Abstract

In order to quantitatively determine the projected electron densities of a sample, one needs to extract the monochromatic fringe phase shifts from the polychromatic fringe phase shifts measured in the grating interferometry with incoherent X-ray sources. In this work the authors propose a novel analytic approach that allows to directly compute the monochromatic fringe shifts from the polychromatic fringe shifts. This approach is validated with numerical simulations of several grating interferometry setups. This work provides a useful tool in quantitative imaging for biomedical and material science applications.

© 2017 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Predicting visibility of interference fringes in X-ray grating interferometry

Aimin Yan, Xizeng Wu, and Hong Liu
Opt. Express 24(14) 15927-15939 (2016)

Quantitative analysis of fringe visibility in grating-based x-ray phase-contrast imaging

Jianheng Huang, Yaohu Lei, Yang Du, Xin Liu, Jinchuan Guo, Ji Li, and Hanben Niu
J. Opt. Soc. Am. A 33(1) 69-73 (2016)

Efficiency of capturing a phase image using cone-beam x-ray Talbot interferometry

Wataru Yashiro, Yoshihiro Takeda, and Atsushi Momose
J. Opt. Soc. Am. A 25(8) 2025-2039 (2008)

References

  • View by:
  • |
  • |
  • |

  1. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, H. Takai, and Y. Suzuki, “Demonstration of x-ray talbot interferometry,” Jpn. J. Appl. Phys. 42, 866 (2003).
    [Crossref]
  2. 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]
  3. A. Momose, “Recent advances in x-ray phase imaging,” Jpn. J. Appl. Phys. 44, 6355–6367 (2005).
    [Crossref]
  4. 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]
  5. 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]
  6. P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc Natl. Acad. Sci. USA 107, 13576–13581 (2010).
    [Crossref] [PubMed]
  7. N. Bevins, J. Zambelli, K. Li, Z. Qi, and G.-H. Chen, “Multicontrast x-ray computed tomography imaging using talbot-lau interferometry without phase stepping,” Med. Phys. 39, 424–428 (2012).
    [Crossref] [PubMed]
  8. X. Tang, Y. Yang, and S. Tang, “Characterization of imaging performance in differential phase contrast ct compared with the conventional ct: Spectrum of noise equivalent quanta neq(k),” Med. Phys. 39, 4367–4382 (2012).
    [Crossref]
  9. T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional gratings-based phase-contrast imaging using a conventional x-ray tube,” Opt. Lett. 36, 3551–3553 (2011).
    [Crossref] [PubMed]
  10. E. Bennett, R. Kopace, A. Stein, and H. Wen, “A grating-based single shot x-ray phase contrast and diffraction method for in vivo imaging,” Med. Phys. 37, 6047–6054 (2010).
    [Crossref] [PubMed]
  11. M. Jiang, C. Wyatt, and G. Wang, “X-ray phase-contrast imaging with three 2d gratings,” Int. J. Biomed. Imaging 58, 827152 (2008).
    [PubMed]
  12. J. Rizzi, T. Weitkamp, N. Guerineau, M. Idir, P. Mercere, G. Druart, G. Vincent, P. Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett. 36, 1398–1400 (2011).
    [Crossref] [PubMed]
  13. H. Itoh, K. Nagai, G. Sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based x-ray phase-contrast imaging using fourier transform phase retrieval,” Opt. Express 19, 3339–3346 (2011).
    [Crossref] [PubMed]
  14. J. Rizzi, P. Mercere, M. Idir, P. D. Silva, G. Vincent, and J. Primot, “X-ray phase contrast imaging and noise evaluation using a single phase grating interferometer,” Opt. Express 21, 17340–17351 (2013).
    [Crossref] [PubMed]
  15. A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Physics in Medicine and Biology 58, R1–R35 (2013).
    [Crossref]
  16. N. Morimoto, S. Fujino, K. Ohshima, J. Harada, T. Hosoi, H. Watanabe, and T. Shimura, “X-ray phase contrast imaging by compact talbot-lau interferometer with a single transmission grating,” Opt. Lett. 39, 4297–4300 (2014).
    [Crossref] [PubMed]
  17. N. Morimoto, S. Fujino, A. Yamazaki, Y. Ito, T. Hosoi, H. Watanabe, and T. Shimura, “Two dimensional x-ray phase imaging using single grating interferometer with embedded x-ray targets,” Opt. Express 23, 16582–16588 (2015).
    [Crossref] [PubMed]
  18. A. Yan, X. Wu, and H. Liu, “A general theory of interference fringes in x-ray phase grating imaging,” Med. Phys. 42, 3036–3047 (2015).
    [Crossref] [PubMed]
  19. A. Yan, X. Wu, and H. Liu, “Predicting visibility of interference fringes in x-ray grating interometry,” Opt. Express 24, 15927–15939 (2016).
    [Crossref] [PubMed]
  20. L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental realization of high-sensitivity laboratory x-ray grating-based phase-contrast computed tomography,” Scientific Reports 6, 24022 (2016).
    [Crossref]
  21. M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” Journal of Microscopy 232, 145–157 (2008).
    [Crossref] [PubMed]
  22. P. Munro and A. Olivo, “X-ray phase-contrast imaging with polychromatic sources and the concept of effective energy,” Physical Review A 87, 053838 (2013).
    [Crossref]
  23. A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based x-ray phase-contrast imaging,” Rev. Sci. Instrum. 85, 103708 (2014).
    [Crossref] [PubMed]
  24. M. Chabior, T. Donath, C. David, O. Bun, M. Schuster, C. Schroer, and F. Pfeiffer, “Beam hardening effects in grating-based x-ray phase-contrast imaging,” Med. Phys 38, 1189–1195 (2011).
    [Crossref] [PubMed]
  25. J. Goodman, Statistical Optics (John Wiley and Sons, Inc., 1985).
  26. A. Ritter, P. Bartl, F. Bayer, K. C. Gödel, W. Haas, T. Michel, G. Pelzer, J. Rieger, T. Weber, A. Zang, and G. Anton, “Simulation framework for coherent and incoherent x-ray imaging and its application in talbot-lau dark-field imaging,” Opt. Express 22, 23276–23289 (2014).
    [Crossref] [PubMed]
  27. A. Hipp, M. Willner, J. Herzen, S. Auweter, M. Chabior, J. Meiser, K. Achterhold, J. Mohr, and F. Pfeiffer, “Energy-resolved visibility analysis of grating interferometers operated at polychromatic x-ray sources,” Opt. Express 22, 30394–30409 (2014).
    [Crossref]

