Abstract

In most of grating x-ray interferometry one needs an absorbing grating as the analyzer to measure high-resolution interference fringes. Dual phase grating interferometry is a technique to get rid of the absorbing grating for radiation dose reduction. The authors present a quantitative theory of dual grating x-ray interferometry. The theory elucidates the fringe formation mechanism. The derived formulas of fringe period and fringe visibility provide useful tools for design optimization of dual phase grating interferometers.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [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]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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2017 (3)

2016 (1)

2015 (2)

2014 (1)

2013 (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]

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 (3)

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 2008, 827152 (2008).
[PubMed]

X. Wu and H. Liu, “Phase-space evolution of x-ray coherence in phase-sensitive imaging,” Appl. Opt. 47, 44–52 (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)

2004 (1)

X. Wu and H. Liu, “A new theory of phase-contrast x-ray imaging based on wigner distributions,” Med. Phys. 31, 2378–2384 (2004).
[Crossref] [PubMed]

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, L866–L868 (2003).
[Crossref]

1992 (1)

Arrizón, V.

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]

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]

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]

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. Phys. 2, 258–261 (2006).
[Crossref]

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]

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. 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.

Fujino, S.

Goodman, J.

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

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, L866–L868 (2003).
[Crossref]

Harada, J.

Hosoi, T.

Ito, Y.

Itoh, H.

Jefimovs, K.

M. Kagias, Z. Wang, K. Jefimovs, and M. Stampanoni, “Dual phase grating interferometer for tunable dark-field sensitivity,” Appl. Phys. Lett. 110, 014105 (2017).
[Crossref]

Jiang, M.

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

Kagias, M.

M. Kagias, Z. Wang, K. Jefimovs, and M. Stampanoni, “Dual phase grating interferometer for tunable dark-field sensitivity,” Appl. Phys. Lett. 110, 014105 (2017).
[Crossref]

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, L866–L868 (2003).
[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]

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, L866–L868 (2003).
[Crossref]

Kuwabara, H.

A. Momose, H. Kuwabara, and W. Yashiro, “X-ray phase imaging using lau effects,” Appl. Phys. Express 4, 066603 (2011).
[Crossref]

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, “Polychromatic x-ray effects on fringe phase shifts in grating interferometry,” Opt. Express 25, 6053–6068 (2017).
[Crossref] [PubMed]

A. Yan, X. Wu, and H. Liu, “Beam hardening correction in polychromatic x-ray grating interferometry,” Opt. Express 25, 24690–24704 (2017).
[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]

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]

X. Wu and H. Liu, “Phase-space evolution of x-ray coherence in phase-sensitive imaging,” Appl. Opt. 47, 44–52 (2008).
[Crossref] [PubMed]

X. Wu and H. Liu, “A new theory of phase-contrast x-ray imaging based on wigner distributions,” Med. Phys. 31, 2378–2384 (2004).
[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]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

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]

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]

Momose, A.

A. Momose, H. Kuwabara, and W. Yashiro, “X-ray phase imaging using lau effects,” Appl. Phys. Express 4, 066603 (2011).
[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]

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, L866–L868 (2003).
[Crossref]

Morimoto, N.

Nagai, K.

Nakamura, T.

Ohshima, K.

Ojeda-Castañeda, J.

Ouchi, C.

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. 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]

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]

Sato, G.

Setomoto, Y.

Shimura, T.

Stampanoni, M.

M. Kagias, Z. Wang, K. Jefimovs, and M. Stampanoni, “Dual phase grating interferometer for tunable dark-field sensitivity,” Appl. Phys. Lett. 110, 014105 (2017).
[Crossref]

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, L866–L868 (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, L866–L868 (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.

Wang, G.

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

Wang, Z.

M. Kagias, Z. Wang, K. Jefimovs, and M. Stampanoni, “Dual phase grating interferometer for tunable dark-field sensitivity,” Appl. Phys. Lett. 110, 014105 (2017).
[Crossref]

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.

Weitkamp, T.

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

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.

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]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Wu, X.

