Abstract

In this Letter, we present a single-shot approach to quantitatively retrieve x-ray absorption and phase shift in grating interferometry. The proposed approach makes use of the energy-resolving capability of x-ray photon-counting detectors. The retrieval method is derived and presented and is tested based on numerical simulations, including photon shot noise. The good agreement between retrieval results and theoretical values confirms the feasibility of the presented approach.

© 2014 Optical Society of America

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References

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  1. A. Bravin, P. Coan, and P. Suortti, Phys. Med. Biol. 58, R1 (2013).
    [CrossRef]
  2. A. Appel, M. Anastasio, and E. Brey, Tissue Eng. 17, 321 (2011).
    [CrossRef]
  3. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
    [CrossRef]
  4. A. Taper, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, Proc. Natl. Acad. Sci. USA 109, 15691 (2012).
    [CrossRef]
  5. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, Opt. Express 13, 6296 (2005).
    [CrossRef]
  6. A. Momose, W. Yashiro, S. Harasse, and H. Kuwabara, Opt. Express 19, 8423 (2011).
    [CrossRef]
  7. P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. Mcdonald, F. Marone, and M. Stampanoni, Proc. Natl. Acad. Sci. USA 107, 13576 (2010).
    [CrossRef]
  8. S. D. Sordo, L. Abbene, E. Caroli, A. M. Mancini, A. Zappettini, and P. Ubertini, Sensors 9, 3491 (2009).
    [CrossRef]
  9. T. Thuering, W. C. Barber, Y. Seo, F. Alhassen, J. S. Iwanczyk, and M. Stampanoni, Appl. Phys. Lett. 102, 191113 (2013).
    [CrossRef]
  10. G. Pelzer, T. Weber, G. Anton, R. Ballabriga, F. Bayer, M. Campbell, T. Gabor, W. Haas, F. Horn, X. Llopart, N. Michel, U. Mollenbauer, J. Rieger, A. Ritter, I. Ritter, P. Sievers, S. Wolfel, W. S. Wong, A. Zang, and T. Michel, Opt. Express 21, 25677 (2013).
    [CrossRef]
  11. D. Gürsoy and M. Das, Opt. Lett. 38, 1461 (2013).
    [CrossRef]
  12. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  13. Z. Qi, J. Zambelli, N. Bevins, and G. Chen, Phys. Med. Biol. 55, 2669 (2010).
    [CrossRef]
  14. R. E. Alvarez and A. Macovski, Phys. Med. Biol. 21, 733 (1976).
    [CrossRef]
  15. J. M. Boone, T. R. Fewell, and R. J. Jennings, Med. Phys. 24, 1863 (1997).
    [CrossRef]
  16. J. H. Hubbell and S. M. Seltzer, Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients from 1 keV to 20 MeV for Elements Z=1 to 92 and 48 Additional Substances of Dosimetric Interest (NIST, 1996).
  17. G. Sato, T. Kondoh, H. Itoh, S. Handa, K. Yamaguchi, T. Nakamura, K. Nagai, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, Opt. Lett. 36, 3551 (2011).
    [CrossRef]
  18. J. Rizzi, T. Weitkamp, N. Guerineau, M. Idir, P. Mercere, G. Druart, G. Vincent, P. Silva, and J. Primot, Opt. Lett. 36, 1398 (2011).
    [CrossRef]

2013

2012

A. Taper, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, Proc. Natl. Acad. Sci. USA 109, 15691 (2012).
[CrossRef]

2011

2010

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. Mcdonald, F. Marone, and M. Stampanoni, Proc. Natl. Acad. Sci. USA 107, 13576 (2010).
[CrossRef]

Z. Qi, J. Zambelli, N. Bevins, and G. Chen, Phys. Med. Biol. 55, 2669 (2010).
[CrossRef]

2009

S. D. Sordo, L. Abbene, E. Caroli, A. M. Mancini, A. Zappettini, and P. Ubertini, Sensors 9, 3491 (2009).
[CrossRef]

2006

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

2005

1997

J. M. Boone, T. R. Fewell, and R. J. Jennings, Med. Phys. 24, 1863 (1997).
[CrossRef]

1976

R. E. Alvarez and A. Macovski, Phys. Med. Biol. 21, 733 (1976).
[CrossRef]

Abbene, L.

