Abstract

The reduction in visibility in x-ray grating interferometry based on the Talbot effect is formulated by the autocorrelation function of spatial fluctuations of a wavefront due to unresolved micron-size structures in samples. The experimental results for microspheres and melamine sponge were successfully explained by this formula with three parameters characterizing the wavefront fluctuations: variance, correlation length, and the Hurst exponent. The ultra-small-angle x-ray scattering of these samples was measured, and the scattering profiles were consistent with the formulation. Furthermore, we discuss the relation between the three parameters and the features of the micron-sized structures. The visibility-reduction contrast observed by x-ray grating interferometry can thus be understood in relation to the structural parameters of the microstructures.

© 2010 Optical Society of America

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  24. Here, we defined the pth Talbot order as it expresses the pth position where the electric field just behind the first grating is perfectly reproduced for any grating. For plane-wave illumination, the pth Talbot order corresponds to the position of zT = pd2/⌊. This definition is convenient because it is independent of what kind of grating is used as the first grating. Note that, once zT is given by pd2/⌊, we can use p for specifying any position behind the first grating instead of zT. The analytical calculations presented in this paper can be applied to any periodic image generated behind the first grating at any position of p (> 0).
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2009 (8)

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, "Bulk magnetic domain structures visualized by neutron dark-field imaging," Appl. Phys. Lett. 93,112504 (2009).
[CrossRef]

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, "X-ray dark-field and phase-contrast imaging using a grating interferometer," J. Appl. Phys. 105,102006 (2009).
[CrossRef]

S. A. MacDonald, F. Marone, C. Hintermüller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, "Advanced phase-contrast imaging using a grating interferometer," J. Synchrotron Radiation 16,562-572 (2009).
[CrossRef]

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, "Fourier x-ray scattering radiography yields bone structural information," Radiography 251,910-918 (2009).

Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, "Quantitative grating-based x-ray dark-field computed tomography," Appl. Phys. Lett. 95,094105 (2009).
[CrossRef]

Z.-F. Huang, K.-J. Kang, L. Zhang, Z.-Q. Chen, F. Ding, Z.-T. Wang, and Q.-G. Fang, "Alternative method for differential phase-contrast imaging with weakly coherent hard x rays," Phys. Rev. A 79,013815 (2009).
[CrossRef]

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, "A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously," Appl. Phys. Lett. 94,044108 (2009).
[CrossRef]

W. Yashiro, Y. Takeda, A. Takeuchi, Y. Suzuki, and A. Momose, "Hard x-ray phase-difference microscopy using a Fresnel zone plate and a transmission grating," Phys. Rev. Lett. 103,180801 (2009).
[CrossRef] [PubMed]

2008 (8)

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei G. Kühne, J. Kohlbrecher, R. Schäfer, P. Lejcek, H. M. R. Ronnow, and F. Pfeiffer, "Neutron decoherence imaging for visualizing bulk magnetic domain structures," Phys. Rev. Lett. 101,025504 (2008).
[CrossRef] [PubMed]

M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, "Neutron dark-field tomography," Phys. Rev. Lett. 101,123902 (2008).
[CrossRef] [PubMed]

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bosecke, and M. Giglio, "X-ray-scattering information obtained from near-field speckle," Nat. Phys. 4,238-243 (2008).
[CrossRef]

Y. I. Nesterets, "On the origins of decoherence and extinction contrast in phase-contrast imaging," Opt. Commun. 281,533-542 (2008).
[CrossRef]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. H. Brönnimann, C. Grünzweig, and C. David, "Hardx-ray dark-field imaging using a grating interferometer," Nat. Mat. 7,134-137 (2008).
[CrossRef]

Y. I. Nesterets and S. W. Wilkins, "Phase-contrast imaging using a scanning-doublegrating configuration," Opt. Express 16,5849-5867 (2008).
[CrossRef] [PubMed]

Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, "Differential phase x-ray imaging microscopy with x-ray Talbot interferometer," Appl. Phys. Express 1,117002 (2008).
[CrossRef]

W. Yashiro, Y. Takeda, and A. Momose, "Efficiency of capturing a phase image using conebeam x-ray Talbot interferometry," J. Opt. Soc. Am. A 25,2025-2039 (2008).
[CrossRef]

