## On the origin of visibility contrast in x-ray Talbot interferometry |

Optics Express, Vol. 18, Issue 16, pp. 16890-16901 (2010)

http://dx.doi.org/10.1364/OE.18.016890

Acrobat PDF (1136 KB)

### Abstract

The reduction in visibility in x-ray grating interferometry based on the Talbot effect is formulated by the autocorrelation function of spatial fluctuations of a wavefront due to unresolved micron-size structures in samples. The experimental results for microspheres and melamine sponge were successfully explained by this formula with three parameters characterizing the wavefront fluctuations: variance, correlation length, and the Hurst exponent. The ultra-small-angle x-ray scattering of these samples was measured, and the scattering profiles were consistent with the formulation. Furthermore, we discuss the relation between the three parameters and the features of the micron-sized structures. The visibility-reduction contrast observed by x-ray grating interferometry can thus be understood in relation to the structural parameters of the microstructures.

© 2010 Optical Society of America

## 1. Introduction

1. R. Fitzgerald, “Phase-sensitive x-ray imaging,” Phys. Today **53**, 23–26 (2000). [CrossRef]

2. A. Momose, “Recent advances in x-ray phase imaging,” Jpn. J. Appl. Phys. **44**, 6355–6367 (2005). [CrossRef]

3. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. **42**, L866–L868 (2003). [CrossRef]

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. A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. **91**, 074106 (2007). [CrossRef]

6. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E.F. Eikenberry, C.H. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mat. **7**, 134–137 (2008). [CrossRef]

7. Y.I. Nesterets and S. W. Wilkins, “Phase-contrast imaging using a scanning-doublegrating configuration,” Opt. Express **16**, 5849–5867 (2008). [CrossRef] [PubMed]

8. Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, “Differential phase x-ray imaging microscopy with x-ray Talbot interferometer,” Appl. Phys. Express **1**, 117002 (2008). [CrossRef]

9. W. Yashiro, Y. Takeda, and A. Momose, “Efficiency of capturing a phase image using conebeam x-ray Talbot interferometry,” J. Opt. Soc. Am. A **25**, 2025–2039 (2008). [CrossRef]

11. Z.-F. Huang, K.-J. Kang, L. Zhang, Z.-Q. Chen, F. Ding, Z.-T. Wang, and Q.-G. Fang, “Alternative method for differential phase-contrast imaging with weakly coherent hard x rays,” Phys. Rev. A **79**, 013815 (2009). [CrossRef]

12. A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. **94**, 044108 (2009). [CrossRef]

*et al.*has recently proposed another approach to forming image contrast, where a relative reduction in the visibility of the moiré image is quantified by defining normalized visibility [6

6. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E.F. Eikenberry, C.H. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mat. **7**, 134–137 (2008). [CrossRef]

14. C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kühne, J. Kohlbrecher, R. Schäfer, P. Lejcek, H. M. R. Ronnow, and F. Pfeiffer, “Neutron decoherence imaging for visualizing bulk magnetic domain structures,” Phys. Rev. Lett. **101**, 025504 (2008). [CrossRef] [PubMed]

15. M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, “Neutron dark-field tomography,” Phys. Rev. Lett. **101**, 123902 (2008). [CrossRef] [PubMed]

16. C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, “Bulk magnetic domain structures visualized by neutron dark-field imaging,” Appl. Phys. Lett. **93**, 112504 (2009). [CrossRef]

17. F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys. **105**, 102006 (2009). [CrossRef]

18. S.A. MacDonald, F. Marone, C. Hintermüller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” J. Synchrotron Radiation **16**, 562–572 (2009). [CrossRef]

20. Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. **95**, 094105 (2009). [CrossRef]

## 2. Theoretical description of the reduced visibility

*λ*is illuminating a sample located just in front of the first grating; the following can easily be extended to the case of polychromatic- and spherical-wave illumination [22].

