Abstract

In grating-based x-ray phase sensitive imaging, dark-field contrast refers to the extinction of the interference fringes due to small-angle scattering. For configurations where the sample is placed before the beamsplitter grating, the dark-field contrast has been quantified with theoretical wave propagation models. Yet when the grating is placed before the sample, the dark-field contrast has only been modeled in the geometric optics regime. Here we attempt to quantify the dark-field effect in the grating-before-sample geometry with first-principle wave calculations and understand the associated particle-size selectivity. We obtain an expression for the dark-field effect in terms of the sample material’s complex refractive index, which can be verified experimentally without fitting parameters. A dark-field computed tomography experiment shows that the particle-size selectivity can be used to differentiate materials of identical x-ray absorption.

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    [CrossRef] [PubMed]
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    [CrossRef]
  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).
    [CrossRef]
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    [CrossRef] [PubMed]
  5. Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Jpn. J. Appl. Phys. 46, L89–L91 (2007).
    [CrossRef]
  6. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  14. E. A. Miller, T. A. White, B. S. McDonald, A. Seifert, and M. J. Flynn, “Phase contrast x-ray imaging signatures for homeland security applications,” in IEEE Nuclear Science Symposium and Medical Imaging Conference (2010 NSS/MIC) (IEEE, 2011), pp. 896–899.
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  16. T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys. 106, 054703 (2009).
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  17. 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]
  18. F. Krejci, J. Jakubek, and M. Kroupa, “Hard x-ray phase contrast imaging using single absorption grating and hybrid semiconductor pixel detector,” Rev. Sci. Instrum. 81, 113702(2010).
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    [CrossRef] [PubMed]
  21. T. J. Davis, “A unified treatment of small-angle x-ray-scattering, x-ray refraction and absorption using the Rytov approximation,” Acta Cryst. 50, 686–690 (1994).
    [CrossRef]
  22. M. Born and E. Wolf, “Kirchhoff’s diffraction theory,” in Principles of Optics (Cambridge Univ. Press, 1999), pp. 421–424.
  23. H. H. Wen, E. E. Bennett, R. Kopace, A. F. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission gratings,” Opt. Lett. 35, 1932–1934 (2010).
    [CrossRef] [PubMed]
  24. X. Z. Wu and H. Liu, “An experimental method of determining relative phase-contrast factor for x-ray imaging systems,” Med. Phys. 31, 997–1002 (2004).
    [CrossRef]
  25. C. K. Kemble, J. Auxier, S. K. Lynch, E. E. Bennett, N. Y. Morgan, and H. Wen, “Grazing angle Mach–Zehnder interferometer using reflective phase gratings and a polychromatic, un-collimated light source,” Opt. Express 18, 27481–27492 (2010).
    [CrossRef]

2011 (2)

E. A. Miller, T. A. White, B. S. McDonald, A. Seifert, and M. J. Flynn, “Phase contrast x-ray imaging signatures for homeland security applications,” in IEEE Nuclear Science Symposium and Medical Imaging Conference (2010 NSS/MIC) (IEEE, 2011), pp. 896–899.

K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “Quantitative x-ray phase-contrast imaging using a single grating of comparable pitch to sample feature size,” Opt. Lett. 36, 55–57 (2011).
[CrossRef] [PubMed]

2010 (7)

A. F. Stein, J. Ilavsky, R. Kopace, E. E. Bennett, and H. Wen, “Selective imaging of nano-particle contrast agents by a single-shot x-ray diffraction technique,” Opt. Express 18, 13271–13278 (2010).
[CrossRef] [PubMed]

F. Krejci, J. Jakubek, and M. Kroupa, “Hard x-ray phase contrast imaging using single absorption grating and hybrid semiconductor pixel detector,” Rev. Sci. Instrum. 81, 113702(2010).
[CrossRef] [PubMed]

H. H. Wen, E. E. Bennett, R. Kopace, A. F. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission gratings,” Opt. Lett. 35, 1932–1934 (2010).
[CrossRef] [PubMed]

