Abstract

We describe a quantitative phase imaging process using an x-ray laboratory-based source with an extremely broad bandwidth spectrum. The thickness of a homogeneous object can be retrieved by using separately spectrally weighted values for the attenuation coefficient and the decrement of the real part of the refractive index. This method is valid for a wide range of object types, including objects with an absorption edge in the used energy range. The accessibility of conventional x-ray laboratory sources makes this method very useful for quantitative phase retrieval of homogeneous objects. We demonstrate the application of this method for quantitative phase retrieval imaging in tomographic measurements.

© 2008 Optical Society of America

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  1. A. Snigirev, I. Snigreva, V. Kohn, S. Kuznetsov, and I. Schelokov, "On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486-5492 (1995).
    [CrossRef]
  2. D. Gao, T. J. Davis, and S. W. Wilkins, "X-ray phase contrast imaging study of voids and fibres in a polymer matrix," Aust. J. Phys. 48, 103-111 (1995).
  3. R. Toth, J. C. Kieffer, S. Fourmaux, and T. Ozaki, "In-line phase contrast imaging with a laser-based hard x-ray source," Rev. Sci. Instrum. 76, 0837011-0837016 (2005).
    [CrossRef]
  4. F. Zernike, "Phase contrast, a new method for the microscopic observation of transparent objects," Physica 9, 686-698 (1942).
    [CrossRef]
  5. B. Zakharin and J. Stricker, "Schlieren systems with coherent illumination for quantitative measurements," Appl. Opt. 43, 4786-4795 (2004).
    [CrossRef] [PubMed]
  6. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard X rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
    [CrossRef] [PubMed]
  7. D. Paganin, S. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. (Paris) 206, 33-40 (2002).
  8. K. A. Nugent, "Partially coherent diffraction patterns and coherence measurement," J. Opt. Soc. Am. A 8, 1574-1579 (1991).
    [CrossRef]
  9. B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, "Phase contrast radiography: Image modelling and optimization," Rev. Sci. Instrum. 75, 5271-5276 (2004).
    [CrossRef]
  10. D. Paganin and K. A. Nugent, "Noninterferometric phase imaging with partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
    [CrossRef]
  11. S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207, 79-96 (2002).
    [CrossRef] [PubMed]
  12. J. M. Cowley, Diffraction Physics, 3rd revised ed. (North-Holland, Amsterdam, 1995).
  13. B. D. Arhatari, F. De Carlo, and A. G. Peele, "Direct quantitative tomographic reconstruction for weakly absorbing homogeneous phase objects," Rev. Sci. Instrum. 78, 0537011-0537015 (2007).
    [CrossRef]
  14. G. R. Myers, S. Mayo, T. E. Gureyev, D. Paganin, and S. W. Wilkins, "Polychromatic cone-beam phase-contrast tomography," Phys.Rev. A 76, 1-4 (2007).
    [CrossRef]
  15. A. G. Peele, F. De Carlo, P. J. McMahon, B. B. Dhal, and K. A. Nugent, "X-ray phase contrast tomography with a bending magnet source," Rev. Sci. Instrum. 76, 837071-837075 (2005).
    [CrossRef]
  16. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).
  17. B. D. Arhatari, A. G. Peele, K. A. Nugent, and F. De Carlo, "Quality of the reconstruction in x-ray phase contrast tomography," Rev. Sci. Instrum. 77, 0637091-0637096 (2006).
    [CrossRef]

2007

B. D. Arhatari, F. De Carlo, and A. G. Peele, "Direct quantitative tomographic reconstruction for weakly absorbing homogeneous phase objects," Rev. Sci. Instrum. 78, 0537011-0537015 (2007).
[CrossRef]

G. R. Myers, S. Mayo, T. E. Gureyev, D. Paganin, and S. W. Wilkins, "Polychromatic cone-beam phase-contrast tomography," Phys.Rev. A 76, 1-4 (2007).
[CrossRef]