S. D. Sordo, L. Abbene, E. Caroli, A. M. Mancini, A. Zappettini, and P. Ubertini, Sensors 9, 3491 (2009).
[CrossRef]

Alhassen, F.

T. Thuering, W. C. Barber, Y. Seo, F. Alhassen, J. S. Iwanczyk, and M. Stampanoni, Appl. Phys. Lett. 102, 191113 (2013).
[CrossRef]

Alvarez, R. E.

R. E. Alvarez and A. Macovski, Phys. Med. Biol. 21, 733 (1976).
[CrossRef]

Anastasio, M.

A. Appel, M. Anastasio, and E. Brey, Tissue Eng. 17, 321 (2011).
[CrossRef]

Anton, G.

Appel, A.

A. Appel, M. Anastasio, and E. Brey, Tissue Eng. 17, 321 (2011).
[CrossRef]

Ballabriga, R.

Barber, W. C.

T. Thuering, W. C. Barber, Y. Seo, F. Alhassen, J. S. Iwanczyk, and M. Stampanoni, Appl. Phys. Lett. 102, 191113 (2013).
[CrossRef]

Bayer, F.

Bech, M.

A. Taper, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, Proc. Natl. Acad. Sci. USA 109, 15691 (2012).
[CrossRef]

Bevins, N.

Z. Qi, J. Zambelli, N. Bevins, and G. Chen, Phys. Med. Biol. 55, 2669 (2010).
[CrossRef]

Boone, J. M.

J. M. Boone, T. R. Fewell, and R. J. Jennings, Med. Phys. 24, 1863 (1997).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Bravin, A.

A. Bravin, P. Coan, and P. Suortti, Phys. Med. Biol. 58, R1 (2013).
[CrossRef]

Brey, E.

A. Appel, M. Anastasio, and E. Brey, Tissue Eng. 17, 321 (2011).
[CrossRef]

Bruyndonckx, P.

A. Taper, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, Proc. Natl. Acad. Sci. USA 109, 15691 (2012).
[CrossRef]

Bunk, O.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

Campbell, M.

Caroli, E.

S. D. Sordo, L. Abbene, E. Caroli, A. M. Mancini, A. Zappettini, and P. Ubertini, Sensors 9, 3491 (2009).
[CrossRef]

Chen, G.

Z. Qi, J. Zambelli, N. Bevins, and G. Chen, Phys. Med. Biol. 55, 2669 (2010).
[CrossRef]

Cloetens, P.

Coan, P.

A. Bravin, P. Coan, and P. Suortti, Phys. Med. Biol. 58, R1 (2013).
[CrossRef]

Das, M.

David, C.

Den, T.

Diaz, A.

Druart, G.

Fewell, T. R.

J. M. Boone, T. R. Fewell, and R. J. Jennings, Med. Phys. 24, 1863 (1997).
[CrossRef]

Gabor, T.

Guerineau, N.

Gürsoy, D.

Haas, W.

Handa, S.

Harasse, S.

Horn, F.

Hubbell, J. H.

J. H. Hubbell and S. M. Seltzer, Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients from 1 keV to 20 MeV for Elements Z=1 to 92 and 48 Additional Substances of Dosimetric Interest (NIST, 1996).

Idir, M.

Itoh, H.

Iwanczyk, J. S.

T. Thuering, W. C. Barber, Y. Seo, F. Alhassen, J. S. Iwanczyk, and M. Stampanoni, Appl. Phys. Lett. 102, 191113 (2013).
[CrossRef]

Jennings, R. J.

J. M. Boone, T. R. Fewell, and R. J. Jennings, Med. Phys. 24, 1863 (1997).
[CrossRef]

Kondoh, T.

Kuwabara, H.

Liu, X.

A. Taper, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, Proc. Natl. Acad. Sci. USA 109, 15691 (2012).
[CrossRef]

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. Mcdonald, F. Marone, and M. Stampanoni, Proc. Natl. Acad. Sci. USA 107, 13576 (2010).
[CrossRef]

Liu, Y.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. Mcdonald, F. Marone, and M. Stampanoni, Proc. Natl. Acad. Sci. USA 107, 13576 (2010).
[CrossRef]

Llopart, X.

Macovski, A.

R. E. Alvarez and A. Macovski, Phys. Med. Biol. 21, 733 (1976).
[CrossRef]

Mancini, A. M.