A. Yan, X. Wu, and H. Liu, “Polychromatic x-ray effects on fringe phase shifts in grating interferometry,” Opt. Express 25, 6053–6068 (2017).
[Crossref] [PubMed]

A. Yan, X. Wu, and H. Liu, “Beam hardening correction in polychromatic x-ray grating interferometry,” Opt. Express 25, 24690–24704 (2017).
[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]

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]

X. Wu and H. Liu, “Phase-space evolution of x-ray coherence in phase-sensitive imaging,” Appl. Opt. 47, 44–52 (2008).
[Crossref] [PubMed]

X. Wu and H. Liu, “A new theory of phase-contrast x-ray imaging based on wigner distributions,” Med. Phys. 31, 2378–2384 (2004).
[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 2008, 827152 (2008).
[PubMed]

Yamaguchi, K.

Yamazaki, A.

Yan, A.

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]

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.

Appl. Opt. (1)

X. Wu and H. Liu, “Phase-space evolution of x-ray coherence in phase-sensitive imaging,” Appl. Opt. 47, 44–52 (2008).
[Crossref] [PubMed]

Appl. Phys. Express (1)

A. Momose, H. Kuwabara, and W. Yashiro, “X-ray phase imaging using lau effects,” Appl. Phys. Express 4, 066603 (2011).
[Crossref]

Appl. Phys. Lett. (1)

M. Kagias, Z. Wang, K. Jefimovs, and M. Stampanoni, “Dual phase grating interferometer for tunable dark-field sensitivity,” Appl. Phys. Lett. 110, 014105 (2017).
[Crossref]

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 2008, 827152 (2008).
[PubMed]

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

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, L866–L868 (2003).
[Crossref]

Med. Phys. (5)

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]

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Figures (4)

Fig. 1
Fig. 1 Schematic of an x-ray dual-phase grating interferometer with a micro-spot X-ray source.
Fig. 2
Fig. 2 Effects of detector pitch size on fringe period. In this simulation, we present the effects of detector pitch on the resolved fringes. In the simulation we assume a 20 keV point x-ray source. The simulated dual π-grating is with period of 2µm. The geometry is set to R1 = R4 = 300mm, and R2 = 15mm. The solid blue lines represent the plot of the simulated intensity fringes, while the dashed red line, in Fig. 2(b), is the sum of all the resolvable diffraction orders given in Eq. (16). It can be seen that when detector pixel size is sufficiently small, the fringes of all periods are present (Fig. 2(a)). When detector pixel size is large enough, Fig. 2(b), only the resolvable fringes, of period pfr/2 = M6p/2 = 41µm, are visible, and the fringes of small periods are diminished due to the average effect given by Eq. (8). The simulation result (the blue line in Fig. 2(b)), agrees well with theoretical values (the red line in Fig. 2(b)).
Fig. 3
Fig. 3 Plot of visibility curve with respect to the G1-G2 spacing R2. In the figure, we assume the dual phase grating has period p = 1µm and phase shift π at design energy ED = 20keV. The 20keV source is a focal spot of width a = 40µm, and the detector pixel size is pD = 25µm. The geometry is set symmetrically with R1 = R4 = 450mm and the dual-grating spacing R2 changes from 2mm to 10mm. The blue line is computed directly from Eq. (20). While the black squares represent the visibility values of the fringe patterns, which were obtained by numerical simulations of Fresnel diffraction at various spacing R2. We can see the simulation results fit the theoretical values very well.
Fig. 4
Fig. 4 We plot the visibility curve of dual π/2-grating with respect to the dual-grating spacing R2. The only difference from Fig. 3 in the system setup is that the dual gratings are of π/2 phase shift at design energy ED. One can see that the maximal visibility occurs at R2 = 10.89mm, as compared to R2 = 3.6mm for the dual π-grating case.

