Abstract

Using a high-efficiency grating interferometer for hard X rays (10–30 keV) and a phase-stepping technique, separate radiographs of the phase and absorption profiles of bulk samples can be obtained from a single set of measurements. Tomographic reconstruction yields quantitative three-dimensional maps of the X-ray refractive index, with a spatial resolution down to a few microns. The method is mechanically robust, requires little spatial coherence and monochromaticity, and can be scaled up to large fields of view, with a detector of correspondingly moderate spatial resolution. These are important prerequisites for use with laboratory X-ray sources.

© 2005 Optical Society of America

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References

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  1. R. Fitzgerald, ??Phase-Sensitive X-Ray Imaging,?? Phys. Today 53(7), 23??27 (2000).
    [CrossRef]
  2. A. Momose, ??Phase-sensitive imaging and phase tomography using X-ray interferometers,?? Opt. Express 11, 2303??2314 (2003). URL <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-19-2303.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-19-2303.</a>
    [CrossRef] [PubMed]
  3. U. Bonse and M. Hart, ??An X-ray interferometer,?? Appl. Phys. Lett. 6, 155??156 (1965).
    [CrossRef]
  4. A. Momose, T. Takeda, Y. Itai, and K. Hirano, ??Phase-contrast X-ray computed tomography for observing biological soft tissues,?? Nature Med. 2, 473??475 (1996).
    [CrossRef] [PubMed]
  5. F. Beckmann, U. Bonse, F. Busch, and O. Günnewig, ??X-ray Microtomography Using Phase Contrast for the Investigation of Organic Matter,?? J. Comp. Assist. Tomography 21, 539??553 (1997).
    [CrossRef]
  6. V. N. Ingal and E. A. Beliaevskaya, ??X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,?? J. Phys. D 28, 2314??2317 (1995).
    [CrossRef]
  7. T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, ??Phase-contrast imaging of weakly absorbing materials using hard X-rays,?? Nature (London) 373, 595??598 (1995).
    [CrossRef]
  8. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, ??Diffraction enhanced x-ray imaging,?? Phys. Med. Biol. 42, 2015??2025 (1997).
    [CrossRef] [PubMed]
  9. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, ??On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,?? Rev. Sci. Instr. 66, 5486??5492 (1995).
    [CrossRef]
  10. S.W.Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A.W. Stevenson, ??Phase-contrast imaging using polychromatic hard X-rays,?? Nature (London) 384, 335??337 (1996).
    [CrossRef]
  11. P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, ??Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,?? Appl. Phys. Lett. 75, 2912??2914 (1999).
    [CrossRef]
  12. C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, ??Differential x-ray phase contrast imaging using a shearing interferometer,?? Appl. Phys. Lett. 81, 3287??3289 (2002).
    [CrossRef]
  13. A. Momose, ??Demonstration of X-Ray Talbot Interferometry,?? Jpn. J. Appl. Phys. 42, L866??L868 (2003).
    [CrossRef]
  14. T.Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, ??X-ray wavefront analysis and optics characterization with a grating interferometer,?? Appl. Phys. Lett. 86, 054,101 (2005).
  15. If the incoming wave is not plane but curved, with a radius of curvature R, the only additional consideration that must be made is that the interference fringe period at a distance z downstream of the grating scales with a factor of M = 1+z/R..
  16. K. Creath, ??Phase-measurement Interferometry Techniques,?? in Progress In Optics XXVI, E. Wolf, ed., pp. 349??393 (Elsevier Science, 1988).
    [CrossRef]
  17. M. Born and E. Wolf, Principles of Optics, sixth ed. (Pergamon Press, Oxford, England, 1993).
  18. It is conceivable to measure the derivatives of Φ along both transverse directions x and y instead of only one as described in this paper. This may make phase reconstruction more robust in the presence of phase wrapping. However, it would require a more complicated measurement procedure and, preferably, two-dimensional gratings, which cannot be fabricated with high aspect ratios using the same technology as for the line gratings.
  19. B. F. McEwen, K. H. Downing, and R. M. Glaeser, ??The relevance of dose-fractionation in tomography of radiation-sensitive specimens,?? Ultramicroscopy 60, 357??373 (1995).
    [CrossRef] [PubMed]
  20. The energy spectrum was obtained by filtering the continuous spectrum from a wiggler source with a combination of absorption foils (Zr 0.1 mm, Si 0.8 mm) and the energy-dependent efficiency of the detector scintillator, a 20-μm-thick yttrium aluminium garnet (YAG) crystal.
  21. For example, the two Kα?lines in the emission spectrum of a molybdenum target are 0.1 keV apart from each other, at a mean energy of 17.4 keV.
  22. T. Weitkamp, ??XWFP: An X-ray wavefront propagation software package for the IDL computer language,?? in Advances in Computational Methods for X-Ray and Neutron Optics, M. Sanchez del Rio, ed., vol. 5536 of Proc. SPIE, pp. 181??189 (2004).
    [CrossRef]

Appl. Phys. Lett. (4)

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

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, ??Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,?? Appl. Phys. Lett. 75, 2912??2914 (1999).
[CrossRef]

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, ??Differential x-ray phase contrast imaging using a shearing interferometer,?? Appl. Phys. Lett. 81, 3287??3289 (2002).
[CrossRef]

T.Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, ??X-ray wavefront analysis and optics characterization with a grating interferometer,?? Appl. Phys. Lett. 86, 054,101 (2005).