C. K. Kemble, J. Auxier, S. K. Lynch, E. E. Bennett, N. Y. Morgan, and H. Wen, “Grazing angle Mach–Zehnder interferometer using reflective phase gratings and a polychromatic, un-collimated light source,” Opt. Express 18, 27481–27492 (2010).
[CrossRef]

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

T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans'l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
[CrossRef]

2009 (3)

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, “Fourier x-ray scattering radiography yields bone structural information,” Radiology 251, 910–918 (2009).
[CrossRef] [PubMed]

T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys. 106, 054703 (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]

2008 (3)

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

H. Wen, E. Bennett, M. M. Hegedus, and S. C. Carroll, “Spatial harmonic imaging of x-ray scattering—initial results,” IEEE Trans. Med. Imaging 27, 997–1002 (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]

2007 (2)

Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Jpn. J. Appl. Phys. 46, L89–L91 (2007).
[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]

2005 (1)

2004 (1)

X. Z. Wu and H. Liu, “An experimental method of determining relative phase-contrast factor for x-ray imaging systems,” Med. Phys. 31, 997–1002 (2004).
[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]

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

2002 (1)

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

1999 (1)

M. Born and E. Wolf, “Kirchhoff’s diffraction theory,” in Principles of Optics (Cambridge Univ. Press, 1999), pp. 421–424.

1994 (1)

T. J. Davis, “A unified treatment of small-angle x-ray-scattering, x-ray refraction and absorption using the Rytov approximation,” Acta Cryst. 50, 686–690 (1994).
[CrossRef]

1971 (1)

Anastasio, M. A.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Aoki, S.

Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Jpn. J. Appl. Phys. 46, L89–L91 (2007).
[CrossRef]

Auxier, J.

Baumann, J.

T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys. 106, 054703 (2009).
[CrossRef]

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]

T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans'l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
[CrossRef]

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

Bennett, E.

H. Wen, E. Bennett, M. M. Hegedus, and S. C. Carroll, “Spatial harmonic imaging of x-ray scattering—initial results,” IEEE Trans. Med. Imaging 27, 997–1002 (2008).
[CrossRef] [PubMed]

Bennett, E. E.

Born, M.

M. Born and E. Wolf, “Kirchhoff’s diffraction theory,” in Principles of Optics (Cambridge Univ. Press, 1999), pp. 421–424.

Brankov, J. G.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Bronnimann, C.

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

Bunk, O.

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]

T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys. 106, 054703 (2009).
[CrossRef]

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

Carroll, S. C.

H. Wen, E. Bennett, M. M. Hegedus, and S. C. Carroll, “Spatial harmonic imaging of x-ray scattering—initial results,” IEEE Trans. Med. Imaging 27, 997–1002 (2008).
[CrossRef] [PubMed]

Chabior, M.

T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys. 106, 054703 (2009).
[CrossRef]

Chapman, D.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[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]

Cloetens, P.

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]

T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans'l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
[CrossRef]

T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys. 106, 054703 (2009).
[CrossRef]

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

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

Davis, T. J.

T. J. Davis, “A unified treatment of small-angle x-ray-scattering, x-ray refraction and absorption using the Rytov approximation,” Acta Cryst. 50, 686–690 (1994).
[CrossRef]

Deyhle, H.

T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans'l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
[CrossRef]

Diaz, A.

Donath, T.

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]

T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys. 106, 054703 (2009).
[CrossRef]

Eikenberry, E. F.

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

T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans'l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
[CrossRef]

Flynn, M. J.

E. A. Miller, T. A. White, B. S. McDonald, A. Seifert, and M. J. Flynn, “Phase contrast x-ray imaging signatures for homeland security applications,” in IEEE Nuclear Science Symposium and Medical Imaging Conference (2010 NSS/MIC) (IEEE, 2011), pp. 896–899.

