Abstract

We report on a two-directional approach for grating based x-ray differential phase contrast imaging. In order to retrieve good quality and artifact-free phase images for quantitative analysis and image processing, particular emphasis is put on the algorithm for proper phase retrieval. Examples of application are discussed that demonstrate the functionality of the method even in cases where the one-dimensional phase integration fails completely.

© 2007 Optical Society of America

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  1. R. Fitzgerald, "Phase-Sensitive X-Ray Imaging," Phys. Today 53, 23-27 (2000).
  2. A. Momose, "Phase-sensitive imaging and phase tomography using X-ray interferometers," Opt. Express 11, 2303-2314 (2003).
    [CrossRef] [PubMed]
  3. A. Momose, "Demonstration of X-Ray Talbot Interferometry," Jpn. J. Appl. Phys. 42, L866-L868 (2003).
    [CrossRef]
  4. V. A. Somenkov, A. K. Tkalich, and S. S. Shilstein, "Refraction contrast in x-ray introscopy" Sov. Phys. Tech. Phys. 61, 197-201 (1991).
  5. V. V. Protopopov, and J. Sobota, "X-ray dark-field refraction-contrast imaging of micro-objects," Opt. Commun. 213, 267-279 (2002).
    [CrossRef]
  6. U. Bonse and M. Hart, "An X-ray interferometer," Appl. Phys. Lett. 6, 155-156 (1965).
    [CrossRef]
  7. A. Momose, T. Takeda, Y. Itai, and K. Hirano, "Phase-contrast X-ray computed tomography for observing biological soft tissues," Nature Med. 2, 473-475 (1996).
    [CrossRef] [PubMed]
  8. V. N. Ingal and E. A. Beliaevskaya, "X-ray plane-wave topography observation of the phase contrast from a non-crystalline object," J. Phys. D 28, 2314-2317 (1995).
    [CrossRef]
  9. T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, "Phase-contrast imaging of weakly absorbing materials using hard X-rays," Nature (London) 373, 595-598 (1995).
    [CrossRef]
  10. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 42, 2015-2025 (1997).
    [CrossRef] [PubMed]
  11. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature (London) 384, 335-337 (1996).
    [CrossRef]
  12. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, "Phase retrieval and differential phase-contrast imaging with low brilliance X-ray sources," Nat. Physics 2, 258 (2006).
    [CrossRef]
  13. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler "X-ray phase imaging with grating interferometer," Opt. Express 13, 6296-6304 (2005).
    [CrossRef] [PubMed]
  14. K. Creath, "Phase-measurement Interferometry Techniques," in Progress In Optics XXVI, E. Wolf, ed., (Elsevier Science, 1988) pp. 349-393.
  15. 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]
  16. 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, 054,101 (2005).
  17. Huber goniometer head type #1003, HUBER Diffraktionstechnik GmbH & Co. KG, Germany.
  18. M. Born and E. Wolf, Principles of Optics, Sixth ed. (Pergamon Press, Oxford, England, 1993).
  19. In fact, the quantity measured is not the absolute displacement ‹Δ›real ∝ ∂Φ/ ∂ x but ‹Δ›x = (‹Δx›real mod p2) because a steep phase gradient can shift the fringes more than one period p2. This phenomenon is known as phase-wrapping.
  20. M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
    [CrossRef] [PubMed]

2006

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, "Phase retrieval and differential phase-contrast imaging with low brilliance X-ray sources," Nat. Physics 2, 258 (2006).
[CrossRef]

2005

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, 054,101 (2005).