2007 (1)

A. Olivo and R. Speller, "A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources," Appl. Phys. Lett. 91,074106 (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," Nat. Phys. 2,258-261 (2006).
[CrossRef]

2005 (1)

A. Momose, "Recent advances in x-ray phase imaging," Jpn. J. Appl. Phys. 44,6355-6367 (2005).
[CrossRef]

2003 (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]

I. A. Vartanyants and I. K. Robinson, "Origins of decoherence in coherent x-ray diffraction experiments," Opt. Commun. 222,29-50 (2003).
[CrossRef]

2000 (1)

R. Fitzgerald, "Phase-sensitive x-ray imaging," Phys. Today 53,23-26 (2000).
[CrossRef]

1988 (1)

S. K. Sinha, E. B. Sirota, and S. Garoff, "X-ray and neutron scattering from rough surfaces," Phys. Rev. B 38,2297-2312 (1988).
[CrossRef]

1974 (1)

Bech, M.

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, "X-ray dark-field and phase-contrast imaging using a grating interferometer," J. Appl. Phys. 105,102006 (2009).
[CrossRef]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. H. Brönnimann, C. Grünzweig, and C. David, "Hardx-ray dark-field imaging using a grating interferometer," Nat. Mat. 7,134-137 (2008).
[CrossRef]

Bennett, E. E.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, "Fourier x-ray scattering radiography yields bone structural information," Radiography 251,910-918 (2009).

Bohndiek, S. E.

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, "A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously," Appl. Phys. Lett. 94,044108 (2009).
[CrossRef]

Bosecke, P.

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bosecke, and M. Giglio, "X-ray-scattering information obtained from near-field speckle," Nat. Phys. 4,238-243 (2008).
[CrossRef]

Brangaccio, D. J.

Brönnimann, C. H.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. H. Brönnimann, C. Grünzweig, and C. David, "Hardx-ray dark-field imaging using a grating interferometer," Nat. Mat. 7,134-137 (2008).
[CrossRef]

Bruning, J. H.

Bunk, O.

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, "X-ray dark-field and phase-contrast imaging using a grating interferometer," J. Appl. Phys. 105,102006 (2009).
[CrossRef]

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, "Bulk magnetic domain structures visualized by neutron dark-field imaging," Appl. Phys. Lett. 93,112504 (2009).
[CrossRef]

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei G. Kühne, J. Kohlbrecher, R. Schäfer, P. Lejcek, H. M. R. Ronnow, and F. Pfeiffer, "Neutron decoherence imaging for visualizing bulk magnetic domain structures," Phys. Rev. Lett. 101,025504 (2008).
[CrossRef] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. H. Brönnimann, C. Grünzweig, and C. David, "Hardx-ray dark-field imaging using a grating interferometer," Nat. Mat. 7,134-137 (2008).
[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]

Cerbino, R.

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bosecke, and M. Giglio, "X-ray-scattering information obtained from near-field speckle," Nat. Phys. 4,238-243 (2008).
[CrossRef]

Chen, Z.-Q.

Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, "Quantitative grating-based x-ray dark-field computed tomography," Appl. Phys. Lett. 95,094105 (2009).
[CrossRef]

Z.-F. Huang, K.-J. Kang, L. Zhang, Z.-Q. Chen, F. Ding, Z.-T. Wang, and Q.-G. Fang, "Alternative method for differential phase-contrast imaging with weakly coherent hard x rays," Phys. Rev. A 79,013815 (2009).
[CrossRef]

David, C.

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, "Bulk magnetic domain structures visualized by neutron dark-field imaging," Appl. Phys. Lett. 93,112504 (2009).
[CrossRef]

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, "X-ray dark-field and phase-contrast imaging using a grating interferometer," J. Appl. Phys. 105,102006 (2009).
[CrossRef]

S. A. MacDonald, F. Marone, C. Hintermüller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, "Advanced phase-contrast imaging using a grating interferometer," J. Synchrotron Radiation 16,562-572 (2009).
[CrossRef]

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei G. Kühne, J. Kohlbrecher, R. Schäfer, P. Lejcek, H. M. R. Ronnow, and F. Pfeiffer, "Neutron decoherence imaging for visualizing bulk magnetic domain structures," Phys. Rev. Lett. 101,025504 (2008).
[CrossRef] [PubMed]

M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, "Neutron dark-field tomography," Phys. Rev. Lett. 101,123902 (2008).
[CrossRef] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. H. Brönnimann, C. Grünzweig, and C. David, "Hardx-ray dark-field imaging using a grating interferometer," Nat. Mat. 7,134-137 (2008).
[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]

Dierolf, M.