Galatsanos, N. P.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Grunzweig, C.

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

Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Jpn. J. Appl. Phys. 46, L89–L91 (2007).
[CrossRef]

Hegedus, M. M.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, “Fourier x-ray scattering radiography yields bone structural information,” Radiology 251, 910–918 (2009).
[CrossRef] [PubMed]

H. Wen, E. Bennett, M. M. Hegedus, and S. C. Carroll, “Spatial harmonic imaging of x-ray scattering—initial results,” IEEE Trans. Med. Imaging 27, 997–1002 (2008).
[CrossRef] [PubMed]

Hempel, E.

T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys. 106, 054703 (2009).
[CrossRef]

Hoheisel, M.

T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys. 106, 054703 (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]

Ilavsky, J.

Jakubek, J.

F. Krejci, J. Jakubek, and M. Kroupa, “Hard x-ray phase contrast imaging using single absorption grating and hybrid semiconductor pixel detector,” Rev. Sci. Instrum. 81, 113702(2010).
[CrossRef] [PubMed]

Jensen, T. H.

T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans'l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
[CrossRef]

Kang, K. J.

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]

Kawabata, K.

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]

Kemble, C. K.

Kopace, R.

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

Krejci, F.

F. Krejci, J. Jakubek, and M. Kroupa, “Hard x-ray phase contrast imaging using single absorption grating and hybrid semiconductor pixel detector,” Rev. Sci. Instrum. 81, 113702(2010).
[CrossRef] [PubMed]

Kroupa, M.

F. Krejci, J. Jakubek, and M. Kroupa, “Hard x-ray phase contrast imaging using single absorption grating and hybrid semiconductor pixel detector,” Rev. Sci. Instrum. 81, 113702(2010).
[CrossRef] [PubMed]

Liu, H.

X. Z. Wu and H. Liu, “An experimental method of determining relative phase-contrast factor for x-ray imaging systems,” Med. Phys. 31, 997–1002 (2004).
[CrossRef]

Lynch, S. K.

McDonald, B. S.

E. A. Miller, T. A. White, B. S. McDonald, A. Seifert, and M. J. Flynn, “Phase contrast x-ray imaging signatures for homeland security applications,” in IEEE Nuclear Science Symposium and Medical Imaging Conference (2010 NSS/MIC) (IEEE, 2011), pp. 896–899.

Miller, E. A.

E. A. Miller, T. A. White, B. S. McDonald, A. Seifert, and M. J. Flynn, “Phase contrast x-ray imaging signatures for homeland security applications,” in IEEE Nuclear Science Symposium and Medical Imaging Conference (2010 NSS/MIC) (IEEE, 2011), pp. 896–899.

Mohr, J.

T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans'l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
[CrossRef]

T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys. 106, 054703 (2009).
[CrossRef]

Momose, A.

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

Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Jpn. J. Appl. Phys. 46, L89–L91 (2007).
[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]

Morgan, K. S.

Morgan, N. Y.

Muehleman, C.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Nesterets, Y. I.

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

Nohammer, B.

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

Olivo, A.

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]

Oltulu, O.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Paganin, D. M.

Pai, V.

Pfeiffer, F.

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]

T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans'l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
[CrossRef]

T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys. 106, 054703 (2009).
[CrossRef]

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

Popescu, S.

T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys. 106, 054703 (2009).
[CrossRef]

Rapacchi, S.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, “Fourier x-ray scattering radiography yields bone structural information,” Radiology 251, 910–918 (2009).
[CrossRef] [PubMed]

Reznikova, E.

T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans'l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
[CrossRef]

T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys. 106, 054703 (2009).
[CrossRef]

Rutishauser, S.

T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans'l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
[CrossRef]

Schuster, M.

T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys. 106, 054703 (2009).
[CrossRef]

Seifert, A.