*δ*+

*iβ*, and the electric field just behind the sample can be written in the projection approximation by

*E*

_{0}exp[−

*α*(

*x,y*)/2]exp[−

*i*Φ(

*x,y*)], where

*E*

_{0}is the amplitude of the incident x-rays, and

*α*and Φ are given by (4

*π*/

*λ*)

*∫ β*(

*x,y,z*)

*dz*and (−2

*π*/

*λ*)

*∫ δ*(

*x,y,z*)

*dz*. To discuss the effect of phase fluctuations, we present

*α*and Φ as superpositions of smooth (resolvable) and fine (unresolvable) features [23

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

*α*=

*α*

_{s}+

*α*

_{f}and Φ = Φ

_{s}+ Φ

_{f}. In the following, we assume that

*α*

_{f}is negligible. We also assume, for the sake of simplicity, that the size of the first grating is infinite.

*z*

_{T}from the first grating, can be given in the paraxial approximation by [10]

*a*′

_{n}is the

*n*th Fourier coefficient of the electric field generated in front of the second grating (

*n*= 0,±1, …), given by

*a*exp(−

_{n}*iπpn*

^{2}), where

*a*is the

_{n}*n*th Fourier coefficient of the amplitude transmission function of the first grating and

*p*is the Talbot order [24

24.
Here, we defined the *p*th Talbot order as it expresses the *p*th position where the electric field just behind the first grating is perfectly reproduced for any grating. For plane-wave illumination, the *p*th Talbot order corresponds to the position of *z*_{T} = *pd*^{2}/*λ*. This definition is convenient because it is independent of what kind of grating is used as the first grating. Note that, once *z*_{T} is given by *pd*^{2}/*λ*, we can use *p* for specifying any position behind the first grating instead of *z*_{T}. The analytical calculations presented in this paper can be applied to any periodic image generated behind the first grating at any position of *p* (> 0).

*d*is the pitch of the gratings, and Φ

_{n}(

*x,y*) is defined by Φ(

*x*−

*npd,y*).

*x,y*) is approximated by

*I*

_{0}≡ ∣

*E*

_{0}∣

^{2},

*c*is the

_{N}*N*th Fourier coefficient of the intensity transmission function of the second grating (

*N*= 0,±1, …),

*µ*is the complex coherence factor of x-rays [25] at two points separated by distance

_{m}*mpd*on the first grating (

*m*= 0,±1, …),

*χ*is the relative displacement of the second grating to the first grating in the

*x*direction, and

*PSF*(

*x,y*) is the normalized point spread function (PSF) of the detector. Note that, in Eq. (2), we have assumed that the two gratings are parallel to the

*y*axis without loss of generality and that the scale resolved by the image system is sufficiently larger than the pitch of the gratings.

*q*

_{1}, in Eq. (2) (corresponding to

*N*= 1) to that of the 0th order,

*q*

_{0}, (corresponding to

*N*= 0), both of which are experimentally obtained by using the fringe-scanning technique [26

26. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital Wavefront Measuring Interferometer for Testing Optical Surfaces and Lenses,” Appl. Opt. **13**, 2693–2703 (1974). [CrossRef] [PubMed]

*q*

_{0}, is given by

*Ã*

_{n,m,0}, is rewritten by

_{f,n}(

*x,y*) ≡ Φ

_{f}(

*x*−

*npd,y*). Here, we have considered that the resolvable feature, Φ

_{s}, is a slowly varying function of

*x*and that the derivative of Φ

_{s}higher than the first can be neglected. If we assume that the unresolvable features, Φ

_{f}, are distributed randomly and that the width of PSF is sufficiently larger than the characteristic scale of the unresolvable features,

*Ã*

_{n,m,0}(

*x,y*) can be further approximated by [7

7. Y.I. Nesterets and S. W. Wilkins, “Phase-contrast imaging using a scanning-doublegrating configuration,” Opt. Express **16**, 5849–5867 (2008). [CrossRef] [PubMed]

*x,y*):

*D*is the width of the PSF. Since the phase Φ

_{f,n}(

*x,y*) − Φ

_{f,n−m}(

*x,y*) in Eq. (7) is random, the terms of

*Ã*

_{n,m,0}(

*x,y*) for

*m*≠ 0 are much smaller than

*Ã*

_{n,0,0}(

*x,y*). Hence,

*q*

_{0}∣ is finally given by

*C*

_{0}≡ ∑

_{n}μ_{0}∣

*a*′

_{n}∣

^{2}

*c*

_{0}.