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

2004

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

2003

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

A. Momose, "Demonstration of X-Ray Talbot Interferometry," Jpn. J. Appl. Phys. 42, L866-L868 (2003).
[CrossRef]

2002

V. V. Protopopov, and J. Sobota, "X-ray dark-field refraction-contrast imaging of micro-objects," Opt. Commun. 213, 267-279 (2002).
[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]

2000

R. Fitzgerald, "Phase-Sensitive X-Ray Imaging," Phys. Today 53, 23-27 (2000).

1997

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 42, 2015-2025 (1997).
[CrossRef] [PubMed]

1996

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature (London) 384, 335-337 (1996).
[CrossRef]

A. Momose, T. Takeda, Y. Itai, and K. Hirano, "Phase-contrast X-ray computed tomography for observing biological soft tissues," Nature Med. 2, 473-475 (1996).
[CrossRef] [PubMed]

1995

V. N. Ingal and E. A. Beliaevskaya, "X-ray plane-wave topography observation of the phase contrast from a non-crystalline object," J. Phys. D 28, 2314-2317 (1995).
[CrossRef]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, "Phase-contrast imaging of weakly absorbing materials using hard X-rays," Nature (London) 373, 595-598 (1995).
[CrossRef]

1991

V. A. Somenkov, A. K. Tkalich, and S. S. Shilstein, "Refraction contrast in x-ray introscopy" Sov. Phys. Tech. Phys. 61, 197-201 (1991).

1965

U. Bonse and M. Hart, "An X-ray interferometer," Appl. Phys. Lett. 6, 155-156 (1965).
[CrossRef]

Arfelli, F.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 42, 2015-2025 (1997).
[CrossRef] [PubMed]

Arnison, M. R.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

Beliaevskaya, E. A.

V. N. Ingal and E. A. Beliaevskaya, "X-ray plane-wave topography observation of the phase contrast from a non-crystalline object," J. Phys. D 28, 2314-2317 (1995).
[CrossRef]

Bonse, U.

U. Bonse and M. Hart, "An X-ray interferometer," Appl. Phys. Lett. 6, 155-156 (1965).
[CrossRef]

Bunk, O.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, "Phase retrieval and differential phase-contrast imaging with low brilliance X-ray sources," Nat. Physics 2, 258 (2006).
[CrossRef]

Chapman, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 42, 2015-2025 (1997).
[CrossRef] [PubMed]

Cloetens, P.

Cogswell, C. J.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

David, C.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, "Phase retrieval and differential phase-contrast imaging with low brilliance X-ray sources," Nat. Physics 2, 258 (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, 054,101 (2005).

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler "X-ray phase imaging with 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]

Davis, T. J.

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, "Phase-contrast imaging of weakly absorbing materials using hard X-rays," Nature (London) 373, 595-598 (1995).
[CrossRef]

Diaz, A.

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

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, 054,101 (2005).

Fitzgerald, R.

R. Fitzgerald, "Phase-Sensitive X-Ray Imaging," Phys. Today 53, 23-27 (2000).

Gao, D.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature (London) 384, 335-337 (1996).
[CrossRef]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, "Phase-contrast imaging of weakly absorbing materials using hard X-rays," Nature (London) 373, 595-598 (1995).
[CrossRef]

Gmür, N.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 42, 2015-2025 (1997).
[CrossRef] [PubMed]

Gureyev, T. E.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature (London) 384, 335-337 (1996).
[CrossRef]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, "Phase-contrast imaging of weakly absorbing materials using hard X-rays," Nature (London) 373, 595-598 (1995).
[CrossRef]

Hart, M.

U. Bonse and M. Hart, "An X-ray interferometer," Appl. Phys. Lett. 6, 155-156 (1965).
[CrossRef]

Hirano, K.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, "Phase-contrast X-ray computed tomography for observing biological soft tissues," Nature Med. 2, 473-475 (1996).
[CrossRef] [PubMed]

Ingal, V. N.

V. N. Ingal and E. A. Beliaevskaya, "X-ray plane-wave topography observation of the phase contrast from a non-crystalline object," J. Phys. D 28, 2314-2317 (1995).
[CrossRef]

Itai, Y.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, "Phase-contrast X-ray computed tomography for observing biological soft tissues," Nature Med. 2, 473-475 (1996).
[CrossRef] [PubMed]

Johnston, R. E.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 42, 2015-2025 (1997).
[CrossRef] [PubMed]

Larkin, K. G.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

Menk, R.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 42, 2015-2025 (1997).
[CrossRef] [PubMed]

Momose, A.