2006

B. D. Arhatari, A. G. Peele, K. A. Nugent, and F. De Carlo, "Quality of the reconstruction in x-ray phase contrast tomography," Rev. Sci. Instrum. 77, 0637091-0637096 (2006).
[CrossRef]

2005

A. G. Peele, F. De Carlo, P. J. McMahon, B. B. Dhal, and K. A. Nugent, "X-ray phase contrast tomography with a bending magnet source," Rev. Sci. Instrum. 76, 837071-837075 (2005).
[CrossRef]

R. Toth, J. C. Kieffer, S. Fourmaux, and T. Ozaki, "In-line phase contrast imaging with a laser-based hard x-ray source," Rev. Sci. Instrum. 76, 0837011-0837016 (2005).
[CrossRef]

2004

B. Zakharin and J. Stricker, "Schlieren systems with coherent illumination for quantitative measurements," Appl. Opt. 43, 4786-4795 (2004).
[CrossRef] [PubMed]

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, "Phase contrast radiography: Image modelling and optimization," Rev. Sci. Instrum. 75, 5271-5276 (2004).
[CrossRef]

2002

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207, 79-96 (2002).
[CrossRef] [PubMed]

D. Paganin, S. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. (Paris) 206, 33-40 (2002).

1998

D. Paganin and K. A. Nugent, "Noninterferometric phase imaging with partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
[CrossRef]

1996

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard X rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

1995

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

D. Gao, T. J. Davis, and S. W. Wilkins, "X-ray phase contrast imaging study of voids and fibres in a polymer matrix," Aust. J. Phys. 48, 103-111 (1995).

1991

1942

F. Zernike, "Phase contrast, a new method for the microscopic observation of transparent objects," Physica 9, 686-698 (1942).
[CrossRef]

Arhatari, B. D.

B. D. Arhatari, F. De Carlo, and A. G. Peele, "Direct quantitative tomographic reconstruction for weakly absorbing homogeneous phase objects," Rev. Sci. Instrum. 78, 0537011-0537015 (2007).
[CrossRef]

B. D. Arhatari, A. G. Peele, K. A. Nugent, and F. De Carlo, "Quality of the reconstruction in x-ray phase contrast tomography," Rev. Sci. Instrum. 77, 0637091-0637096 (2006).
[CrossRef]

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, "Phase contrast radiography: Image modelling and optimization," Rev. Sci. Instrum. 75, 5271-5276 (2004).
[CrossRef]

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard X rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard X rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Davis, T. J.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207, 79-96 (2002).
[CrossRef] [PubMed]

D. Gao, T. J. Davis, and S. W. Wilkins, "X-ray phase contrast imaging study of voids and fibres in a polymer matrix," Aust. J. Phys. 48, 103-111 (1995).

De Carlo, F.

B. D. Arhatari, F. De Carlo, and A. G. Peele, "Direct quantitative tomographic reconstruction for weakly absorbing homogeneous phase objects," Rev. Sci. Instrum. 78, 0537011-0537015 (2007).
[CrossRef]

B. D. Arhatari, A. G. Peele, K. A. Nugent, and F. De Carlo, "Quality of the reconstruction in x-ray phase contrast tomography," Rev. Sci. Instrum. 77, 0637091-0637096 (2006).
[CrossRef]

A. G. Peele, F. De Carlo, P. J. McMahon, B. B. Dhal, and K. A. Nugent, "X-ray phase contrast tomography with a bending magnet source," Rev. Sci. Instrum. 76, 837071-837075 (2005).
[CrossRef]

Dhal, B. B.

A. G. Peele, F. De Carlo, P. J. McMahon, B. B. Dhal, and K. A. Nugent, "X-ray phase contrast tomography with a bending magnet source," Rev. Sci. Instrum. 76, 837071-837075 (2005).
[CrossRef]

Fourmaux, S.