E. A. Miller, T. A. White, B. S. McDonald, A. Seifert, and M. J. Flynn, “Phase contrast x-ray imaging signatures for homeland security applications,” in IEEE Nuclear Science Symposium and Medical Imaging Conference (2010 NSS/MIC) (IEEE, 2011), pp. 896–899.

Siu, K. K. W.

Solak, H. H.

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

Stampanoni, M.

Stein, A. F.

Suzuki, T.

Suzuki, Y.

Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Jpn. J. Appl. Phys. 46, L89–L91 (2007).
[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.

Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Jpn. J. Appl. Phys. 46, L89–L91 (2007).
[CrossRef]

Terui, Y.

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]

Weitkamp, T.

T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans'l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
[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.

Wen, H. H.

Wernick, M. N.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

White, T. A.

E. A. Miller, T. A. White, B. S. McDonald, A. Seifert, and M. J. Flynn, “Phase contrast x-ray imaging signatures for homeland security applications,” in IEEE Nuclear Science Symposium and Medical Imaging Conference (2010 NSS/MIC) (IEEE, 2011), pp. 896–899.

Wirjadi, O.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, “Kirchhoff’s diffraction theory,” in Principles of Optics (Cambridge Univ. Press, 1999), pp. 421–424.

Wu, X. Z.

X. Z. Wu and H. Liu, “An experimental method of determining relative phase-contrast factor for x-ray imaging systems,” Med. Phys. 31, 997–1002 (2004).
[CrossRef]

Yang, Y. Y.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Yashiro, W.

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

Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Jpn. J. Appl. Phys. 46, L89–L91 (2007).
[CrossRef]

Yokozeki, S.

Zanette, I.

T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans'l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
[CrossRef]

Zhong, Z.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Ziegler, E.

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[CrossRef] [PubMed]

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

Acta Cryst. (1)

T. J. Davis, “A unified treatment of small-angle x-ray-scattering, x-ray refraction and absorption using the Rytov approximation,” Acta Cryst. 50, 686–690 (1994).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

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

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]

IEEE Trans. Med. Imaging (1)

H. Wen, E. Bennett, M. M. Hegedus, and S. C. Carroll, “Spatial harmonic imaging of x-ray scattering—initial results,” IEEE Trans. Med. Imaging 27, 997–1002 (2008).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys. 106, 054703 (2009).
[CrossRef]

Jpn. J. Appl. Phys. (2)

Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Jpn. J. Appl. Phys. 46, L89–L91 (2007).
[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]

Med. Phys. (1)

X. Z. Wu and H. Liu, “An experimental method of determining relative phase-contrast factor for x-ray imaging systems,” Med. Phys. 31, 997–1002 (2004).
[CrossRef]

Nat. Mater. (1)

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

Opt. Commun. (1)

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

Opt. Express (4)

Opt. Lett. (2)

Phys. Med. Biol. (2)

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

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]

Phys. Rev. B (1)

T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans'l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
[CrossRef]

Radiology (1)

H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, “Fourier x-ray scattering radiography yields bone structural information,” Radiology 251, 910–918 (2009).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

F. Krejci, J. Jakubek, and M. Kroupa, “Hard x-ray phase contrast imaging using single absorption grating and hybrid semiconductor pixel detector,” Rev. Sci. Instrum. 81, 113702(2010).
[CrossRef] [PubMed]

Other (2)

E. A. Miller, T. A. White, B. S. McDonald, A. Seifert, and M. J. Flynn, “Phase contrast x-ray imaging signatures for homeland security applications,” in IEEE Nuclear Science Symposium and Medical Imaging Conference (2010 NSS/MIC) (IEEE, 2011), pp. 896–899.

M. Born and E. Wolf, “Kirchhoff’s diffraction theory,” in Principles of Optics (Cambridge Univ. Press, 1999), pp. 421–424.

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Figures (5)

Fig. 1
Fig. 1

Two different configurations of grating interferometers. (a) In the SbG geometry, a plane wave illuminates the sample and then the gratings. (b) In the GbS geometry, the plane wave is split by the grating into different directions of propagation and then transmitted through the sample.