2016 (2)

A. Yan, X. Wu, and H. Liu, “Predicting visibility of interference fringes in x-ray grating interometry,” Opt. Express 24, 15927–15939 (2016).
[Crossref] [PubMed]

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental realization of high-sensitivity laboratory x-ray grating-based phase-contrast computed tomography,” Scientific Reports 6, 24022 (2016).
[Crossref]

2015 (2)

2014 (4)

2013 (3)

P. Munro and A. Olivo, “X-ray phase-contrast imaging with polychromatic sources and the concept of effective energy,” Physical Review A 87, 053838 (2013).
[Crossref]

J. Rizzi, P. Mercere, M. Idir, P. D. Silva, G. Vincent, and J. Primot, “X-ray phase contrast imaging and noise evaluation using a single phase grating interferometer,” Opt. Express 21, 17340–17351 (2013).
[Crossref] [PubMed]

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Physics in Medicine and Biology 58, R1–R35 (2013).
[Crossref]

2012 (2)

N. Bevins, J. Zambelli, K. Li, Z. Qi, and G.-H. Chen, “Multicontrast x-ray computed tomography imaging using talbot-lau interferometry without phase stepping,” Med. Phys. 39, 424–428 (2012).
[Crossref] [PubMed]

X. Tang, Y. Yang, and S. Tang, “Characterization of imaging performance in differential phase contrast ct compared with the conventional ct: Spectrum of noise equivalent quanta neq(k),” Med. Phys. 39, 4367–4382 (2012).
[Crossref]

2011 (4)

2010 (3)

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]

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[Crossref] [PubMed]

E. Bennett, R. Kopace, A. Stein, and H. Wen, “A grating-based single shot x-ray phase contrast and diffraction method for in vivo imaging,” Med. Phys. 37, 6047–6054 (2010).
[Crossref] [PubMed]

2008 (2)

M. Jiang, C. Wyatt, and G. Wang, “X-ray phase-contrast imaging with three 2d gratings,” Int. J. Biomed. Imaging 58, 827152 (2008).
[PubMed]

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” Journal of Microscopy 232, 145–157 (2008).
[Crossref] [PubMed]

2006 (1)

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

2005 (2)

2003 (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, H. Takai, and Y. Suzuki, “Demonstration of x-ray talbot interferometry,” Jpn. J. Appl. Phys. 42, 866 (2003).
[Crossref]

Achterhold, K.

Anton, G.

Auweter, S.

Bartl, P.

Baumann, J.

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” Journal of Microscopy 232, 145–157 (2008).
[Crossref] [PubMed]

Bayer, F.

Bennett, E.

E. Bennett, R. Kopace, A. Stein, and H. Wen, “A grating-based single shot x-ray phase contrast and diffraction method for in vivo imaging,” Med. Phys. 37, 6047–6054 (2010).
[Crossref] [PubMed]

Bequé, D.

A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based x-ray phase-contrast imaging,” Rev. Sci. Instrum. 85, 103708 (2014).
[Crossref] [PubMed]

Bevins, N.