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, "Bulk magnetic domain structures visualized by neutron dark-field imaging," Appl. Phys. Lett. 93,112504 (2009).
[CrossRef]

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei G. Kühne, J. Kohlbrecher, R. Schäfer, P. Lejcek, H. M. R. Ronnow, and F. Pfeiffer, "Neutron decoherence imaging for visualizing bulk magnetic domain structures," Phys. Rev. Lett. 101,025504 (2008).
[CrossRef] [PubMed]

Ding, F.

Z.-F. Huang, K.-J. Kang, L. Zhang, Z.-Q. Chen, F. Ding, Z.-T. Wang, and Q.-G. Fang, "Alternative method for differential phase-contrast imaging with weakly coherent hard x rays," Phys. Rev. A 79,013815 (2009).
[CrossRef]

Donath, T.

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, "X-ray dark-field and phase-contrast imaging using a grating interferometer," J. Appl. Phys. 105,102006 (2009).
[CrossRef]

Eikenberry, E. F.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. H. Brönnimann, C. Grünzweig, and C. David, "Hardx-ray dark-field imaging using a grating interferometer," Nat. Mat. 7,134-137 (2008).
[CrossRef]

Fang, Q.-G.

Z.-F. Huang, K.-J. Kang, L. Zhang, Z.-Q. Chen, F. Ding, Z.-T. Wang, and Q.-G. Fang, "Alternative method for differential phase-contrast imaging with weakly coherent hard x rays," Phys. Rev. A 79,013815 (2009).
[CrossRef]

Fitzgerald, R.

R. Fitzgerald, "Phase-sensitive x-ray imaging," Phys. Today 53,23-26 (2000).
[CrossRef]

Frei, G.

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, "Bulk magnetic domain structures visualized by neutron dark-field imaging," Appl. Phys. Lett. 93,112504 (2009).
[CrossRef]

Gallagher, J. E.

Garoff, S.

S. K. Sinha, E. B. Sirota, and S. Garoff, "X-ray and neutron scattering from rough surfaces," Phys. Rev. B 38,2297-2312 (1988).
[CrossRef]

Giglio, M.

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bosecke, and M. Giglio, "X-ray-scattering information obtained from near-field speckle," Nat. Phys. 4,238-243 (2008).
[CrossRef]

Griffiths, J. A.

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, "A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously," Appl. Phys. Lett. 94,044108 (2009).
[CrossRef]

Grünzweig, C.

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, "Bulk magnetic domain structures visualized by neutron dark-field imaging," Appl. Phys. Lett. 93,112504 (2009).
[CrossRef]

M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, "Neutron dark-field tomography," Phys. Rev. Lett. 101,123902 (2008).
[CrossRef] [PubMed]

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei G. Kühne, J. Kohlbrecher, R. Schäfer, P. Lejcek, H. M. R. Ronnow, and F. Pfeiffer, "Neutron decoherence imaging for visualizing bulk magnetic domain structures," Phys. Rev. Lett. 101,025504 (2008).
[CrossRef] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. H. Brönnimann, C. Grünzweig, and C. David, "Hardx-ray dark-field imaging using a grating interferometer," Nat. Mat. 7,134-137 (2008).
[CrossRef]

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.

Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, "Differential phase x-ray imaging microscopy with x-ray Talbot interferometer," Appl. Phys. Express 1,117002 (2008).
[CrossRef]

Hegedus, M. M.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, "Fourier x-ray scattering radiography yields bone structural information," Radiography 251,910-918 (2009).

Henrich, B.

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, "X-ray dark-field and phase-contrast imaging using a grating interferometer," J. Appl. Phys. 105,102006 (2009).
[CrossRef]

Herriott, D. R.

Hilger, A.

M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, "Neutron dark-field tomography," Phys. Rev. Lett. 101,123902 (2008).
[CrossRef] [PubMed]

Hintermüller, C.