A. Momose, "Demonstration of X-Ray Talbot Interferometry," Jpn. J. Appl. Phys. 42, L866-L868 (2003).
[CrossRef]

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

A. Momose, T. Takeda, Y. Itai, and K. Hirano, "Phase-contrast X-ray computed tomography for observing biological soft tissues," Nature Med. 2, 473-475 (1996).
[CrossRef] [PubMed]

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, 054,101 (2005).

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]

Pfeiffer, F.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, "Phase retrieval and differential phase-contrast imaging with low brilliance X-ray sources," Nat. Physics 2, 258 (2006).
[CrossRef]

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

Pisano, E.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 42, 2015-2025 (1997).
[CrossRef] [PubMed]

Pogany, A.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature (London) 384, 335-337 (1996).
[CrossRef]

Protopopov, V. V.

V. V. Protopopov, and J. Sobota, "X-ray dark-field refraction-contrast imaging of micro-objects," Opt. Commun. 213, 267-279 (2002).
[CrossRef]

Sayers, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 42, 2015-2025 (1997).
[CrossRef] [PubMed]

Sheppard, C. J. R.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

Shilstein, S. S.

V. A. Somenkov, A. K. Tkalich, and S. S. Shilstein, "Refraction contrast in x-ray introscopy" Sov. Phys. Tech. Phys. 61, 197-201 (1991).

Smith, N. I.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

Sobota, J.

V. V. Protopopov, and J. Sobota, "X-ray dark-field refraction-contrast imaging of micro-objects," Opt. Commun. 213, 267-279 (2002).
[CrossRef]

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]

Somenkov, V. A.

V. A. Somenkov, A. K. Tkalich, and S. S. Shilstein, "Refraction contrast in x-ray introscopy" Sov. Phys. Tech. Phys. 61, 197-201 (1991).

Stampanoni, M.

Stevenson, A. W.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature (London) 384, 335-337 (1996).
[CrossRef]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, "Phase-contrast imaging of weakly absorbing materials using hard X-rays," Nature (London) 373, 595-598 (1995).
[CrossRef]

Takeda, T.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, "Phase-contrast X-ray computed tomography for observing biological soft tissues," Nature Med. 2, 473-475 (1996).
[CrossRef] [PubMed]

Thomlinson, W.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 42, 2015-2025 (1997).
[CrossRef] [PubMed]

Tkalich, A. K.

V. A. Somenkov, A. K. Tkalich, and S. S. Shilstein, "Refraction contrast in x-ray introscopy" Sov. Phys. Tech. Phys. 61, 197-201 (1991).

Washburn, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 42, 2015-2025 (1997).
[CrossRef] [PubMed]

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, "Phase retrieval and differential phase-contrast imaging with low brilliance X-ray sources," Nat. Physics 2, 258 (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, 054,101 (2005).

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

Wilkins, S. W.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature (London) 384, 335-337 (1996).
[CrossRef]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, "Phase-contrast imaging of weakly absorbing materials using hard X-rays," Nature (London) 373, 595-598 (1995).
[CrossRef]

Zhong, Z.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 42, 2015-2025 (1997).
[CrossRef] [PubMed]

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, 054,101 (2005).

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler "X-ray phase imaging with 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. Phys. Lett.

U. Bonse and M. Hart, "An X-ray interferometer," Appl. Phys. Lett. 6, 155-156 (1965).
[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]

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, 054,101 (2005).

J. Microsc.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

J. Phys. D

V. N. Ingal and E. A. Beliaevskaya, "X-ray plane-wave topography observation of the phase contrast from a non-crystalline object," J. Phys. D 28, 2314-2317 (1995).
[CrossRef]

Jpn. J. Appl. Phys.