N. Bevins, J. Zambelli, K. Li, Z. Qi, and G.-H. Chen, “Multicontrast x-ray computed tomography imaging using talbot-lau interferometry without phase stepping,” Med. Phys. 39, 424–428 (2012).
[Crossref] [PubMed]

Birnbacher, L.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental realization of high-sensitivity laboratory x-ray grating-based phase-contrast computed tomography,” Scientific Reports 6, 24022 (2016).
[Crossref]

Bravin, A.

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Physics in Medicine and Biology 58, R1–R35 (2013).
[Crossref]

Bun, O.

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

Bunk, O.

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” Journal of Microscopy 232, 145–157 (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]

Chabior, M.

A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based x-ray phase-contrast imaging,” Rev. Sci. Instrum. 85, 103708 (2014).
[Crossref] [PubMed]

A. Hipp, M. Willner, J. Herzen, S. Auweter, M. Chabior, J. Meiser, K. Achterhold, J. Mohr, and F. Pfeiffer, “Energy-resolved visibility analysis of grating interferometers operated at polychromatic x-ray sources,” Opt. Express 22, 30394–30409 (2014).
[Crossref]

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

Chen, G.-H.

N. Bevins, J. Zambelli, K. Li, Z. Qi, and G.-H. Chen, “Multicontrast x-ray computed tomography imaging using talbot-lau interferometry without phase stepping,” Med. Phys. 39, 424–428 (2012).
[Crossref] [PubMed]

Cloetens, P.

Coan, P.

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Physics in Medicine and Biology 58, R1–R35 (2013).
[Crossref]

Coman, J.

A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based x-ray phase-contrast imaging,” Rev. Sci. Instrum. 85, 103708 (2014).
[Crossref] [PubMed]

Cozzini, C.

A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based x-ray phase-contrast imaging,” Rev. Sci. Instrum. 85, 103708 (2014).
[Crossref] [PubMed]

David, C.

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

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” Journal of Microscopy 232, 145–157 (2008).
[Crossref] [PubMed]

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

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

Den, T.

Diaz, A.

Donath, T.

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

Druart, G.

Engelhardt, M.

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” Journal of Microscopy 232, 145–157 (2008).
[Crossref] [PubMed]

Fujino, S.

Gödel, K. C.

Goodman, J.

J. Goodman, Statistical Optics (John Wiley and Sons, Inc., 1985).

Guerineau, N.

Haas, W.

Hamaishi, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, H. Takai, and Y. Suzuki, “Demonstration of x-ray talbot interferometry,” Jpn. J. Appl. Phys. 42, 866 (2003).
[Crossref]

Harada, J.

Herzen, J.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental realization of high-sensitivity laboratory x-ray grating-based phase-contrast computed tomography,” Scientific Reports 6, 24022 (2016).
[Crossref]

A. Hipp, M. Willner, J. Herzen, S. Auweter, M. Chabior, J. Meiser, K. Achterhold, J. Mohr, and F. Pfeiffer, “Energy-resolved visibility analysis of grating interferometers operated at polychromatic x-ray sources,” Opt. Express 22, 30394–30409 (2014).
[Crossref]

Hipp, A.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental realization of high-sensitivity laboratory x-ray grating-based phase-contrast computed tomography,” Scientific Reports 6, 24022 (2016).
[Crossref]

A. Hipp, M. Willner, J. Herzen, S. Auweter, M. Chabior, J. Meiser, K. Achterhold, J. Mohr, and F. Pfeiffer, “Energy-resolved visibility analysis of grating interferometers operated at polychromatic x-ray sources,” Opt. Express 22, 30394–30409 (2014).
[Crossref]

Hosoi, T.

Idir, M.

Ito, Y.

Itoh, H.

Jiang, M.

M. Jiang, C. Wyatt, and G. Wang, “X-ray phase-contrast imaging with three 2d gratings,” Int. J. Biomed. Imaging 58, 827152 (2008).
[PubMed]

Kawabata, K.

Kawamoto, S.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, H. Takai, and Y. Suzuki, “Demonstration of x-ray talbot interferometry,” Jpn. J. Appl. Phys. 42, 866 (2003).
[Crossref]

Koch, F.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental realization of high-sensitivity laboratory x-ray grating-based phase-contrast computed tomography,” Scientific Reports 6, 24022 (2016).
[Crossref]

Kondoh, T.

Kopace, R.

E. Bennett, R. Kopace, A. Stein, and H. Wen, “A grating-based single shot x-ray phase contrast and diffraction method for in vivo imaging,” Med. Phys. 37, 6047–6054 (2010).
[Crossref] [PubMed]

Kottler, C.

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” Journal of Microscopy 232, 145–157 (2008).
[Crossref] [PubMed]

Koyama, I.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, H. Takai, and Y. Suzuki, “Demonstration of x-ray talbot interferometry,” Jpn. J. Appl. Phys. 42, 866 (2003).
[Crossref]

Kunka, D.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental realization of high-sensitivity laboratory x-ray grating-based phase-contrast computed tomography,” Scientific Reports 6, 24022 (2016).
[Crossref]

Langner, O.

