2009).
[Crossref]

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, XXVII, E. Wolf, ed. (Elsevier, 1989) pp. 1–108.
[Crossref]

Pfeiffer, F.

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]

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[Crossref] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. 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]

Redus, R. H.

R. H. Redus, J. A. Pantazis, T. J. Pantazis, A. C. Huber, and B. J. Cross, “Characterization of CdTe detectors for quantitative X-ray spectroscopy,” IEEE Trans. Nucl. Sci. 56, 2524–2532 (2009).
[Crossref]

Rosenfeld, D. P.

Schreiber, H.

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley Interscience, 2007), pp. 547–666.
[Crossref]

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]

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]

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1989).

Solak, H. H.

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
[Crossref]

Stampanoni, M.

Suzuki, Y.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[Crossref]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[Crossref]

Takai, K.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[Crossref]

Takeda, Y.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[Crossref]

A. Momose, W. Yashiro, and Y. Takeda, “X-ray phase imaging with Talbot interferometry,” in, Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems, Y. Censor, M. Jiang, and G. Wang, eds. (Medical Physics Publishing, 2010), pp. 281–320.

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).

Terui, Y.

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[Crossref]

T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (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 13, 6296–6304 (2005).
[Crossref] [PubMed]

White, A. D.

Yashiro, W.

W. Yashiro and A. Momose, “Effects of unresolvable edges in grating-based X-ray differential phase imaging,” Opt. Express 23, 9233–9251 (2015).
[Crossref] [PubMed]

W. Yashiro, Y. Terui, K. Kawabara, and A. Momose, “On the origin of visibility contrast in x-ray Talbot interferometry,” Opt. Express 18, 16890–16901 (2010).
[Crossref] [PubMed]

A. Momose, W. Yashiro, H. Kuwabara, and K. Kawabata, “Grating-based X-ray phase imaging using multiline X-ray source,” Jpn. J. Appl. Phys. 48, 076512 (2009).
[Crossref]

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[Crossref]

A. Momose, W. Yashiro, and Y. Takeda, “X-ray phase imaging with Talbot interferometry,” in, Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems, Y. Censor, M. Jiang, and G. Wang, eds. (Medical Physics Publishing, 2010), pp. 281–320.

Ziegler, E.

T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (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 13, 6296–6304 (2005).
[Crossref] [PubMed]

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

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IEEE Trans. Nucl. Sci. (1)

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Jpn. J. Appl. Phys. (3)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
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A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
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A. Momose, W. Yashiro, H. Kuwabara, and K. Kawabata, “Grating-based X-ray phase imaging using multiline X-ray source,” Jpn. J. Appl. Phys. 48, 076512 (2009).
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Med. Phys. (1)

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).
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Nat. Mater. (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
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Nat. Phys. (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
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Philos. Mag. (1)

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[Crossref]

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley Interscience, 2007), pp. 547–666.
[Crossref]

A. Momose, W. Yashiro, and Y. Takeda, “X-ray phase imaging with Talbot interferometry,” in, Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems, Y. Censor, M. Jiang, and G. Wang, eds. (Medical Physics Publishing, 2010), pp. 281–320.

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Figures (6)

Fig. 1
Fig. 1 Experimental setup of X-ray Talbot-Lau interferometer.
Fig. 2
Fig. 2 Left figures: (a) transmittance (log(I/I0)), (b) moiré-phase, and (c) normalized-visibility (log(V/V0)) images near an edge of a single-crystalline silicon block with two orthogonal faces inclined at an angle of 45 degrees with respect to the optical axis, as shown in Fig. 1. Right graphs: thickness dependences experimentally obtained from the line profiles along the red lines in the left images (filled circles) and results of numerical calculations (green lines).
Fig. 3
Fig. 3 Dependences of normalized visibility (V/V0) on transmittance (I/I0) for several materials. Black line and blue line: experiment and calculation results for silicon shown in Section 2; red, yellow, green, and gray broken lines: results of calculations for gold, copper, PMMA, and silver.
Fig. 4
Fig. 4 (a) Simple models of intensity S(λ) averaged over a fringe scanning and the visibility V0(λ) of the intensity oscillation during the fringe scanning with no sample present (λ : X-ray wavelength; both S(λ) and V0(λ) can be given for a pixel), and (b) T″(λ0)/T(λ0) as a function of ρtC.
Fig. 5
Fig. 5 V/V0-normalized I/I0 graph for the results of the calculations shown in Fig. 3 for silicon, gold, copper, PMMA, and silver.
Fig. 6
Fig. 6 Experimentally obtained thickness dependence of normalized visibility for silicon (black filled circles) and that after beam-hardening correction by using the universal curve in Fig. 5 (orange filled circles).

Tables (1)

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Table 1 Critical thicknesses for several materials for λ0 = 0.5Å.

Equations (13)

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V V 0 = V 0 ( λ ) T S V 0 ( λ ) S ,
1 1 2 T ( λ 0 ) T ( λ 0 ) ( ( Δ λ ) 2 S ( Δ λ ) 2 V 0 S ) ,
f ( λ ) G f ( λ ) G ( λ ) d λ / G ( λ ) d λ ,
Δ λ λ λ 0 ,
C var ( Δ λ ) 2 S ( Δ λ ) 2 V 0 S
T ( λ 0 ) exp [ ρ t C λ 0 3 ] ,
T ( λ 0 ) T ( λ 0 ) 6 ρ t C λ 0 3 9 ( ρ t C λ 0 3 ) 2 λ 0 2 ,
I I 0 T ( λ 0 ) ( 1 + 1 2 T ( λ 0 ) T ( λ 0 ) ( Δ λ ) 2 S ) .
T ( λ 0 ) = 0 ,
I I 0 T ( λ 0 ) ,
e 2 / 3 ,
0.5 .
T ( λ 0 ; t c ) = 0.5 .

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