_{f,n}(

*x,y*) − Φ

_{f,n−m}(

*x,y*), in Eq. (15) is random, the terms,

*Ã*

_{n,m,1}(

*x,y*), for

*m*≠ −1 are much smaller than

*Ã*

_{n,−1,1}(

*x,y*). Hence,

*a*′

_{0}

*a*′

^{*}

_{1}by |

*a*′

_{0}

*a*′

^{*}

_{1}|exp[

*i*Ω], and using the fact

*a*′

_{−1}

*a*′

^{*}

_{0}= (

*a*′

_{0}

*a*′

^{*}

_{1})

^{*}, we can finally obtain an approximate form of ∣

*q*

_{1}∣:

*C*

_{1}≡ 2∣

*μ*

_{−1}∣∣

*c*

_{1}∣∣

*a*′

_{0}

*a*′

^{*}

_{1}∣ cosΩ and ∆Φ

_{f}(

*x,y*;

*pd*) ≡ Φ

_{f,0}(

*x,y*) − Φ

_{f,1}(

*x,y*). Here, we have assumed that the average phase,

*D*was assumed to be much larger than

*pd*.

*V*) to that without the sample (

*V*

_{0}), is given by

_{f}could be modeled as a random Gaussian process [23

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

27. I. A. Vartanyants and I. K. Robinson, “Origins of decoherence in coherent x-ray diffraction experiments,” Opt. Commun. **222**, 29–50 (2003). [CrossRef]

*σ*. The term,

*γ*, is the normalized autocorrelation function given by

*γ*that was proposed by Sinha

*et al.*to deal with height fluctuations on surfaces [28

28. S.K. Sinha, E.B. Sirota, and S. Garoff, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B **38**, 2297–2312 (1988). [CrossRef]

*H*is the Hurst exponent (0<

*H*<1) and

*ξ*is the correlation length of phase fluctuations. Thus, we can generally formulate the reduced visibility in terms of the autocorrelation function of spatial fluctuations of a wavefront. Note that, from Eq. (22),

*V*/

*V*

_{0}approaches zero when

*σ*increases. In addition, for a given sample (with a given ‘scattering power’),

*V*/

*V*

_{0}is a function of

*pd*.

## 3. Verification of formulation through experiments

### 3.1. Dependence of V/V0 on pd

*V*/

*V*

_{0}on

*pd*to corroborate Eq. (22) by using monochromatic x-rays. The experiment was carried out with synchrotron x-rays at the beamline 14C, Photon Factory (PF), Japan. Cross-linked PMMA microspheres (EPOSTAR MA, Nippon Shokubai Co., Ltd.) with various radii and a melamine sponge were used as the samples. Microspheres with a fixed radius were dispersed in glycerin liquid, which was dispensed into a 10 mm-thick plastic cell. The volume fraction of the microspheres was fixed at 0.057. The melamine sponge was 1.6 mm thick. Gold gratings with a Ronchi ruling (a

*π*/2-phase grating for the first grating and a 30

*µ*m-thick absorption grating) were used. They were aligned parallel to each other. The images were recorded using a charge-coupled device (CCD)-based x-ray image detector (Spectral Instruments), where the CCD (4096 × 4096 pixels) was connected to a 40

*µ*m GOS screen with a 2:1 fiber coupling. The effective pixel size was 18

*µ*m, and the width of the line spread function was 70

*µ*m.

*µ*m-pitch gratings were used and the Talbot order,

*p*, was changed by changing the distance between the first and second gratings. The curves in the figure plot the results of least-squares fitting to the experimental data by using Eqs. (22) and (24). The experimental data are in good agreement with the fitting curves. This means that the description of the reduced visibility obtained by using Eqs. (22) and (24) is valid.

*µ*m-P43, Gd

_{2}O

_{2}S:Tb+ fine powders), a relay lens, and a CCD camera (Hamamatsu Photonics C4742-98-24A, 1344 × 1024 pixels) was used. The effective pixel size of the detector was 3.14

*µ*m. The open and filled circles in Fig. 2 (b) plot the results for

*λ*= 1.0 and 0.5 Å (SPring-8), while the triangles and squares plot those for another sample with the same thickness for

*d*= 5.3 and 8.0

*µ*m at 0.7 Å (PF). At SPring-8, 20 × 20 pixel binning was implemented to make

*D*almost the same as that of the detector used at PF. The good agreement between the experimental results for the 5.3

*µ*m and 8.0

*µ*m pitches corroborates that Eq. (22) is correct. In addition, the experimental data were well fitted by using Eqs. (22) and (24) for all the wavelengths. Note that the results did not depend on the direction of the sample. The fitting results in Figs. 2(a) and 2(b) are summarized in Table 1.