J. Comp. Assist. Tomography (1)

F. Beckmann, U. Bonse, F. Busch, and O. Günnewig, ??X-ray Microtomography Using Phase Contrast for the Investigation of Organic Matter,?? J. Comp. Assist. Tomography 21, 539??553 (1997).
[CrossRef]

J. Phys. D (1)

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

Jpn. J. Appl. Phys. (1)

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

Nature (London) (2)

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

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

Nature Med. (1)

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

Opt. Express (1)

Phys. Med. Biol. (1)

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

Phys. Today (1)

R. Fitzgerald, ??Phase-Sensitive X-Ray Imaging,?? Phys. Today 53(7), 23??27 (2000).
[CrossRef]

Proc. SPIE (1)

T. Weitkamp, ??XWFP: An X-ray wavefront propagation software package for the IDL computer language,?? in Advances in Computational Methods for X-Ray and Neutron Optics, M. Sanchez del Rio, ed., vol. 5536 of Proc. SPIE, pp. 181??189 (2004).
[CrossRef]

Progress In Optics (1)

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

Rev. Sci. Instr. (1)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, ??On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,?? Rev. Sci. Instr. 66, 5486??5492 (1995).
[CrossRef]

Ultramicroscopy (1)

B. F. McEwen, K. H. Downing, and R. M. Glaeser, ??The relevance of dose-fractionation in tomography of radiation-sensitive specimens,?? Ultramicroscopy 60, 357??373 (1995).
[CrossRef] [PubMed]

Other (5)

The energy spectrum was obtained by filtering the continuous spectrum from a wiggler source with a combination of absorption foils (Zr 0.1 mm, Si 0.8 mm) and the energy-dependent efficiency of the detector scintillator, a 20-μm-thick yttrium aluminium garnet (YAG) crystal.

For example, the two Kα?lines in the emission spectrum of a molybdenum target are 0.1 keV apart from each other, at a mean energy of 17.4 keV.

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

It is conceivable to measure the derivatives of Φ along both transverse directions x and y instead of only one as described in this paper. This may make phase reconstruction more robust in the presence of phase wrapping. However, it would require a more complicated measurement procedure and, preferably, two-dimensional gratings, which cannot be fabricated with high aspect ratios using the same technology as for the line gratings.

If the incoming wave is not plane but curved, with a radius of curvature R, the only additional consideration that must be made is that the interference fringe period at a distance z downstream of the grating scales with a factor of M = 1+z/R..

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Figures (5)

Fig. 1.
Fig. 1.

Grating-based hard X-ray interferometer. (a) Principle: the beam splitter grating (G1) splits the incident beam into essentially two diffraction orders, which form a periodic interference pattern in the plane of the analyzer grating. A phase object in the incident beam will cause slight refraction, which results in changes of the locally transmitted intensity through the analyzer. (b,c) Scanning electron micrographs of cross sections through the gratings used. The silicon beam-splitter grating (b) has a pitch of 4 μm, i.e., twice that of the analyzer grating (c), which was made by filling the grooves of a silicon grating with gold by electroplating.

Fig. 2.
Fig. 2.

Principle of phase stepping. (a-d) Interferograms of polystyrene spheres (100 and 200 μm diameter), taken at the different relative positions x g =x 1,…,x 4 of the two interferometer gratings. (e) Intensity oscillation in two different detector pixels i = 1,2 as a function of x g. For each pixel, the oscillation phase φi and the average intensity ai over one grating period can be determined. (f) Image of the oscillation phase φ for all pixels. (g) Wave-front phase Φ retrieved from φ by integration. (h) Image of the averaged intensity a for all pixels, equivalent to a non-interferometric image. The length of the scale bar is 50 μm

Fig. 3.
Fig. 3.

Radiographs and tomograms of a reference sample consisting of two polymer fibers (polyamide, PA, and polybutylene terephthalate, PBT) and a boron fiber with a tungsten core, acquired with broadband radiation of (17.5 ±0.5) keV photon energy. (a) Non-interferometric projection image. (b) Tomographic slice, corresponding to the position indicated by the horizontal line in (a). (c) Reconstructed phase projection. (d) Tomographic slice through the refractive-index distribution. (e,f) Section profiles through the fiber centers in, respectively, the non-interferometric tomogram (e) and the phase tomogram (f) (solid: B/W, dashed: PA, dash-dotted: PBT, dotted: literature values). In (e), the lines for PA and B/W are displaced along the ordinate axis for clarity.

Fig. 4.
Fig. 4.

Three-dimensional density-projection rendering of the reconstructed refractive index of a small spider, supported by two polyamide fibers. These data were taken at a photon energy of 14.4 keV with gratings of pitch g 1 = 4 μm,g 2 = 2 μm, and an intergrating distance of d = 23.2 mm.

Fig. 5.
Fig. 5.

Left: Contrast transfer for fixed grating periods and intergrating distance as a function of wavelength λ, for ideal gratings. The half-width Δλ between zero-contrast wavelengths λ 1 ,λ 2 around the design wavelength λ 0 is a measure of the efficient energy range for a given setup. Right: Measured and simulated efficiency of a grating interferometer, taking into account variations of grating efficiency with photon energy. Here, a photon energy of 14.4 keV was used with an interferometer with grating periods of g 1 = 4μm and g 2 =2μm, at an intergrating distance of d = 69.7 mm. The simulation was made using the XWFP computer code [22].

Equations (3)

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φ = λd g 2 Φ x .
d m = ( m 1 2 ) g 1 2 4 λ , with m = 1,2,3,
Δλ = λ 0 2 m 1 .

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