S. A. MacDonald, F. Marone, C. Hintermüller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, "Advanced phase-contrast imaging using a grating interferometer," J. Synchrotron Radiation 16,562-572 (2009).
[CrossRef]

Huang, Z.-F.

Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, "Quantitative grating-based x-ray dark-field computed tomography," Appl. Phys. Lett. 95,094105 (2009).
[CrossRef]

Z.-F. Huang, K.-J. Kang, L. Zhang, Z.-Q. Chen, F. Ding, Z.-T. Wang, and Q.-G. Fang, "Alternative method for differential phase-contrast imaging with weakly coherent hard x rays," Phys. Rev. A 79,013815 (2009).
[CrossRef]

Kang, K.-J.

Z.-F. Huang, K.-J. Kang, L. Zhang, Z.-Q. Chen, F. Ding, Z.-T. Wang, and Q.-G. Fang, "Alternative method for differential phase-contrast imaging with weakly coherent hard x rays," Phys. Rev. A 79,013815 (2009).
[CrossRef]

Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, "Quantitative grating-based x-ray dark-field computed tomography," Appl. Phys. Lett. 95,094105 (2009).
[CrossRef]

Kardjilov, N.

M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, "Neutron dark-field tomography," Phys. Rev. Lett. 101,123902 (2008).
[CrossRef] [PubMed]

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]

Konstantinidis, A.

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, "A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously," Appl. Phys. Lett. 94,044108 (2009).
[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]

Kraft, P.

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, "X-ray dark-field and phase-contrast imaging using a grating interferometer," J. Appl. Phys. 105,102006 (2009).
[CrossRef]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. H. Brönnimann, C. Grünzweig, and C. David, "Hardx-ray dark-field imaging using a grating interferometer," Nat. Mat. 7,134-137 (2008).
[CrossRef]

Kuhne, G.

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, "Bulk magnetic domain structures visualized by neutron dark-field imaging," Appl. Phys. Lett. 93,112504 (2009).
[CrossRef]

MacDonald, S. A.

S. A. MacDonald, F. Marone, C. Hintermüller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, "Advanced phase-contrast imaging using a grating interferometer," J. Synchrotron Radiation 16,562-572 (2009).
[CrossRef]

Manke, I.

M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, "Neutron dark-field tomography," Phys. Rev. Lett. 101,123902 (2008).
[CrossRef] [PubMed]

Marone, F.

S. A. MacDonald, F. Marone, C. Hintermüller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, "Advanced phase-contrast imaging using a grating interferometer," J. Synchrotron Radiation 16,562-572 (2009).
[CrossRef]

Mikuljan, G.

S. A. MacDonald, F. Marone, C. Hintermüller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, "Advanced phase-contrast imaging using a grating interferometer," J. Synchrotron Radiation 16,562-572 (2009).
[CrossRef]

Momose, A.

W. Yashiro, Y. Takeda, A. Takeuchi, Y. Suzuki, and A. Momose, "Hard x-ray phase-difference microscopy using a Fresnel zone plate and a transmission grating," Phys. Rev. Lett. 103,180801 (2009).
[CrossRef] [PubMed]

W. Yashiro, Y. Takeda, and A. Momose, "Efficiency of capturing a phase image using conebeam x-ray Talbot interferometry," J. Opt. Soc. Am. A 25,2025-2039 (2008).
[CrossRef]

Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, "Differential phase x-ray imaging microscopy with x-ray Talbot interferometer," Appl. Phys. Express 1,117002 (2008).
[CrossRef]

A. Momose, "Recent advances in x-ray phase imaging," Jpn. J. Appl. Phys. 44,6355-6367 (2005).
[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]

Nesterets, Y. I.

Y. I. Nesterets and S. W. Wilkins, "Phase-contrast imaging using a scanning-doublegrating configuration," Opt. Express 16,5849-5867 (2008).
[CrossRef] [PubMed]

Y. I. Nesterets, "On the origins of decoherence and extinction contrast in phase-contrast imaging," Opt. Commun. 281,533-542 (2008).
[CrossRef]

Olivo, A.

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, "A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously," Appl. Phys. Lett. 94,044108 (2009).
[CrossRef]

A. Olivo and R. Speller, "A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources," Appl. Phys. Lett. 91,074106 (2007).
[CrossRef]

Peverini, L.