A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based x-ray phase-contrast imaging,” Rev. Sci. Instrum. 85, 103708 (2014).
[Crossref] [PubMed]

Li, K.

N. Bevins, J. Zambelli, K. Li, Z. Qi, and G.-H. Chen, “Multicontrast x-ray computed tomography imaging using talbot-lau interferometry without phase stepping,” Med. Phys. 39, 424–428 (2012).
[Crossref] [PubMed]

Liu, H.

A. Yan, X. Wu, and H. Liu, “Predicting visibility of interference fringes in x-ray grating interometry,” Opt. Express 24, 15927–15939 (2016).
[Crossref] [PubMed]

A. Yan, X. Wu, and H. Liu, “A general theory of interference fringes in x-ray phase grating imaging,” Med. Phys. 42, 3036–3047 (2015).
[Crossref] [PubMed]

Liu, X.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[Crossref] [PubMed]

Liu, Y.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[Crossref] [PubMed]

Marone, F.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[Crossref] [PubMed]

Marschner, M.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental realization of high-sensitivity laboratory x-ray grating-based phase-contrast computed tomography,” Scientific Reports 6, 24022 (2016).
[Crossref]

McDonald, S.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[Crossref] [PubMed]

Meiser, J.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental realization of high-sensitivity laboratory x-ray grating-based phase-contrast computed tomography,” Scientific Reports 6, 24022 (2016).
[Crossref]

A. Hipp, M. Willner, J. Herzen, S. Auweter, M. Chabior, J. Meiser, K. Achterhold, J. Mohr, and F. Pfeiffer, “Energy-resolved visibility analysis of grating interferometers operated at polychromatic x-ray sources,” Opt. Express 22, 30394–30409 (2014).
[Crossref]

Mercere, P.

Michel, T.

Mohr, J.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental realization of high-sensitivity laboratory x-ray grating-based phase-contrast computed tomography,” Scientific Reports 6, 24022 (2016).
[Crossref]

A. Hipp, M. Willner, J. Herzen, S. Auweter, M. Chabior, J. Meiser, K. Achterhold, J. Mohr, and F. Pfeiffer, “Energy-resolved visibility analysis of grating interferometers operated at polychromatic x-ray sources,” Opt. Express 22, 30394–30409 (2014).
[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]

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, H. Takai, and Y. Suzuki, “Demonstration of x-ray talbot interferometry,” Jpn. J. Appl. Phys. 42, 866 (2003).
[Crossref]

Morimoto, N.

Munro, P.

P. Munro and A. Olivo, “X-ray phase-contrast imaging with polychromatic sources and the concept of effective energy,” Physical Review A 87, 053838 (2013).
[Crossref]

Nagai, K.

Nakamura, T.

Ohshima, K.

Olivo, A.

P. Munro and A. Olivo, “X-ray phase-contrast imaging with polychromatic sources and the concept of effective energy,” Physical Review A 87, 053838 (2013).
[Crossref]

Ouchi, C.

Pelzer, G.

Pfeiffer, F.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental realization of high-sensitivity laboratory x-ray grating-based phase-contrast computed tomography,” Scientific Reports 6, 24022 (2016).
[Crossref]

A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based x-ray phase-contrast imaging,” Rev. Sci. Instrum. 85, 103708 (2014).
[Crossref] [PubMed]

A. Hipp, M. Willner, J. Herzen, S. Auweter, M. Chabior, J. Meiser, K. Achterhold, J. Mohr, and F. Pfeiffer, “Energy-resolved visibility analysis of grating interferometers operated at polychromatic x-ray sources,” Opt. Express 22, 30394–30409 (2014).
[Crossref]

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

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” Journal of Microscopy 232, 145–157 (2008).
[Crossref] [PubMed]

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

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

Primot, J.

Qi, Z.

N. Bevins, J. Zambelli, K. Li, Z. Qi, and G.-H. Chen, “Multicontrast x-ray computed tomography imaging using talbot-lau interferometry without phase stepping,” Med. Phys. 39, 424–428 (2012).
[Crossref] [PubMed]

Rieger, J.

Ritter, A.

Rizzi, J.

Ruiz-Yaniz, M.

A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based x-ray phase-contrast imaging,” Rev. Sci. Instrum. 85, 103708 (2014).
[Crossref] [PubMed]

Sarapata, A.

A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based x-ray phase-contrast imaging,” Rev. Sci. Instrum. 85, 103708 (2014).
[Crossref] [PubMed]

Sato, G.

Schroer, C.

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

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” Journal of Microscopy 232, 145–157 (2008).
[Crossref] [PubMed]

Schröter, T.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental realization of high-sensitivity laboratory x-ray grating-based phase-contrast computed tomography,” Scientific Reports 6, 24022 (2016).
[Crossref]

Schuster, M.