Equations (33)

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G 1 ( x ) = l a l exp [ i 2 π m x p 1 ] , G 2 ( x ) = r b r exp [ i 2 π n x p 2 ] ,
W ( x , u ) = J ( x + Δ 2 , x Δ 2 ) exp [ i 2 π u Δ ] d Δ ,
W R 1 ( x , u ; y ) = J in ( x + Δ 2 , x Δ 2 ) G 1 ( x + Δ 2 ) G 1 * ( x Δ 2 ) exp [ i 2 π u Δ ] d Δ ,
W R 1 + R 2 0 ( x , u ; y ) = W R 1 ( x λ R 2 u , u ; y ) .
I R 1 + R 2 + R 4 ( x , y ) = W R 1 + R 2 + R 4 ( x , u ; y ) d u .
I ( x , y ) = I in M 5 2 l , n , r , s μ ( λ R 4 M 1 [ l M 5 p 1 + r M D p 2 ] l λ R 2 M 1 p 1 ) a l + n a n * b r + s b s * × × exp [ i 2 π ( l ( l + 2 n ) λ R 2 2 M 1 p 1 2 ) ] exp [ i 2 π ( ( l + 2 n ) λ R 4 2 M 1 p 1 ) ( l M 5 p 1 + r M D p 2 ) ] × × exp [ i 2 π ( ( r + 2 s ) λ R 4 2 p 2 ) ( l M 5 p 1 + r M D p 2 ) ] exp [ i 2 π ( l M 5 p 1 + r M D p 2 ) x ] .
M 1 = R 1 + R 2 R 1 ; M D = R 1 + R 2 + R 4 R 1 + R 2 ; M 5 = R 1 + R 2 + R 4 R 1 .
a v = 1 p D p D / 2 p D / 2 exp [ i 2 π l x p fr ] d x = sin ( π l p D / p fr ) π l p D / p fr = sinc ( l p D p fr ) .
p 1 = p 2 = p p D , and R 1 = R 4 R 2 .
p fr = M 6 p = ( 1 + R 1 + R 4 R 2 ) p .
I res ( x , y ) = I in M 5 2 l , n , s , μ ( λ R 2 M 5 p ) sinc ( l p D M 6 p ) a l + n a n * b s l b s * × × exp [ i 2 π l ( l + 2 n ) λ R 1 2 M 6 p 2 ] exp [ i 2 π l ( l + 2 s ) λ R 4 2 M 6 p 2 ] exp [ i 2 π l x M 6 P ] ,
D G 1 ( l ) = n = a l + n a n * exp [ i 2 π l ( l + 2 n ) λ R 1 2 M 6 p 2 ] .
I ( x ) = l = n = g l + n g n * exp [ i 2 π l ( l + 2 n ) λ d 2 p g 2 ] exp [ i 2 π l x p g ] = l = C l ( d , λ , p g , Δ ϕ g ) exp [ i 2 π l x p g ] .
C l ( d , λ , p g , Δ ϕ g ) = { 1 , if l = 0 , ( 1 cos Δ ϕ g ) ( 1 ) 4 k λ d / p g 2 sin ( 4 k 2 π λ d / p g 2 ) k π , if l = 2 k 0 , i 2 π sin Δ ϕ g sin ( 4 k λ d / p g 2 ( k + 1 / 2 ) 2 ) π ( 2 k + 1 ) , if l = 2 k + 1 .
D G 2 ( l ) = s = b l + s b s * exp [ i 2 π l ( l + 2 s ) λ R 4 2 M 6 p 2 ] .
I res ( x , y ) = I in M 5 2 l = μ ( λ R 1 M 6 p ) sinc ( l p D M 6 p ) × × C l ( R 1 M 6 , λ , p , Δ ϕ 1 ) C l ( R 4 M 6 , λ , p , Δ ϕ 2 ) exp [ i 2 π l x M 6 P ] .
C l ( R 1 M 6 , E D , p , π ) C l ( R 4 M 6 , E D , p , π ) = 16 π 2 l 2 sin 2 ( l 2 π R 1 / M 6 8 Z π ) .