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bosecke, and M. Giglio, "X-ray-scattering information obtained from near-field speckle," Nat. Phys. 4,238-243 (2008).
[CrossRef]

Pfeiffer, F.

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, "Bulk magnetic domain structures visualized by neutron dark-field imaging," Appl. Phys. Lett. 93,112504 (2009).
[CrossRef]

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, "X-ray dark-field and phase-contrast imaging using a grating interferometer," J. Appl. Phys. 105,102006 (2009).
[CrossRef]

S. A. MacDonald, F. Marone, C. Hintermüller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, "Advanced phase-contrast imaging using a grating interferometer," J. Synchrotron Radiation 16,562-572 (2009).
[CrossRef]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. H. Brönnimann, C. Grünzweig, and C. David, "Hardx-ray dark-field imaging using a grating interferometer," Nat. Mat. 7,134-137 (2008).
[CrossRef]

M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, "Neutron dark-field tomography," Phys. Rev. Lett. 101,123902 (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]

Pofahl, S.

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, "Bulk magnetic domain structures visualized by neutron dark-field imaging," Appl. Phys. Lett. 93,112504 (2009).
[CrossRef]

Potenza, M. A. C.

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bosecke, and M. Giglio, "X-ray-scattering information obtained from near-field speckle," Nat. Phys. 4,238-243 (2008).
[CrossRef]

Rapacchi, S.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, "Fourier x-ray scattering radiography yields bone structural information," Radiography 251,910-918 (2009).

Robert, A.

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bosecke, and M. Giglio, "X-ray-scattering information obtained from near-field speckle," Nat. Phys. 4,238-243 (2008).
[CrossRef]

Robinson, I. K.

I. A. Vartanyants and I. K. Robinson, "Origins of decoherence in coherent x-ray diffraction experiments," Opt. Commun. 222,29-50 (2003).
[CrossRef]

Ronnow, H. M. R.

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, "Bulk magnetic domain structures visualized by neutron dark-field imaging," Appl. Phys. Lett. 93,112504 (2009).
[CrossRef]

Rosenfeld, D. P.

Schafer, R.

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, "Bulk magnetic domain structures visualized by neutron dark-field imaging," Appl. Phys. Lett. 93,112504 (2009).
[CrossRef]

Sinha, S. K.

S. K. Sinha, E. B. Sirota, and S. Garoff, "X-ray and neutron scattering from rough surfaces," Phys. Rev. B 38,2297-2312 (1988).
[CrossRef]

Sirota, E. B.

S. K. Sinha, E. B. Sirota, and S. Garoff, "X-ray and neutron scattering from rough surfaces," Phys. Rev. B 38,2297-2312 (1988).
[CrossRef]

Speller, R.

A. Olivo and R. Speller, "A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources," Appl. Phys. Lett. 91,074106 (2007).
[CrossRef]

Speller, R. D.

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, "A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously," Appl. Phys. Lett. 94,044108 (2009).
[CrossRef]

Stampanoni, M.

S. A. MacDonald, F. Marone, C. Hintermüller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, "Advanced phase-contrast imaging using a grating interferometer," J. Synchrotron Radiation 16,562-572 (2009).
[CrossRef]

Strobl, M.

M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, "Neutron dark-field tomography," Phys. Rev. Lett. 101,123902 (2008).
[CrossRef] [PubMed]

Suzuki, Y.

W. Yashiro, Y. Takeda, A. Takeuchi, Y. Suzuki, and A. Momose, "Hard x-ray phase-difference microscopy using a Fresnel zone plate and a transmission grating," Phys. Rev. Lett. 103,180801 (2009).
[CrossRef] [PubMed]

Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, "Differential phase x-ray imaging microscopy with x-ray Talbot interferometer," Appl. Phys. Express 1,117002 (2008).
[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.

W. Yashiro, Y. Takeda, A. Takeuchi, Y. Suzuki, and A. Momose, "Hard x-ray phase-difference microscopy using a Fresnel zone plate and a transmission grating," Phys. Rev. Lett. 103,180801 (2009).
[CrossRef] [PubMed]

W. Yashiro, Y. Takeda, and A. Momose, "Efficiency of capturing a phase image using conebeam x-ray Talbot interferometry," J. Opt. Soc. Am. A 25,2025-2039 (2008).
[CrossRef]

Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, "Differential phase x-ray imaging microscopy with x-ray Talbot interferometer," Appl. Phys. Express 1,117002 (2008).
[CrossRef]

Takeuchi, A.