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

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” Journal of Microscopy 232, 145–157 (2008).
[Crossref] [PubMed]

Setomoto, Y.

Shimura, T.

Silva, P.

Silva, P. D.

Sperl, J. I.

A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based x-ray phase-contrast imaging,” Rev. Sci. Instrum. 85, 103708 (2014).
[Crossref] [PubMed]

Stampanoni, M.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[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]

Stein, A.

E. Bennett, R. Kopace, A. Stein, and H. Wen, “A grating-based single shot x-ray phase contrast and diffraction method for in vivo imaging,” Med. Phys. 37, 6047–6054 (2010).
[Crossref] [PubMed]

Suortti, P.

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Physics in Medicine and Biology 58, R1–R35 (2013).
[Crossref]

Suzuki, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, H. Takai, and Y. Suzuki, “Demonstration of x-ray talbot interferometry,” Jpn. J. Appl. Phys. 42, 866 (2003).
[Crossref]

Takai, H.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, H. Takai, and Y. Suzuki, “Demonstration of x-ray talbot interferometry,” Jpn. J. Appl. Phys. 42, 866 (2003).
[Crossref]

Tang, S.

X. Tang, Y. Yang, and S. Tang, “Characterization of imaging performance in differential phase contrast ct compared with the conventional ct: Spectrum of noise equivalent quanta neq(k),” Med. Phys. 39, 4367–4382 (2012).
[Crossref]

Tang, X.

X. Tang, Y. Yang, and S. Tang, “Characterization of imaging performance in differential phase contrast ct compared with the conventional ct: Spectrum of noise equivalent quanta neq(k),” Med. Phys. 39, 4367–4382 (2012).
[Crossref]

Terui, Y.

Teshima, T.

Velroyen, A.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental realization of high-sensitivity laboratory x-ray grating-based phase-contrast computed tomography,” Scientific Reports 6, 24022 (2016).
[Crossref]

Vincent, G.

Wang, G.

M. Jiang, C. Wyatt, and G. Wang, “X-ray phase-contrast imaging with three 2d gratings,” Int. J. Biomed. Imaging 58, 827152 (2008).
[PubMed]

Wang, Z.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[Crossref] [PubMed]

Watanabe, H.

Weber, T.

Weitkamp, T.

Wen, H.

E. Bennett, R. Kopace, A. Stein, and H. Wen, “A grating-based single shot x-ray phase contrast and diffraction method for in vivo imaging,” Med. Phys. 37, 6047–6054 (2010).
[Crossref] [PubMed]

Willner, M.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental realization of high-sensitivity laboratory x-ray grating-based phase-contrast computed tomography,” Scientific Reports 6, 24022 (2016).
[Crossref]

A. Hipp, M. Willner, J. Herzen, S. Auweter, M. Chabior, J. Meiser, K. Achterhold, J. Mohr, and F. Pfeiffer, “Energy-resolved visibility analysis of grating interferometers operated at polychromatic x-ray sources,” Opt. Express 22, 30394–30409 (2014).
[Crossref]

Wu, X.

A. Yan, X. Wu, and H. Liu, “Predicting visibility of interference fringes in x-ray grating interometry,” Opt. Express 24, 15927–15939 (2016).
[Crossref] [PubMed]

A. Yan, X. Wu, and H. Liu, “A general theory of interference fringes in x-ray phase grating imaging,” Med. Phys. 42, 3036–3047 (2015).
[Crossref] [PubMed]

Wu, Z.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[Crossref] [PubMed]

Wyatt, C.

M. Jiang, C. Wyatt, and G. Wang, “X-ray phase-contrast imaging with three 2d gratings,” Int. J. Biomed. Imaging 58, 827152 (2008).
[PubMed]

Yamaguchi, K.

Yamazaki, A.

Yan, A.

A. Yan, X. Wu, and H. Liu, “Predicting visibility of interference fringes in x-ray grating interometry,” Opt. Express 24, 15927–15939 (2016).
[Crossref] [PubMed]

A. Yan, X. Wu, and H. Liu, “A general theory of interference fringes in x-ray phase grating imaging,” Med. Phys. 42, 3036–3047 (2015).
[Crossref] [PubMed]

Yang, Y.

X. Tang, Y. Yang, and S. Tang, “Characterization of imaging performance in differential phase contrast ct compared with the conventional ct: Spectrum of noise equivalent quanta neq(k),” Med. Phys. 39, 4367–4382 (2012).
[Crossref]

Yashiro, W.

Zambelli, J.

N. Bevins, J. Zambelli, K. Li, Z. Qi, and G.-H. Chen, “Multicontrast x-ray computed tomography imaging using talbot-lau interferometry without phase stepping,” Med. Phys. 39, 424–428 (2012).
[Crossref] [PubMed]

Zanette, I.