Fig. 2
Fig. 2

Illustration of the fringe amplitude A and average intensity J of a fringe pattern on the detector screen. The fringe visibility V is defined as A / J .

Fig. 3
Fig. 3

To model the DFEC in the GbS geometry, the imaged sample is sectioned into thin slices along the Z axis. The effect of a slice of thickness t at position z is analyzed by first removing all material beyond z, and then adding only the slice. The gap between the two surfaces of the section plane at z is artificially added to illustrate the entry wave into the slice.

Fig. 4
Fig. 4

Experimental and theoretical values of the unitless DFEC defined in Eq. (68). Because all parameters are known a priori, no fitting was involved.

Fig. 5
Fig. 5

(a) Cross section of the volume CT data of absorption coefficient of two vials holding a black acrylic paint (on left) and 80 mg / ml KI solution, respectively, in units of cm 1 . (b) The same cross section showing the DFECs of the vials. (c) and (d) are semitransparent volume renditions of the absorption and DFEC, respectively. The vials have the same level of absorption, but x-ray scattering by the paint pigments causes a more visible dark-field effect.

Tables (1)

Tables Icon

Table 1 Experimental Conditions

Equations (71)

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A / A = exp ( μ a T μ d T ) ,
V = A / J .
n = 1 + χ ,
χ = δ i β ,
χ = χ s + χ f .
k = 2 π / λ ,
k z = k 2 k x 2 k y 2 .
V ( z + t ) V ( z ) = t μ d ( z ) V ( z ) ,
d V ( z ) d z = μ d ( z ) V ( z ) .
V ( z ) = V ( 0 ) exp [ 0 z μ d ( τ ) d τ ] .
E ( x , y , z ) = d k x d k y a ( k x , k y ) exp ( i k x x i k y y i k z z ) .
E ( x , y , z + t ) = d k x d k y a ( k x , k y ) exp ( i k x x i k y y i k z z ) exp [ i Φ ( k x , k y , x , y , z ) ] ,
Φ ( k x , k y , x , y , z ) = 0 t d τ k 2 k z χ ( x k x k τ , y k y k τ , z + τ ) .
Φ ( k x , k y , x , y , z ) = Φ s ( k x , k y , x , y , z ) + Φ f ( k x , k y , x , y , z ) ,
Φ f ( k x , k y , x , y , z ) = 0 t d τ k 2 k z χ f ( x k x k τ , y k y k τ , z + τ ) .
χ f ( x k x k τ , y k y k τ , z + τ ) χ f ( x , y , z + τ ) k x k τ χ f x k y k τ χ f y .
k x k , k y k λ D 1 .
χ f ( x k x k τ , y k y k τ , z + τ ) = χ f ( x , y , z + τ ) [ 1 + O ( λ t D 2 ) ] ,
k 2 k z k [ 1 + 1 2 ( k x 2 + k y 2 k 2 ) ] = k [ 1 + O ( λ 2 D 2 ) ] .
Φ f ( k x , k y , x , y , z ) = 0 t d τ k χ f ( x , y , z + τ ) [ 1 + O ( λ 2 D 2 ) + O ( λ t D 2 ) ] .