W. Yashiro, Y. Takeda, A. Takeuchi, Y. Suzuki, and A. Momose, "Hard x-ray phase-difference microscopy using a Fresnel zone plate and a transmission grating," Phys. Rev. Lett. 103,180801 (2009).
[CrossRef] [PubMed]

Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, "Differential phase x-ray imaging microscopy with x-ray Talbot interferometer," Appl. Phys. Express 1,117002 (2008).
[CrossRef]

Vartanyants, I. A.

I. A. Vartanyants and I. K. Robinson, "Origins of decoherence in coherent x-ray diffraction experiments," Opt. Commun. 222,29-50 (2003).
[CrossRef]

Wang, Z.-T.

Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, "Quantitative grating-based x-ray dark-field computed tomography," Appl. Phys. Lett. 95,094105 (2009).
[CrossRef]

Z.-F. Huang, K.-J. Kang, L. Zhang, Z.-Q. Chen, F. Ding, Z.-T. Wang, and Q.-G. Fang, "Alternative method for differential phase-contrast imaging with weakly coherent hard x rays," Phys. Rev. A 79,013815 (2009).
[CrossRef]

Weitkamp, T.

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]

Wen, H.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, "Fourier x-ray scattering radiography yields bone structural information," Radiography 251,910-918 (2009).

White, A. D.

Wilkins, S. W.

Yashiro, W.

W. Yashiro, Y. Takeda, A. Takeuchi, Y. Suzuki, and A. Momose, "Hard x-ray phase-difference microscopy using a Fresnel zone plate and a transmission grating," Phys. Rev. Lett. 103,180801 (2009).
[CrossRef] [PubMed]

Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, "Differential phase x-ray imaging microscopy with x-ray Talbot interferometer," Appl. Phys. Express 1,117002 (2008).
[CrossRef]

W. Yashiro, Y. Takeda, and A. Momose, "Efficiency of capturing a phase image using conebeam x-ray Talbot interferometry," J. Opt. Soc. Am. A 25,2025-2039 (2008).
[CrossRef]

Zhang, L.

Z.-F. Huang, K.-J. Kang, L. Zhang, Z.-Q. Chen, F. Ding, Z.-T. Wang, and Q.-G. Fang, "Alternative method for differential phase-contrast imaging with weakly coherent hard x rays," Phys. Rev. A 79,013815 (2009).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Express (1)

Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, "Differential phase x-ray imaging microscopy with x-ray Talbot interferometer," Appl. Phys. Express 1,117002 (2008).
[CrossRef]

Appl. Phys. Lett. (4)

A. Olivo and R. Speller, "A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources," Appl. Phys. Lett. 91,074106 (2007).
[CrossRef]

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, "A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously," Appl. Phys. Lett. 94,044108 (2009).
[CrossRef]

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, "Bulk magnetic domain structures visualized by neutron dark-field imaging," Appl. Phys. Lett. 93,112504 (2009).
[CrossRef]

Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, "Quantitative grating-based x-ray dark-field computed tomography," Appl. Phys. Lett. 95,094105 (2009).
[CrossRef]

J. Appl. Phys. (1)

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, "X-ray dark-field and phase-contrast imaging using a grating interferometer," J. Appl. Phys. 105,102006 (2009).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Synchrotron Radiation (1)

S. A. MacDonald, F. Marone, C. Hintermüller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, "Advanced phase-contrast imaging using a grating interferometer," J. Synchrotron Radiation 16,562-572 (2009).
[CrossRef]

Jpn. J. Appl. Phys. (2)

A. Momose, "Recent advances in x-ray phase imaging," Jpn. J. Appl. Phys. 44,6355-6367 (2005).
[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]

Nat. Mat. (1)

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. H. Brönnimann, C. Grünzweig, and C. David, "Hardx-ray dark-field imaging using a grating interferometer," Nat. Mat. 7,134-137 (2008).
[CrossRef]

Nat. Phys. (2)