A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based x-ray phase-contrast imaging,” Rev. Sci. Instrum. 85, 103708 (2014).
[Crossref] [PubMed]

Zang, A.

Zhang, K.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[Crossref] [PubMed]

Zhu, P.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[Crossref] [PubMed]

Ziegler, E.

Int. J. Biomed. Imaging (1)

M. Jiang, C. Wyatt, and G. Wang, “X-ray phase-contrast imaging with three 2d gratings,” Int. J. Biomed. Imaging 58, 827152 (2008).
[PubMed]

Journal of Microscopy (1)

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” Journal of Microscopy 232, 145–157 (2008).
[Crossref] [PubMed]

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, H. Takai, and Y. Suzuki, “Demonstration of x-ray talbot interferometry,” Jpn. J. Appl. Phys. 42, 866 (2003).
[Crossref]

Med. Phys (1)

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

Med. Phys. (4)

N. Bevins, J. Zambelli, K. Li, Z. Qi, and G.-H. Chen, “Multicontrast x-ray computed tomography imaging using talbot-lau interferometry without phase stepping,” Med. Phys. 39, 424–428 (2012).
[Crossref] [PubMed]

X. Tang, Y. Yang, and S. Tang, “Characterization of imaging performance in differential phase contrast ct compared with the conventional ct: Spectrum of noise equivalent quanta neq(k),” Med. Phys. 39, 4367–4382 (2012).
[Crossref]

E. Bennett, R. Kopace, A. Stein, and H. Wen, “A grating-based single shot x-ray phase contrast and diffraction method for in vivo imaging,” Med. Phys. 37, 6047–6054 (2010).
[Crossref] [PubMed]

A. Yan, X. Wu, and H. Liu, “A general theory of interference fringes in x-ray phase grating imaging,” Med. Phys. 42, 3036–3047 (2015).
[Crossref] [PubMed]

Nat. Phys. (1)

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

Opt. Express (8)

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]

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

A. Yan, X. Wu, and H. Liu, “Predicting visibility of interference fringes in x-ray grating interometry,” Opt. Express 24, 15927–15939 (2016).
[Crossref] [PubMed]

H. Itoh, K. Nagai, G. Sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based x-ray phase-contrast imaging using fourier transform phase retrieval,” Opt. Express 19, 3339–3346 (2011).
[Crossref] [PubMed]

J. Rizzi, P. Mercere, M. Idir, P. D. Silva, G. Vincent, and J. Primot, “X-ray phase contrast imaging and noise evaluation using a single phase grating interferometer,” Opt. Express 21, 17340–17351 (2013).
[Crossref] [PubMed]

A. Ritter, P. Bartl, F. Bayer, K. C. Gödel, W. Haas, T. Michel, G. Pelzer, J. Rieger, T. Weber, A. Zang, and G. Anton, “Simulation framework for coherent and incoherent x-ray imaging and its application in talbot-lau dark-field imaging,” Opt. Express 22, 23276–23289 (2014).
[Crossref] [PubMed]

A. Hipp, M. Willner, J. Herzen, S. Auweter, M. Chabior, J. Meiser, K. Achterhold, J. Mohr, and F. Pfeiffer, “Energy-resolved visibility analysis of grating interferometers operated at polychromatic x-ray sources,” Opt. Express 22, 30394–30409 (2014).
[Crossref]

N. Morimoto, S. Fujino, A. Yamazaki, Y. Ito, T. Hosoi, H. Watanabe, and T. Shimura, “Two dimensional x-ray phase imaging using single grating interferometer with embedded x-ray targets,” Opt. Express 23, 16582–16588 (2015).
[Crossref] [PubMed]

Opt. Lett. (3)

Physical Review A (1)

P. Munro and A. Olivo, “X-ray phase-contrast imaging with polychromatic sources and the concept of effective energy,” Physical Review A 87, 053838 (2013).
[Crossref]

Physics in Medicine and Biology (1)

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Physics in Medicine and Biology 58, R1–R35 (2013).
[Crossref]

Proc Natl. Acad. Sci. USA (1)

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc Natl. Acad. Sci. USA 107, 13576–13581 (2010).
[Crossref] [PubMed]

Rev. Sci. Instrum. (1)

A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based x-ray phase-contrast imaging,” Rev. Sci. Instrum. 85, 103708 (2014).
[Crossref] [PubMed]

Scientific Reports (1)

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental realization of high-sensitivity laboratory x-ray grating-based phase-contrast computed tomography,” Scientific Reports 6, 24022 (2016).
[Crossref]

Other (1)

J. Goodman, Statistical Optics (John Wiley and Sons, Inc., 1985).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Schematic of an x-ray phase grating interferometer with a microfocus source.

Fig. 2
Fig. 2

Profile of phantom thickness.