A. Momose, "Demonstration of X-Ray Talbot Interferometry," Jpn. J. Appl. Phys. 42, L866-L868 (2003).
[CrossRef]

Nat. Physics

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, "Phase retrieval and differential phase-contrast imaging with low brilliance X-ray sources," Nat. Physics 2, 258 (2006).
[CrossRef]

Nature (London)

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, "Phase-contrast imaging of weakly absorbing materials using hard X-rays," Nature (London) 373, 595-598 (1995).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature (London) 384, 335-337 (1996).
[CrossRef]

Nature Med.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, "Phase-contrast X-ray computed tomography for observing biological soft tissues," Nature Med. 2, 473-475 (1996).
[CrossRef] [PubMed]

Opt. Commun.

V. V. Protopopov, and J. Sobota, "X-ray dark-field refraction-contrast imaging of micro-objects," Opt. Commun. 213, 267-279 (2002).
[CrossRef]

Opt. Express

Phys. Med. Biol.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 42, 2015-2025 (1997).
[CrossRef] [PubMed]

Phys. Today

R. Fitzgerald, "Phase-Sensitive X-Ray Imaging," Phys. Today 53, 23-27 (2000).

Sov. Phys. Tech. Phys.

V. A. Somenkov, A. K. Tkalich, and S. S. Shilstein, "Refraction contrast in x-ray introscopy" Sov. Phys. Tech. Phys. 61, 197-201 (1991).

Other

K. Creath, "Phase-measurement Interferometry Techniques," in Progress In Optics XXVI, E. Wolf, ed., (Elsevier Science, 1988) pp. 349-393.

Huber goniometer head type #1003, HUBER Diffraktionstechnik GmbH & Co. KG, Germany.

M. Born and E. Wolf, Principles of Optics, Sixth ed. (Pergamon Press, Oxford, England, 1993).

In fact, the quantity measured is not the absolute displacement ‹Δ›real ∝ ∂Φ/ ∂ x but ‹Δ›x = (‹Δx›real mod p2) because a steep phase gradient can shift the fringes more than one period p2. This phenomenon is known as phase-wrapping.

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Figures (4)

Fig. 1.
Fig. 1.

Schematic overview of the interferometer set-up consisting of an x-ray tube, a source grating G 0, a phase grating G 1, an analyzer grating G 2 and an imaging detector. DPC images along the x- and y-axis can be measured by alignment of the gratings in vertical (bottom) and horizontal (top) direction, respectively. In order to measure DPC images for both directions x and y, in our experiments the sample was turned by 90° instead of the gratings.

Fig. 2.
Fig. 2.

DPC image of a flower-like shape made of plastic for x- (a) and y-direction (b) of the gradient. (d) and (e) show the corresponding integrated phase images, respectively. Since some of the petals do not completely fit into the field of view shadow-like artifacts arise therein upon integration . (c) shows the image obtained with the 2D algorithm.

Fig. 3.
Fig. 3.

DPC images of two concentrically arranged plastic rings acquired with high (a) and reduced (b) counting statistics, both of them for x-direction of the gradient. The corresponding integrated phase images are shown in (d) and (e), respectively. (c) shows the image obtained with the 2D algorithm applied on the lower statistics data.

Fig. 4.
Fig. 4.

DPC image of a conch for x- (a) and y-direction (b) of the gradient. (d) and (e) show their integrated phase images, respectively. (f) shows the image from the 2D algorithm and in comparison (c) shows the absorption image.

Equations (4)

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α ( x ) = λ 2 π Φ ( x ) x ,
g ( x , y ) = Φ x ( x , y ) + i Φ y ( x , y ) Φ x ( x , y ) + i Φ y ( x , y ) .
[ g ( x , y ) ] ( k , l ) + g ( x , y ) e 2 πi ( kx + ly ) dxdy = 2 πi ( k + il ) [ Φ ( x , y ) ] ( k , l ) ,
Φ ( x , y ) = 1 [ [ Φ x + i Φ y ] ( k , l ) 2 πi ( k + il ) ] ( x , y ) .

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