sinc ( l a M 6 p ) C l ( R 1 M 6 , E D , p , π ) C l ( R 4 M 6 , E D , p , π ) sinc ( l p D M 6 p ) 16 l 4 π 4 ( M 6 p ) 2 a p D ,
I res ( x ) = I in M 5 2 [ 1 + 8 π 2 sinc ( 2 a M 6 p ) sinc ( 2 p D M 6 p ) sin 2 ( π R 1 / M 6 2 Z π ) cos ( 2 π x M 6 p / 2 ) ] .
V = 8 π 2 sinc ( 2 a M 6 p ) sinc ( 2 p D M 6 p ) sin 2 ( π R 1 / M 6 2 Z π ) , Z π = p 2 8 λ D .
C l ( R 1 M 6 , E D , p , π 2 ) C l ( R 4 M 6 , E D , p , π 2 ) = 4 l 2 π 2 sin 2 ( l 2 π R 1 / M 6 2 Z π / 2 ) ,
I res ( x ) = I in M 5 2 [ 1 + 8 π 2 sinc ( a M 6 p ) sinc ( p D M 6 p ) sin 2 ( π R 1 / M 6 2 Z π / 2 ) cos ( 2 π x M 6 p ) ] .
V = 8 π 2 sinc ( a M 6 p ) sinc ( p D M 6 p ) sin 2 ( π R 1 / M 6 2 Z π / 2 ) , Z π / 2 = p 2 8 λ D .
W R 1 ( x , u ; y ) = J in ( x + Δ 2 , x Δ 2 ) G 1 ( x + Δ 2 ) G 1 * ( x Δ 2 ) exp [ i 2 π u Δ ] d Δ ,
J in ( x + Δ 2 , x Δ 2 ) = I in exp ( i 2 π Δ x λ R 1 ) μ in ( Δ ) ,
μ in ( Δ ) = I source ( s ) exp [ i 2 π Δ s / ( λ R 1 ) ] d s I source ( s ) d s .
W R 1 ( x , u ; y ) = I in l , n = a l + n a n * exp ( i 2 π l x p 1 ) × × exp [ i 2 π Δ x λ R 1 ] μ in ( Δ ) exp [ i 2 π l + 2 n p 1 Δ ] exp [ i 2 π u Δ ] d Δ .
W R 1 + R 2 0 ( x , u ; y ) = W R 1 ( x λ R 2 u , u ; y ) .
J R 1 + R 2 ( x + Δ 2 2 , x Δ 2 2 ; y ) = W R 1 + R 2 0 ( x , u ; y ) exp [ i 2 π u Δ 2 ) d u .
J R 1 + R 2 ( x + Δ 2 2 , x Δ 2 2 ; y ) = I in M 1 2 l , n = a l + n a n * exp ( i 2 π l x M 1 p 1 ) × × exp [ i 2 π Δ 2 x M 1 λ R 1 ] × exp [ i 2 π l ( l + 2 n ) λ R 2 2 M 1 p 1 2 ] exp [ i 2 π ( l + 2 n ) Δ 2 2 M 1 p 1 ] × × μ in ( Δ 2 M 1 l λ R 2 M 1 p 1 )
W R 1 + R 2 + 0 ( x , u ; y ) = J z = R 1 + R 2 ( x + Δ 2 2 , x Δ 2 2 ; y ) G 2 ( x + Δ 2 2 ) × × G 2 * ( x Δ 2 2 ) exp [ i 2 π u Δ 2 ] d Δ 2 .
W R 1 + R 2 + 0 ( x , u ; y ) = I in M 1 2 l , n , r , s = a l + n a n * b r + s b r * × × exp [ i 2 π x ( l M 1 p 1 + r p 2 l ( l + 2 n ) λ R 2 2 M 1 p 2 ) ] × × exp [ i 2 π ( l + 2 n ) Δ 2 2 M 1 p 1 exp [ i 2 π ( r + 2 s ) Δ 2 2 p 2 ] ] × × μ in ( Δ 2 M 1 l λ R 2 M 1 p 1 ) exp [ i 2 π Δ 2 x M 1 λ R 1 ] exp [ i 2 π u Δ 2 ] d Δ 2 .
I R 1 + R 2 + R 4 ( x , y ) = W R 1 + R 2 + 0 ( x λ R u , u ; y ) d u .

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