Fig. 3
Fig. 3

35 kVp x-ray effective spectrum employed in the simulations.

Fig. 4
Fig. 4

Plots of the simulation results: The design energy is set to 18 keV with π-grating interferometer. In this simulation, the true theoretical values of ϕ 2 , D Theory ( x , y ) (the solid blue line) are set to a relatively small value of 5 degrees. The dash-dot green line, ϕ2,Poly(x, y), is the polychromatic fringe phase shifts retrieved from the simulated fringe pattern by using the Fourier analysis method. The dotted cyan line is the 1st order approximate solution ϕ 2 , D ( 1 ) ( x , y ) using Eq. (13). The dashed red line is the higher order approximate solution ϕ 2 , D ( q ) ( x , y ) using Eq. (14).

Fig. 5
Fig. 5

Plots of the simulation results: The design energy is set to 18 keV with π-grating interferometer. In this simulation, the true theoretical values of ϕ 2 , D Theory ( x , y ) (the solid blue line) are set to a large value of 45 degress. The dash-dot green line, ϕ2,Poly(x, y), is the polychromatic fringe phase shifts retrieved from the simulated fringe pattern by using the Fourier analysis method. The dotted cyan line is the 1st order approximate solution ϕ 2 , D ( 1 ) ( x , y ) using Eq. (13). The dashed red line is the higher order approximate solution ϕ 2 , D ( q ) ( x , y ) using Eq. (14).

Fig. 6
Fig. 6

Plots of the simulation results: The design energy is set to 18 keV with π-grating interferometer. In this simulation, the true theoretical values of ϕ 2 , D Theory ( x , y ) (the solid blue line) are set to 45 degress. Gaussian white noise (20%) was added to the fringe intensity image Ipoly and the grating only image Ig. In the plot, The dash-dot green line, ϕ2,Poly(x, y), is the polychromatic fringe phase shifts retrieved from the simulated noisy fringe pattern by using the Fourier analysis method. The dotted cyan line is the 1st order approximate solution ϕ 2 , D ( 1 ) ( x , y ) using Eq. (13). The dashed red line is the higher order approximate solution ϕ 2 , D ( q ) ( x , y ) using Eq. (14). The signal noise ratios (SNRs) at the left bump for ϕ2,Poly(x, y), ϕ 2 , D ( 1 ) ( x , y ), and ϕ 2 , D ( q ) ( x , y ) are 64.3, 45.7, and 63.8 respectively.

Fig. 7
Fig. 7

Plots of the simulation results: The design energy is set to 26.5 keV with π/2-grating interferometer. In this simulation, the true theoretical values of ϕ 1 , D Theory ( x , y ) (the solid blue line) are set to 5 degrees. The dash-dot green line, ϕ1,Poly(x, y), is the polychromatic fringe phase shifts retrieved from the simulated fringe pattern by using the Fourier analysis method. The dotted cyan line is the 1st order approximate solution ϕ 1 , D ( 1 ) ( x , y ) using Eq. (16). The dashed red line is the higher order approximate solution ϕ 1 , D ( q ) ( x , y ) using Eq. (17).

Fig. 8
Fig. 8

Plots of the simulation results: The design energy is set to 18 keV with π/2-grating interferometer. In this simulation, the true theoretical values of ϕ 1 , D Theory ( x , y ) (the solid blue line) are set to 5 degrees. The dash-dot green line, ϕ2,Poly(x, y), is the polychromatic fringe phase shifts retrieved from the simulated fringe pattern by using the Fourier analysis method. The dotted cyan line is the 1st order approximate solution ϕ 1 , D ( 1 ) ( x , y ) using Eq. (16). The dashed red line is the higher order approximate solution ϕ 1 , D ( q ) ( x , y ) using Eq. (17).

Tables (1)

Tables Icon

Table 1 Decay of the coefficients 1 n ! Q m ( n ) Q m ( 1 ) with increasing n, in the iterative Eqs. (14) and (17). Here the effective spectrum is assumed 35 kVp shown in Fig. 3, and the Talbot distance was set to the first and third order (j = 1, 3) Talbot distances respectively.

Equations (17)