R. Toth, J. C. Kieffer, S. Fourmaux, and T. Ozaki, "In-line phase contrast imaging with a laser-based hard x-ray source," Rev. Sci. Instrum. 76, 0837011-0837016 (2005).
[CrossRef]

Gao, D.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207, 79-96 (2002).
[CrossRef] [PubMed]

D. Gao, T. J. Davis, and S. W. Wilkins, "X-ray phase contrast imaging study of voids and fibres in a polymer matrix," Aust. J. Phys. 48, 103-111 (1995).

Gureyev, T. E.

G. R. Myers, S. Mayo, T. E. Gureyev, D. Paganin, and S. W. Wilkins, "Polychromatic cone-beam phase-contrast tomography," Phys.Rev. A 76, 1-4 (2007).
[CrossRef]

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207, 79-96 (2002).
[CrossRef] [PubMed]

D. Paganin, S. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. (Paris) 206, 33-40 (2002).

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard X rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Kieffer, J. C.

R. Toth, J. C. Kieffer, S. Fourmaux, and T. Ozaki, "In-line phase contrast imaging with a laser-based hard x-ray source," Rev. Sci. Instrum. 76, 0837011-0837016 (2005).
[CrossRef]

Kohn, V.

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

Kuznetsov, S.

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

Mancuso, A. P.

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, "Phase contrast radiography: Image modelling and optimization," Rev. Sci. Instrum. 75, 5271-5276 (2004).
[CrossRef]

Mayo, S.

G. R. Myers, S. Mayo, T. E. Gureyev, D. Paganin, and S. W. Wilkins, "Polychromatic cone-beam phase-contrast tomography," Phys.Rev. A 76, 1-4 (2007).
[CrossRef]

D. Paganin, S. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. (Paris) 206, 33-40 (2002).

Mayo, S. C.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207, 79-96 (2002).
[CrossRef] [PubMed]

McMahon, P. J.

A. G. Peele, F. De Carlo, P. J. McMahon, B. B. Dhal, and K. A. Nugent, "X-ray phase contrast tomography with a bending magnet source," Rev. Sci. Instrum. 76, 837071-837075 (2005).
[CrossRef]

Miller, P. R.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207, 79-96 (2002).
[CrossRef] [PubMed]

D. Paganin, S. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. (Paris) 206, 33-40 (2002).

Myers, G. R.

G. R. Myers, S. Mayo, T. E. Gureyev, D. Paganin, and S. W. Wilkins, "Polychromatic cone-beam phase-contrast tomography," Phys.Rev. A 76, 1-4 (2007).
[CrossRef]

Nugent, K. A.

B. D. Arhatari, A. G. Peele, K. A. Nugent, and F. De Carlo, "Quality of the reconstruction in x-ray phase contrast tomography," Rev. Sci. Instrum. 77, 0637091-0637096 (2006).
[CrossRef]

A. G. Peele, F. De Carlo, P. J. McMahon, B. B. Dhal, and K. A. Nugent, "X-ray phase contrast tomography with a bending magnet source," Rev. Sci. Instrum. 76, 837071-837075 (2005).
[CrossRef]

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, "Phase contrast radiography: Image modelling and optimization," Rev. Sci. Instrum. 75, 5271-5276 (2004).
[CrossRef]

D. Paganin and K. A. Nugent, "Noninterferometric phase imaging with partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard X rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

K. A. Nugent, "Partially coherent diffraction patterns and coherence measurement," J. Opt. Soc. Am. A 8, 1574-1579 (1991).
[CrossRef]

Ozaki, T.

R. Toth, J. C. Kieffer, S. Fourmaux, and T. Ozaki, "In-line phase contrast imaging with a laser-based hard x-ray source," Rev. Sci. Instrum. 76, 0837011-0837016 (2005).
[CrossRef]

Paganin, D.