## 4. Relation between fitting parameters and structural parameters of sample

*x*′ ≡ ∆

*x*/(2

*a*), ∆

*ρ*is the number density of electrons, and

*r*

_{e}is the classical electron radius. Here we assumed that

*TNπa*

^{2}≫ 1. These results were also confirmed by numerical calculations for a system where microspheres disperse. It should be noted that the first factor on the right hand side of Eq. (26) corresponds to the number of trials in the random walk problem, while the second factor originates from the phase shift by a microsphere. Since the number of trials is proportional to

*N*and

*T*and the phase shift by a particle is proportional to

*λ*far from an absorption edge,

*σ*

^{2}and ln(

*V*/

*V*

_{0}) are proportional to

*TNλ*

^{2}for any homogeneous sample. It can be seen that

*σ*

^{2}is proportional to

*λ*

^{2}for the melamine sponge (see Table 1). We also confirmed that ln(

*V*/

*V*

_{0}) is proportional to the thickness of the melamine sponge.

*H*~ 1 for microspheres, which was consistent with our calculations. On the other hand

*H*~ 0.7 for the melamine sponge, which corresponds to

*β*~ 0.7. This was also consistent with its real shape; it has a thin fiber structure (see Fig. 2 (b)). Our results of

*ξ*for microspheres indicate that the possibility of microspheres aggregating increases when their radius is smaller. This is also supported by the results from ultra-smallangle scattering in Fig. 3 because otherwise the experimentally obtained angular broadening, representing the inverse size of microstructures, cannot be explained. The results of

*ξ*for the melamine sponge were also consistent with observations through an optical microscope.

*V*/

*V*

_{0}at a point (

*x,y*) on the detector is given by

*j*represents the contribution from the

*j*th domain on the path to the point (

*x,y*) along the

*z*-axis. Because

*σ*

^{2}

_{j}should be proportional to the thickness of the

*j*th domain, Eq. (29) has a similar form to the Beer-Lambert law. Hence, −ln [

*V*/

*V*

_{0}] can be given by

*σ*and

*γ*are expressed as a function of

*z*. Note that ∂ (

*σ*

^{2})/∂

*z*is proportional to

*λ*

^{2}and (∆

*ρ*)

^{2}. Thus, we can also carry out tomography and determine the three-dimensional distribution of (1 −

*γ*)∂ (

*σ*

^{2})/∂

*z*. Obtaining tomograms for ∂(

*σ*

^{2})/∂

_{z},

*ξ*, and

*H*is also possible by performing scans for at least three

*pd*s.

## 5. Conclusion

*γ*)∂ (

*σ*

^{2})/∂

*z*, we can also carry out tomography if

*γ*is not regarded as being dependent on

*z*in each domain. Obtaining tomograms for ∂ (

*σ*

^{2})/∂

*z*,

*ξ*, and

*H*is also possible by performing scans for at least three

*pd*s. Furthermore, we can use two-dimensional gratings to obtain structural parameters in both

*x*and

*y*directions. Our approach can directly provide twoand three-dimensional structural information on unresolved microstructures, and we expect it to be broadly applied to medical, biological, and material sciences.

## Acknowledgements

## References and links

1. | R. Fitzgerald, “Phase-sensitive x-ray imaging,” Phys. Today |

2. | A. Momose, “Recent advances in x-ray phase imaging,” Jpn. J. Appl. Phys. |

3. | A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. |

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. |

5. | A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. |

6. | F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E.F. Eikenberry, C.H. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mat. |

7. | Y.I. Nesterets and S. W. Wilkins, “Phase-contrast imaging using a scanning-doublegrating configuration,” Opt. Express |

8. | Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, “Differential phase x-ray imaging microscopy with x-ray Talbot interferometer,” Appl. Phys. Express |

9. | W. Yashiro, Y. Takeda, and A. Momose, “Efficiency of capturing a phase image using conebeam x-ray Talbot interferometry,” J. Opt. Soc. Am. A |

10. | A. Momose, W. Yashiro, and Y. Takeda, “X-ray phase imaging with Talbot interferometry,” in |