S. D. Sordo, L. Abbene, E. Caroli, A. M. Mancini, A. Zappettini, and P. Ubertini, Sensors 9, 3491 (2009).
[CrossRef]

Marone, F.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. Mcdonald, F. Marone, and M. Stampanoni, Proc. Natl. Acad. Sci. USA 107, 13576 (2010).
[CrossRef]

Mcdonald, S. A.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. Mcdonald, F. Marone, and M. Stampanoni, Proc. Natl. Acad. Sci. USA 107, 13576 (2010).
[CrossRef]

Meiser, J.

A. Taper, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, Proc. Natl. Acad. Sci. USA 109, 15691 (2012).
[CrossRef]

Mercere, P.

Michel, N.

Michel, T.

Mohr, J.

A. Taper, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, Proc. Natl. Acad. Sci. USA 109, 15691 (2012).
[CrossRef]

Mollenbauer, U.

Momose, A.

Nagai, K.

Nakamura, T.

Ouchi, C.

Pauwels, B.

A. Taper, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, Proc. Natl. Acad. Sci. USA 109, 15691 (2012).
[CrossRef]

Pelzer, G.

Pfeiffer, F.

A. Taper, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, Proc. Natl. Acad. Sci. USA 109, 15691 (2012).
[CrossRef]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, Opt. Express 13, 6296 (2005).
[CrossRef]

Primot, J.

Qi, Z.

Z. Qi, J. Zambelli, N. Bevins, and G. Chen, Phys. Med. Biol. 55, 2669 (2010).
[CrossRef]

Rieger, J.

Ritter, A.

Ritter, I.

Rizzi, J.

Sasov, A.

A. Taper, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, Proc. Natl. Acad. Sci. USA 109, 15691 (2012).
[CrossRef]

Sato, G.

Schulz, J.

A. Taper, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, Proc. Natl. Acad. Sci. USA 109, 15691 (2012).
[CrossRef]

Seltzer, S. M.

J. H. Hubbell and S. M. Seltzer, Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients from 1 keV to 20 MeV for Elements Z=1 to 92 and 48 Additional Substances of Dosimetric Interest (NIST, 1996).

Seo, Y.

T. Thuering, W. C. Barber, Y. Seo, F. Alhassen, J. S. Iwanczyk, and M. Stampanoni, Appl. Phys. Lett. 102, 191113 (2013).
[CrossRef]

Setomoto, Y.

Sievers, P.

Silva, P.

Sordo, S. D.

S. D. Sordo, L. Abbene, E. Caroli, A. M. Mancini, A. Zappettini, and P. Ubertini, Sensors 9, 3491 (2009).
[CrossRef]

Stampanoni, M.

T. Thuering, W. C. Barber, Y. Seo, F. Alhassen, J. S. Iwanczyk, and M. Stampanoni, Appl. Phys. Lett. 102, 191113 (2013).
[CrossRef]

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. Mcdonald, F. Marone, and M. Stampanoni, Proc. Natl. Acad. Sci. USA 107, 13576 (2010).
[CrossRef]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, Opt. Express 13, 6296 (2005).
[CrossRef]

Suortti, P.

A. Bravin, P. Coan, and P. Suortti, Phys. Med. Biol. 58, R1 (2013).
[CrossRef]

Taper, A.

A. Taper, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, Proc. Natl. Acad. Sci. USA 109, 15691 (2012).
[CrossRef]

Teshima, T.

Thuering, T.

T. Thuering, W. C. Barber, Y. Seo, F. Alhassen, J. S. Iwanczyk, and M. Stampanoni, Appl. Phys. Lett. 102, 191113 (2013).
[CrossRef]

Ubertini, P.

S. D. Sordo, L. Abbene, E. Caroli, A. M. Mancini, A. Zappettini, and P. Ubertini, Sensors 9, 3491 (2009).
[CrossRef]

Velroyen, A.

A. Taper, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, Proc. Natl. Acad. Sci. USA 109, 15691 (2012).
[CrossRef]

Vincent, G.

Walter, M.

A. Taper, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, Proc. Natl. Acad. Sci. USA 109, 15691 (2012).
[CrossRef]

Wang, Z.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. Mcdonald, F. Marone, and M. Stampanoni, Proc. Natl. Acad. Sci. USA 107, 13576 (2010).
[CrossRef]

Weber, T.

Weitkamp, T.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Wolfel, S.