G. R. Myers, S. Mayo, T. E. Gureyev, D. Paganin, and S. W. Wilkins, "Polychromatic cone-beam phase-contrast tomography," Phys.Rev. A 76, 1-4 (2007).
[CrossRef]

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207, 79-96 (2002).
[CrossRef] [PubMed]

D. Paganin, S. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. (Paris) 206, 33-40 (2002).

D. Paganin and K. A. Nugent, "Noninterferometric phase imaging with partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard X rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Parry, D. J.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207, 79-96 (2002).
[CrossRef] [PubMed]

Peele, A. G.

B. D. Arhatari, F. De Carlo, and A. G. Peele, "Direct quantitative tomographic reconstruction for weakly absorbing homogeneous phase objects," Rev. Sci. Instrum. 78, 0537011-0537015 (2007).
[CrossRef]

B. D. Arhatari, A. G. Peele, K. A. Nugent, and F. De Carlo, "Quality of the reconstruction in x-ray phase contrast tomography," Rev. Sci. Instrum. 77, 0637091-0637096 (2006).
[CrossRef]

A. G. Peele, F. De Carlo, P. J. McMahon, B. B. Dhal, and K. A. Nugent, "X-ray phase contrast tomography with a bending magnet source," Rev. Sci. Instrum. 76, 837071-837075 (2005).
[CrossRef]

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, "Phase contrast radiography: Image modelling and optimization," Rev. Sci. Instrum. 75, 5271-5276 (2004).
[CrossRef]

Pogany, A.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207, 79-96 (2002).
[CrossRef] [PubMed]

Schelokov, I.

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

Snigirev, A.

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

Snigreva, I.

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

Stevenson, A. W.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207, 79-96 (2002).
[CrossRef] [PubMed]

Stricker, J.

Toth, R.

R. Toth, J. C. Kieffer, S. Fourmaux, and T. Ozaki, "In-line phase contrast imaging with a laser-based hard x-ray source," Rev. Sci. Instrum. 76, 0837011-0837016 (2005).
[CrossRef]

Wilkins, S. W.

G. R. Myers, S. Mayo, T. E. Gureyev, D. Paganin, and S. W. Wilkins, "Polychromatic cone-beam phase-contrast tomography," Phys.Rev. A 76, 1-4 (2007).
[CrossRef]

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207, 79-96 (2002).
[CrossRef] [PubMed]

D. Paganin, S. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. (Paris) 206, 33-40 (2002).

D. Gao, T. J. Davis, and S. W. Wilkins, "X-ray phase contrast imaging study of voids and fibres in a polymer matrix," Aust. J. Phys. 48, 103-111 (1995).

Zakharin, B.

Zernike, F.

F. Zernike, "Phase contrast, a new method for the microscopic observation of transparent objects," Physica 9, 686-698 (1942).
[CrossRef]

Appl. Opt.

Aust. J. Phys.

D. Gao, T. J. Davis, and S. W. Wilkins, "X-ray phase contrast imaging study of voids and fibres in a polymer matrix," Aust. J. Phys. 48, 103-111 (1995).

J. Microsc.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, and A. W. Stevenson, "Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging," J. Microsc. 207, 79-96 (2002).
[CrossRef] [PubMed]

J. Microsc. (Paris)

D. Paganin, S. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, "Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object," J. Microsc. (Paris) 206, 33-40 (2002).

J. Opt. Soc. Am. A

Phys. Rev. Lett.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard X rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

D. Paganin and K. A. Nugent, "Noninterferometric phase imaging with partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
[CrossRef]

Phys.Rev. A

G. R. Myers, S. Mayo, T. E. Gureyev, D. Paganin, and S. W. Wilkins, "Polychromatic cone-beam phase-contrast tomography," Phys.Rev. A 76, 1-4 (2007).
[CrossRef]

Physica

F. Zernike, "Phase contrast, a new method for the microscopic observation of transparent objects," Physica 9, 686-698 (1942).
[CrossRef]

Rev. Sci. Instrum.