11. | Z.-F. Huang, K.-J. Kang, L. Zhang, Z.-Q. Chen, F. Ding, Z.-T. Wang, and Q.-G. Fang, “Alternative method for differential phase-contrast imaging with weakly coherent hard x rays,” Phys. Rev. A |

12. | A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. |

13. | W. Yashiro, Y. Takeda, A. Takeuchi, Y. Suzuki, and A. Momose, “Hard x-ray phase-difference microscopy using a Fresnel zone plate and a transmission grating,” Phys. Rev. Lett. |

14. | C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kühne, J. Kohlbrecher, R. Schäfer, P. Lejcek, H. M. R. Ronnow, and F. Pfeiffer, “Neutron decoherence imaging for visualizing bulk magnetic domain structures,” Phys. Rev. Lett. |

15. | M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, “Neutron dark-field tomography,” Phys. Rev. Lett. |

16. | C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, “Bulk magnetic domain structures visualized by neutron dark-field imaging,” Appl. Phys. Lett. |

17. | F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys. |

18. | S.A. MacDonald, F. Marone, C. Hintermüller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” J. Synchrotron Radiation |

19. | H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, “Fourier x-ray scattering radiography yields bone structural information,” Radiography |

20. | Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. |

21. | R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bosecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nat. Phys. |

22. | K. Patorski, “Self-imaging and its applications,” in |

23. | Y.I. Nesterets, “On the origins of decoherence and extinction contrast in phase-contrast imaging,” Opt. Commun. |

24. |
Here, we defined the |

25. | J. W. Goodman, |

26. | J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital Wavefront Measuring Interferometer for Testing Optical Surfaces and Lenses,” Appl. Opt. |

27. | I. A. Vartanyants and I. K. Robinson, “Origins of decoherence in coherent x-ray diffraction experiments,” Opt. Commun. |

28. | S.K. Sinha, E.B. Sirota, and S. Garoff, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B |

**OCIS Codes**

(110.6760) Imaging systems : Talbot and self-imaging effects

(170.0170) Medical optics and biotechnology : Medical optics and biotechnology

(340.7450) X-ray optics : X-ray interferometry

**ToC Category:**

X-ray Optics

**History**

Original Manuscript: April 2, 2010

Revised Manuscript: June 11, 2010

Manuscript Accepted: July 3, 2010

Published: July 26, 2010

**Virtual Issues**

Vol. 5, Iss. 12 *Virtual Journal for Biomedical Optics*

**Citation**

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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-16890

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### References

- R. Fitzgerald, "Phase-sensitive x-ray imaging," Phys. Today 53,23-26 (2000). [CrossRef]
- 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, K. Takai, and Y. Suzuki, "Demonstration of x-ray Talbot interferometry," Jpn. J. Appl. Phys. 42,L866-L868 (2003). [CrossRef]
- 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]
- A. Olivo and R. Speller, "A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources," Appl. Phys. Lett. 91,074106 (2007). [CrossRef]
- F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. H. Brönnimann, C. Grünzweig, and C. David, "Hardx-ray dark-field imaging using a grating interferometer," Nat. Mat. 7,134-137 (2008). [CrossRef]
- Y. I. Nesterets and S. W. Wilkins, "Phase-contrast imaging using a scanning-doublegrating configuration," Opt. Express 16,5849-5867 (2008). [CrossRef] [PubMed]
- Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, "Differential phase x-ray imaging microscopy with x-ray Talbot interferometer," Appl. Phys. Express 1,117002 (2008). [CrossRef]
- W. Yashiro, Y. Takeda, and A. Momose, "Efficiency of capturing a phase image using conebeam x-ray Talbot interferometry," J. Opt. Soc. Am. A 25,2025-2039 (2008). [CrossRef]
- A. Momose, W. Yashiro, and Y. Takeda, "X-ray phase imaging with Talbot interferometry," in Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems, Y. Censor, M. Jiang, and G. Wang, eds., (Medical Physics Publishing, Madison, Wisconsin, USA, 2009).
- Z.-F. Huang, K.-J. Kang, L. Zhang, Z.-Q. Chen, F. Ding, Z.-T. Wang, and Q.-G. Fang, "Alternative method for differential phase-contrast imaging with weakly coherent hard x rays," Phys. Rev. A 79,013815 (2009). [CrossRef]
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