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

I E ( M g x , M g y ) = I in M g 2 × { m C m ( E ) γ ( m ) exp [ i 2 π m x p 1 ] } .
I E ( M g x , M g y ) = I in M g 2 × m C m ( E ) γ ( m ) A 2 ( x , y ; E ) exp [ i ( ϕ m ( x , y ; E ) + 2 π m x p 1 ) ] .
ϕ m ( x , y ; E ) = m r e c 2 h 2 ( R 2 / M g ) p 1 1 E 2 ρ e , p ( x , y ) x ,
I Poly ( M g x , M g y ) = I in M g 2 × m γ ( m ) exp [ i 2 π m x p 1 ] × × [ E D ( E ) C m ( E ) exp [ i E D 2 E 2 ϕ m , D ( x , y ) ] dE ] ,
ϕ m , Poly ( x , y ) = Arg { D ( E ) C m ( E ) exp [ i E D 2 E 2 ϕ m , D ( x , y ) ] dE D ( E ) C m ( E ) dE } ,
C ( m ; λ ) = { 1 , if m = 0 , ( 1 cos Δ ϕ ) × ( 1 ) 4 k λ R 2 / M g p 1 2 sin | 4 k 2 π λ R 2 / M g p 1 2 | k π , if m = 2 k , k 0 , i sin Δ ϕ × sin [ ( 4 π λ R 2 / M g p 1 2 ) × ( ( k + 1 / 2 ) 2 ] ) π ( k + 1 / 2 ) , if m = 2 k + 1 .
tan [ ϕ m , Poly ( x , y ) ] = k = 0 ( 1 ) k ( 2 k + 1 ) ! × Q m ( 2 k + 1 ) Q m ( 0 ) × ϕ m , D 2 k + 1 ( x , y ) k = 0 ( 1 ) k ( 2 k ) ! × Q m ( 2 k ) Q m ( 0 ) × ϕ m , D 2 k ( x , y ) ,
Q m ( n ) D ( E ) C m ( E ) × E D 2 n E 2 n dE , n = 1 , 2 , 3 .
ϕ m , D ( 1 ) ( x , y ) = Q m ( 0 ) Q m ( 1 ) × tan [ ϕ m , Poly ( x , y ) ] .
ρ e , p ( x , y ) x = p 1 m r e c 2 h 2 ( R 2 / M g ) × E D 2 × [ Q m ( 0 ) Q m ( 1 ) tan [ ϕ m , Poly ( x , y ) ] ] .
ϕ m , D ( q + 1 ) ( x , y ) = tan [ ϕ m , Poly ( x , y ) ] × [ k = 0 q ( 1 ) k ( 2 k ) ! × Q m ( 2 k ) Q m ( 1 ) × ( ϕ m , D ( q ) ( x , y ) ) ( 2 k ) ] k = 1 q ( 1 ) k ( 2 k + 1 ) ! × Q m ( 2 k + 1 ) Q m ( 1 ) × ( ϕ m , D ( q ) ( x , y ) ) ( 2 k + 1 ) .  
Q 2 ( n ) = 1 π D ( E ) × [ 1 cos ( π E D E ) ] × | sin ( π j 2 E D E ) | × E D 2 n E 2 n dE ,
ϕ 2 , D ( 1 ) ( x , y ) = Q 2 ( 0 ) Q 2 ( 1 ) × tan [ ϕ 2 , Poly ( x , y ) ] , Q 2 ( 0 ) Q 2 ( 1 ) = D ( E ) × [ 1 cos ( π E D / E ) ] × | sin ( π j E D / 2 E ) | dE D ( E ) × [ 1 cos ( π E D / E ) ] × | sin ( π j E D / 2 E ) | × E D 2 / E 2 dE .
ϕ 2 , D ( q + 1 ) ( x , y ) = tan [ ϕ 2 , Poly ( x , y ) ] × [ k = 0 q ( 1 ) k ( 2 k ) ! × Q 2 ( 2 k ) Q 2 ( 1 ) × ( ϕ 2 , D ( q ) ( x , y ) ) ( 2 k ) ] [ k = 1 q ( 1 ) k ( 2 k + 1 ) ! × Q 2 ( 2 k + 1 ) Q 2 ( 1 ) × ( ϕ 2 , D ( q ) ( x , y ) ) ( 2 k + 1 ) ] .
Q 1 ( n ) = i 2 π D ( E ) × sin ( π E D 2 E ) × sin ( π j E D 2 E ) × E D 2 n E 2 n dE , n = 1 , 2 , 3 , .
ϕ 1 , D ( 1 ) ( x , y ) = Q 1 ( 0 ) Q 1 ( 1 ) × tan [ ϕ 1 , Poly ( x , y ) ] , Q 1 ( 0 ) Q 1 ( 1 ) = D ( E ) sin ( π E D / 2 E ) × sin ( j π E D / 2 E ) dE D ( E ) sin ( π E D / 2 E ) × sin ( j π E D / 2 E ) × ( E D 2 / E 2 ) dE .
ϕ 1 , D ( q + 1 ) ( x , y ) = tan [ ϕ 1 , Poly ( x , y ) ] × [ k = 0 q ( 1 ) k ( 2 k ) ! × Q 1 ( 2 k ) Q 1 ( 1 ) × ( ϕ 1 , D ( q ) ( x , y ) ) 2 k ] [ k = 1 q ( 1 ) k ( 2 k + 1 ) ! × Q 1 ( 2 k + 1 ) Q 1 ( 1 ) × ( ϕ 1 , D ( q ) ( x , y ) ) 2 k + 1 ] .

Metrics