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

R. Toth, J. C. Kieffer, S. Fourmaux, and T. Ozaki, "In-line phase contrast imaging with a laser-based hard x-ray source," Rev. Sci. Instrum. 76, 0837011-0837016 (2005).
[CrossRef]

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, "Phase contrast radiography: Image modelling and optimization," Rev. Sci. Instrum. 75, 5271-5276 (2004).
[CrossRef]

A. G. Peele, F. De Carlo, P. J. McMahon, B. B. Dhal, and K. A. Nugent, "X-ray phase contrast tomography with a bending magnet source," Rev. Sci. Instrum. 76, 837071-837075 (2005).
[CrossRef]

B. D. Arhatari, A. G. Peele, K. A. Nugent, and F. De Carlo, "Quality of the reconstruction in x-ray phase contrast tomography," Rev. Sci. Instrum. 77, 0637091-0637096 (2006).
[CrossRef]

B. D. Arhatari, F. De Carlo, and A. G. Peele, "Direct quantitative tomographic reconstruction for weakly absorbing homogeneous phase objects," Rev. Sci. Instrum. 78, 0537011-0537015 (2007).
[CrossRef]

Other

J. M. Cowley, Diffraction Physics, 3rd revised ed. (North-Holland, Amsterdam, 1995).

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

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Figures (4)

Fig. 1.
Fig. 1.

Combined detector response function and spectrum of the beam for a Tungsten target at 40kV(left) and 150kV(right) tube voltage respectively. The detector artefacts – low energy cutoff and fluorescence peaks – can be seen to have a small effect on the overall spectrum.

Fig. 2.
Fig. 2.

The recorded intensity plot measured for a polychromatic x-ray source at 40keV (left) and the retrieved thickness plot (right) for the Kapton step sample. The horizontal dotted lines show the actual thickness, the solid line is the retrieved thickness obtained using Eq. (11), the dashed line is the retrieved thickness obtained by using an effective energy, and the dash-dot line is the retrieved thickness obtained by using an energy corresponding to the peak in the spectral weighting distribution shown in Fig. 1.

Fig. 3.
Fig. 3.

The retrieved thickness of a 100 nm thick gold sample (left) and a plot (right) of the retrieved thickness along the horizontal line indicated by the arrow.

Fig. 4.
Fig. 4.

A reconstructed slice for 42 µm polystyrene spheres (left) imaged using the full spectrum of a laboratory x-ray tube source. The plot (mid) along a horizontal line indicated by the arrow shows the obtained values for δpoly . The image on the right shows a surface rendering for the reconstructed data volume demonstrating that good segmentation for the spheres can be obtained.

Equations (12)

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I λ z ( r ) z = λ 2 π · [ I λ 0 ( r ) φ λ ( r ) ] ,
I λ z ( r ) = I λ 0 ( r ) [ 1 z λ 2 π 2 φ λ ( r ) ] .
I λ 0 ( r ) = I λ in ( r ) e μ λ T ( r ) , φ λ ( r ) = 2 π λ δ λ T ( r ) ,
I λ z ( r ) = I λ in ( r ) e μ λ T ( r ) [ 1 + z δ λ 2 T ( r ) ] .
I λ z ( r ) = I λ in ( r ) [ 1 μ λ T ( r ) + z δ λ 2 T ( r ) ] .
I poly z ( r ) = I λ in ( r ) D ( λ ) [ 1 μ λ T ( r ) + z δ λ 2 T ( r ) ] d λ ,
I poly in ( r ) = I λ in ( r ) D ( λ ) d ( λ ) .
μ poly = μ λ I λ in ( r ) D ( λ ) d λ I λ in ( r ) D ( λ ) d λ ,
I poly z ( r ) = I poly in [ 1 μ poly T ( r ) + z δ poly 2 T ( r ) ]
F [ I poly z ( r ) I poly in ( r ) 1 ] = ( μ poly z δ poly u 2 ) F [ T ( r ) ] ,
T ( r ) = F 1 ( 1 μ poly + z δ poly u 2 F [ I poly z ( r ) I poly in ( r ) 1 ] )
δ ( r ) = FBP [ δ poly T θ ( r ) ]

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