Φ f ( k x , k y , x , y , z ) Φ f ( x , y , z ) ,
Φ f ( x , y , z ) = k 0 t d τ χ f ( x , y , z + τ ) .
E ( x , y , z + t ) = exp [ i Φ f ( x , y , z ) ] d k x d k y a ( k x , k y ) exp ( i k x x i k y y i k z z ) exp [ i Φ s ( k x , k y , x , y , z ) ] .
E s ( x , y , z + t ) = d k x d k y a ( k x , k y ) exp ( i k x x i k y y i k z z ) exp [ i Φ s ( k x , k y , x , y , z ) ] .
E ( x , y , z + t ) = exp [ i Φ f ( x , y , z ) ] E s ( x , y , z + t ) .
A = 2 Area detector d x d d y d I ( r d ) exp ( i g y d ) ,
I ( r d ) = E ( r d ) E * ( r d ) .
E ( r d ) = z   plane d x d y E ( r ) P ( r d r ) .
A ( z ) = 2 Area detector d x d d y d E ( r d ) E * ( r d ) exp ( i g y d ) = 2 z   plane d x 1 d y 1 d x 2 d y 2 E ( r 1 ) E * ( r 2 ) 1 Area detector d x d d y d P ( r d r 1 ) P * ( r d r 2 ) exp ( i g y d ) = 2 z   plane d x 1 d y 1 d x 2 d y 2 E ( r 1 ) E * ( r 2 ) Q ( r 1 , r 2 ) ,
Q ( r 1 , r 2 ) = 1 Area detector d x d d y d P ( r d r 1 ) P * ( r d r 2 ) exp ( i g y d ) .
P ( r d r ) = i λ exp ( i k | r d r | ) | r d r | .
P ( r d r ) i λ 1 ( z d z ) exp [ i k ( z d z ) i k 1 Z d z ( x d 2 + y d 2 + x 2 + y 2 2 x d x y d y ) ] ,
Q ( r 1 , r 2 ) = 1 Area detector d x d d y d 1 λ 2 ( z d z ) 2 exp { i k 1 z d z [ x 1 2 + y 1 2 x 2 2 y 2 2 2 x d ( x 1 x 2 ) y d ( y 1 y 2 d ) ] } ,
d = g k ( z d z ) = λ p ( z d z ) .
Q ( r 1 , r 2 ) = 1 Area δ ( x 1 x 2 ) δ ( y 1 y 2 d ) exp [ i g ( y 2 + d 2 ) ] .
A ( z ) = 2 Area z   plane d x d y E ( r + d 2 y ^ ) E * ( r d 2 y ^ ) exp ( i g y ) .
A ( z + t ) = 2 Area z + t   plane d x d y E ( x , y + d 2 , z + t ) E * ( x , y d 2 , z + t ) exp ( i g y ) .
A ( z + t ) = 2 Area z + t   plane d x d y exp [ i Φ f ( x , y + d 2 , z ) + i Φ f * ( x , y d 2 , z ) ] E s ( x , y + d 2 , z + t ) E s * ( x , y d 2 , z + t ) exp ( i g y ) .
Δ Φ f ( x , y , z ) = Φ f ( x , y + d 2 , z ) Φ f * ( x , y d 2 , z ) .
A ( z + t ) 2 Area z + t   plane d x d y [ 1 i Δ Φ f ( x , y , z ) 1 2 Δ Φ f 2 ( x , y , z ) ] E s ( x , y + d 2 , z + t ) E s * ( x , y d 2 , z + t ) exp ( i g y ) .
A s ( z + t ) = 2 Area z + t   plane d x d y E s ( x , y + d 2 , z + t ) E s * ( x , y d 2 , z + t ) exp ( i g y ) .
Δ A 1 = i 2 Area z + t   plane d x d y Δ Φ f ( x , y , z ) E s ( x , y + d 2 , z + t ) E s * ( x , y d 2 , z + t ) exp ( i g y ) .