Wong, W. S.

Wu, Z.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. Mcdonald, F. Marone, and M. Stampanoni, Proc. Natl. Acad. Sci. USA 107, 13576 (2010).
[CrossRef]

Yamaguchi, K.

Yashiro, W.

Zambelli, J.

Z. Qi, J. Zambelli, N. Bevins, and G. Chen, Phys. Med. Biol. 55, 2669 (2010).
[CrossRef]

Zang, A.

Zappettini, A.

S. D. Sordo, L. Abbene, E. Caroli, A. M. Mancini, A. Zappettini, and P. Ubertini, Sensors 9, 3491 (2009).
[CrossRef]

Zhang, K.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. Mcdonald, F. Marone, and M. Stampanoni, Proc. Natl. Acad. Sci. USA 107, 13576 (2010).
[CrossRef]

Zhu, P.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. Mcdonald, F. Marone, and M. Stampanoni, Proc. Natl. Acad. Sci. USA 107, 13576 (2010).
[CrossRef]

Ziegler, E.

Appl. Phys. Lett.

T. Thuering, W. C. Barber, Y. Seo, F. Alhassen, J. S. Iwanczyk, and M. Stampanoni, Appl. Phys. Lett. 102, 191113 (2013).
[CrossRef]

Med. Phys.

J. M. Boone, T. R. Fewell, and R. J. Jennings, Med. Phys. 24, 1863 (1997).
[CrossRef]

Nat. Phys.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Med. Biol.

Z. Qi, J. Zambelli, N. Bevins, and G. Chen, Phys. Med. Biol. 55, 2669 (2010).
[CrossRef]

R. E. Alvarez and A. Macovski, Phys. Med. Biol. 21, 733 (1976).
[CrossRef]

A. Bravin, P. Coan, and P. Suortti, Phys. Med. Biol. 58, R1 (2013).
[CrossRef]

Proc. Natl. Acad. Sci. USA

A. Taper, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, Proc. Natl. Acad. Sci. USA 109, 15691 (2012).
[CrossRef]

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. Mcdonald, F. Marone, and M. Stampanoni, Proc. Natl. Acad. Sci. USA 107, 13576 (2010).
[CrossRef]

Sensors

S. D. Sordo, L. Abbene, E. Caroli, A. M. Mancini, A. Zappettini, and P. Ubertini, Sensors 9, 3491 (2009).
[CrossRef]

Tissue Eng.

A. Appel, M. Anastasio, and E. Brey, Tissue Eng. 17, 321 (2011).
[CrossRef]

Other

J. H. Hubbell and S. M. Seltzer, Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients from 1 keV to 20 MeV for Elements Z=1 to 92 and 48 Additional Substances of Dosimetric Interest (NIST, 1996).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

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Figures (2)

Fig. 1.
Fig. 1.

(a) Schematic setup of x-ray grating interferometer with a microfocus source, including the coordinate system. (b) Normalized intensity as a function of the relative grating position.

Fig. 2.
Fig. 2.

Profiles of the retrieved values with the proposed approach and the true values of (a) x-ray absorption and (b) phase shift at 45 keV.

Tables (1)

Tables Icon

Table 1. Calculated Relative Errors of Retrieved Absorption and Phase Shift of the Phantom with Respect to the True Values

Equations (10)

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

I(E)=I0(E)M2T(E)nancos[2πnxg+dTΔθ(E)p2]=I0(E)M2T(E)S[xg+dTΔθ(E)],
Δθ(E)=λ(E)2πΦ(E)x
ln[M2I(E)I0(E)]=μ(E)dzln{S[xg+dTΔθ(E)]}.
lnS[xg+dTΔθ(E)]lnS(xg)K1+S˙(xg)S(xg)dTΔθ(E)K2.
ln[M2I(E)I0(E)]+K1=μ(E)dzK2Δθ(E).
μ(E)=KρeZm/E3+ρefKN(E),
Φ(E)=reλ(E)ρedz,
ln[M2I(E)I0(E)]+K1=1E3KZmρedza1+[fKN(E)K2reλ2(E)2πx]ρedza2,
ln{M2[I(E)]i[I0(E)]i}+K1=1B¯i3a1+[fKN(B¯i)K2reλ2(B¯i)2πx]a2i=1,,N,
A=(CTWC)1CTW(I+K1),

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