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]

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bosecke, and M. Giglio, "X-ray-scattering information obtained from near-field speckle," Nat. Phys. 4,238-243 (2008).
[CrossRef]

Opt. Commun. (2)

Y. I. Nesterets, "On the origins of decoherence and extinction contrast in phase-contrast imaging," Opt. Commun. 281,533-542 (2008).
[CrossRef]

I. A. Vartanyants and I. K. Robinson, "Origins of decoherence in coherent x-ray diffraction experiments," Opt. Commun. 222,29-50 (2003).
[CrossRef]

Opt. Express (1)

Phys. Rev. A (1)

Z.-F. Huang, K.-J. Kang, L. Zhang, Z.-Q. Chen, F. Ding, Z.-T. Wang, and Q.-G. Fang, "Alternative method for differential phase-contrast imaging with weakly coherent hard x rays," Phys. Rev. A 79,013815 (2009).
[CrossRef]

Phys. Rev. B (1)

S. K. Sinha, E. B. Sirota, and S. Garoff, "X-ray and neutron scattering from rough surfaces," Phys. Rev. B 38,2297-2312 (1988).
[CrossRef]

Phys. Rev. Lett. (3)

W. Yashiro, Y. Takeda, A. Takeuchi, Y. Suzuki, and A. Momose, "Hard x-ray phase-difference microscopy using a Fresnel zone plate and a transmission grating," Phys. Rev. Lett. 103,180801 (2009).
[CrossRef] [PubMed]

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei G. Kühne, J. Kohlbrecher, R. Schäfer, P. Lejcek, H. M. R. Ronnow, and F. Pfeiffer, "Neutron decoherence imaging for visualizing bulk magnetic domain structures," Phys. Rev. Lett. 101,025504 (2008).
[CrossRef] [PubMed]

M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, "Neutron dark-field tomography," Phys. Rev. Lett. 101,123902 (2008).
[CrossRef] [PubMed]

Phys. Today (1)

R. Fitzgerald, "Phase-sensitive x-ray imaging," Phys. Today 53,23-26 (2000).
[CrossRef]

Radiography (1)

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, "Fourier x-ray scattering radiography yields bone structural information," Radiography 251,910-918 (2009).

Other (4)

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, Madison, Wisconsin, USA, 2009).

Here, we defined the pth Talbot order as it expresses the pth position where the electric field just behind the first grating is perfectly reproduced for any grating. For plane-wave illumination, the pth Talbot order corresponds to the position of zT = pd2/⌊. This definition is convenient because it is independent of what kind of grating is used as the first grating. Note that, once zT is given by pd2/⌊, we can use p for specifying any position behind the first grating instead of zT. The analytical calculations presented in this paper can be applied to any periodic image generated behind the first grating at any position of p (> 0).

J. W. Goodman, Statistical Optics, (A Wiley-Interscience Publication, New York, 2000).

K. Patorski, "Self-imaging and its applications," in Progress in Optics XXVII, E. Wolf, ed., (Elsevier Science Publishers B.V., Amsterdam, 1989).

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Figures (4)

Fig. 1.
Fig. 1.

Experimental setup for x-ray Talbot interferometry.

Fig. 2.
Fig. 2.

Dependence of −ln (V/V 0) on pd. (a) Results for microspheres with various radii a and (b) those for melamine sponge with its optical microscope image. Plots are experimental data and curves are results of least-squares fitting to experimental data.

Fig. 3.
Fig. 3.

(a) Experimental setup for (ultra) small-angle scattering measurement and (b) results of measured angular distribution for microspheres with various radii.

Fig. 4.
Fig. 4.

Hurst exponent is plotted against ratio of diameter to height for cylinder particles. Correlation length ξ normalized by average radius ā is also plotted.

Tables (1)

Tables Icon

Table 1. Results of least-squares fitting to −ln (V/V 0) data in Fig. 2

Equations (37)

Equations on this page are rendered with MathJax. Learn more.