Δ A 1 = i < Δ Φ f ( x , y , z ) > x y 2 Area z + t   plane d x d y E s ( x , y + d 2 , z + t ) E s * ( x , y d 2 , z + t ) exp ( i g y ) .
Δ A 1 = i < Δ Φ f ( x , y , z ) > x y A s ( z + t ) .
< Δ Φ f ( x , y , z ) > x y = 0 .
Δ A 1 = 0 .
Δ A 2 = 1 Area z + t   plane d x d y Δ Φ f 2 ( x , y , z ) E s ( x , y + d 2 , z + t ) E s * ( x , y d 2 , z + t ) exp ( i g y ) .
Δ A 2 = < Δ Φ f 2 ( x , y , z ) > x y 1 Area z + t   plane d x d y E s ( x , y + d 2 , z + t ) E s * ( x , y d 2 , z + t ) exp ( i g y ) .
Δ A 2 = 1 2 < Δ Φ f 2 ( x , y , z ) > x y A s ( z + t ) .
< Δ Φ f 2 ( x , y , z ) > x y = k 2 < [ 0 t d τ 1 χ f ( x , y + d 2 , z + τ 1 ) 0 t d τ 2 χ f * ( x , y d 2 , z + τ 2 ) ] 2 > x y .
C ( d , z ) = 1 2 t < [ 0 t d τ 1 χ f ( x , y + d 2 , z + τ 1 ) 0 t d τ 2 χ f * ( x , y d 2 , z + τ 2 ) ] 2 > x y ,
< Δ Φ f 2 ( x , y , z ) > x y = 2 k 2 C ( d , z ) .
Δ A 2 = t k 2 C ( d , z ) A s ( z + t ) .
A ( z + t ) = A s ( z + t ) [ 1 t k 2 C ( d , z ) ] .
J = 1 Area detector d x d d y d I ( r d ) .
J ( z + t ) = J s ( z + t ) [ 1 t k 2 C ( 0 , z ) ] .
V ( z + t ) = A s ( z + t ) J s ( z + t ) [ 1 t k 2 C ( d , z ) ] [ 1 t k 2 C ( 0 , z ) ] V s ( z + t ) { 1 t k 2 [ C ( d , z ) C ( 0 , z ) ] } ,
V s ( z + t ) = V ( z ) .
V ( z + t ) V ( z ) { 1 t k 2 [ C ( d , z ) C ( 0 , z ) ] } ,
V ( z + t ) V ( z ) t k 2 [ C ( d , z ) C ( 0 , z ) ] V ( z ) .
d V ( z ) d z = k 2 [ C ( d , z ) C ( 0 , z ) ] V ( z ) ,
V ( z ) = V ( 0 ) exp { 0 z k 2 [ C ( d , τ ) C ( 0 , τ ) ] d τ } .
μ d ( z ) = k 2 [ C ( d , z ) C ( 0 , z ) ] .
μ d ( z ) = k 2 2 t < [ 0 t d τ 1 χ f ( x , y + d 2 , z + τ 1 ) 0 t d τ 2 χ f * ( x , y d 2 , z + τ 2 ) ] 2 [ 0 t d τ 1 χ f ( x , y , z + τ 1 ) 0 t d τ 2 χ f * ( x , y , z + τ 2 ) ] 2 > x y = k 2 { 1 t < 0 t d τ 1 0 t d τ 2 χ f ( x , y , z + τ 1 ) χ f * ( x , y , z + τ 2 ) > x y 1 t < 0 t d τ 1 0 t d τ 2 χ f ( x , y + d 2 , z + τ 1 ) χ f * ( x , y d 2 , z + τ 2 ) > x y } .
R ( d , z ) = 1 t < 0 t d τ 1 0 t d τ 2 χ f ( x , y + d 2 , z + τ 1 ) χ f * ( x , y d 2 , z + τ 2 ) > x y ,
μ d ( z ) = 4 π 2 λ 2 [ R ( 0 , z ) R ( d , z ) ] . ̲
μ d = 1 T ln ( V / V ) ,
μ d = 3 π 2 λ 2 f | Δ χ | 2 d { D D 2 1 ( 1 + D 2 / 2 ) + ( D 1 D 3 / 4 ) ln [ ( D + D 2 1 ) / ( D D 2 1 ) ] , for     D > d ; D , for     D d ;
Δ χ = r e λ 2 Δ ρ / 2 π i λ Δ μ / 2 π ,
f = ( μ μ water ) / ( μ silica μ water ) .
μ d = μ d λ 2 / ( 3 π 2 f | Δ χ | 2 d ) .

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