E ( x , y ) E 0 exp [ α s ( x , y ) 2 ] Σ n a n exp [ i { 2 π n x d + Φ n ( x , y ) } ] .
I pixel ( x , y ) I 0 exp [ α s ( x , y ) ] Σ n , m , N μ m a n a n m * c N exp [ 2 π i N χ d ] A ˜ n , m , N ( x , y ) ,
A ˜ n , m , N ( x , y ) PSF ( x x , y y ) exp [ 2 π i ( m + N ) x d ]
× exp [ i { Φ n ( x , y ) Φ n m ( x , y ) } ] d x d y .
q 0 = I 0 e α s ( x , y ) Σ n , m μ m a n a n m * c 0 A ˜ n , m , 0 ( x , y ) .
A ˜ n , m , 0 ( x , y ) = PSF ( x x , y y ) exp [ 2 π i m x d ]
× exp [ i { Φ n ( x , y ) Φ n m ( x , y ) } ] d x d y ,
exp [ impd Φ s ( x , y ) x ]
× PSF ( x x , y y ) exp [ 2 π i m x d ]
× exp [ i { Φ f , n ( x , y ) Φ f , n m ( x , y ) } ] d x d y ,
A ˜ n , m , 0 ( x , y ) exp [ impd Φ s ( x , y ) x ] × exp [ 2 π i m x d ] exp [ i { Φ f , n ( x , y ) Φ f , n m ( x , y ) } ] .
f ( x , y ) ¯ ( 1 D 2 ) D 2 D 2 D 2 D 2 f ( x x , y y ) d x d y ,
Σ m μ m a n a n m * A ˜ n , m , 0 ( x , y ) μ 0 a n a n * A ˜ n , 0,0 ( x , y ) ,
= μ 0 a n 2 .
q 0 I 0 e α s ( x , y ) C 0 ,
q 1 = I 0 e α s ( x , y ) c 1 Σ n , m μ m a n a n m * A ˜ n , m , 1 ( x , y ) .
A ˜ n , m , 1 ( x , y ) = PSF ( x x , y y ) exp [ 2 π i m + 1 d x ]
× exp [ i { Φ n ( x , y ) Φ n m ( x , y ) } ] d x d y ,
exp [ impd Φ s ( x , y ) x ]
× PSF ( x x , y y ) exp [ 2 π i m + 1 d x ]
× exp [ i { Φ f , n ( x , y ) Φ f , n m ( x , y ) } ] d x d y ,
exp [ impd Φ s ( x , y ) x ] × exp [ 2 π i m + 1 d x ] exp [ i { Φ f , n ( x , y ) Φ f , n m ( x , y ) } ] .
Σ n , m μ m a n a n m * A ˜ n , m , 1 Σ n μ 1 a n a n + 1 * A ˜ n , 1 , 1 .
Σ n a n a n + 1 * A ˜ n , 1 , 1 = a 0 a 1 * A ˜ 0 , 1 , 1 + a 1 a 0 * A ˜ 1 , 1 , 1 .
q 1 I 0 e α s μ 1 c 1 ( a 0 a 1 * A ˜ 0 , 1 , 1 + a 1 a 0 * A ˜ 1 , 1 , 1 ) ,
I 0 e α s ( x , y ) C 1 exp [ i Δ Φ f ( x , y ; pd ) ] ¯ ,
exp [ i { Φ f , 1 ( x , y ) Φ f , 0 ( x , y ) } ] ¯ exp [ i { Φ f , 0 ( x , y ) Φ f , 1 ( x , y ) } ] ¯ ,
V V 0 exp [ i Δ Φ f ( x , y ; pd ) ] ¯ ,
exp [ σ 2 ( x , y ) { 1 γ ( x , y ; pd ) } ] .
γ ( x , y ; Δ x ) Φ f ( x , y ) Φ f ( x + Δ x , y ) ¯ σ 2 .
γ ( x , y ; Δ x ) exp [ { Δ x ξ ( x , y ) } 2 H ] ,
+ [ 2 Δ x 2 Δ x 4 2 ] ln [ x 1 + 1 Δ x 2 ] ,
σ 2 NT π a 2 · ( Δ ρ ) 2 r e 2 λ 2 ( 2 a 2 ) ,
V V 0 exp [ i Σ j Δ Φ f , j ] ¯ ,
= Π j exp [ i Δ Φ f , j ] ¯ ,
exp [ Σ j σ j 2 { 1 γ j } ] ,
ln [ V V 0 ] ( σ ( z ) 2 ) z { 1 γ ( z ) } dz ,

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