Abstract

In-line, x-ray phase-contrast imaging is responsive to both phase changes and absorption as the x radiation traverses a body. Expressions are derived for phase-contrast imaging of objects having transmission functions separable in Cartesian coordinates. Starting from the Fresnel–Kirchhoff integral formula for image formation, an expression is found for the phase-contrast image produced by an x-ray source with nonvanishing dimensions. This expression is evaluated in limiting cases where the source-to-object distance is large, where the source acts as a point source, and where the weak phase approximation is valid. The integral expression for the image is evaluated for objects with simple geometrical shapes, showing the influence of the source dimensions on the visibility of phase-contrast features. The expressions derived here are evaluated for cases where the magnification is substantially greater than one as would be employed in biological imaging. Experiments are reported using the in-line phase-contrast imaging method with a microfocus x-ray source and a CCD camera.

© 2007 Optical Society of America

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  1. P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D 29, 133-146 (1996).
    [CrossRef]
  2. S. W. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard x-rays," Nature 384, 335-338 (1996).
    [CrossRef]
  3. F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
    [CrossRef] [PubMed]
  4. 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]
  5. S. Zabler, P. Cloetens, J. P. Guigay, and J. Baruchel, "Optimization of phase contrast imaging using hard x-rays," Rev. Sci. Instrum. 76, 073705 (2005).
    [CrossRef]
  6. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  7. A. Pogany, D. Gao, and S. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
    [CrossRef]
  8. J. M. Cowley, Diffraction Physics (North-Holland, 1984). The use of the Fresnel and the small-angle approximations require that parameters of the form (x?X)2/R be small compared with unity; hence, in experiments, the coordinates of the object and the image should be restricted to small distances from the axis of propagation of the x radiation.
  9. M. V. Klein, Optics (Wiley, 1970).
  10. The electric field is taken as proportional to 1/r. The intensity is expressed in units where the dielectric constant and permittivity are taken as unity so that the intensity becomes proportional to 1/r2.
  11. We use the convention used in Ref. where the field is attenuated proportional to exp[??(z)]. The intensity therefore is attenuated as exp[?2?(z)]. The parameter ?0 is taken as an intensity absorption coefficient.
  12. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, 1965).
  13. The spectrum of the x-ray tube, which uses only a Be window, can be found in Ref. , p. 99.
  14. J. Bushberg, J. Seibert, and E. Leidholdt, Jr., The Essential Physics of Medical Imaging (Williams and Wilkins, 1994).
  15. B. D. Arhatari, A. P. Mancuso, A. G. Peele, and D. A. Nugent, "Phase contrast radiography: image modeling and optimization," Rev. Sci. Instrum. 75, 5271-5276 (2004).
    [CrossRef]
  16. For objects where the weak phase approximation is not applicable, Fourier transformation (with respect to ?) of functions of the form exp[iphiv(?+?Df/2)], where D is a length parameter and f is spatial frequency, is required to determine the intensity in the frequency domain; that is, for each value of f in the frequency domain, a Fourier transformation with respect to ? must first be carried out. See Appendix A2 in Ref. .

2005

S. Zabler, P. Cloetens, J. P. Guigay, and J. Baruchel, "Optimization of phase contrast imaging using hard x-rays," Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

2004

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

1999

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]

1998

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

1997

A. Pogany, D. Gao, and S. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
[CrossRef]

1996

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D 29, 133-146 (1996).
[CrossRef]

S. W. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard x-rays," Nature 384, 335-338 (1996).
[CrossRef]

1994

J. Bushberg, J. Seibert, and E. Leidholdt, Jr., The Essential Physics of Medical Imaging (Williams and Wilkins, 1994).

1984

J. M. Cowley, Diffraction Physics (North-Holland, 1984). The use of the Fresnel and the small-angle approximations require that parameters of the form (x?X)2/R be small compared with unity; hence, in experiments, the coordinates of the object and the image should be restricted to small distances from the axis of propagation of the x radiation.

1980

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

1970

M. V. Klein, Optics (Wiley, 1970).

1965

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, 1965).

Arfelli, F.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Arhatari, B. D.

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

Assante, M.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Barrett, R.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D 29, 133-146 (1996).
[CrossRef]

Baruchel, J.

S. Zabler, P. Cloetens, J. P. Guigay, and J. Baruchel, "Optimization of phase contrast imaging using hard x-rays," Rev. Sci. Instrum. 76, 073705 (2005).
[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]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D 29, 133-146 (1996).
[CrossRef]

Bonvicini, V.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Bravin, A.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Bushberg, J.

J. Bushberg, J. Seibert, and E. Leidholdt, Jr., The Essential Physics of Medical Imaging (Williams and Wilkins, 1994).

Cantatore, G.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Castelli, E.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Cloetens, P.

S. Zabler, P. Cloetens, J. P. Guigay, and J. Baruchel, "Optimization of phase contrast imaging using hard x-rays," Rev. Sci. Instrum. 76, 073705 (2005).
[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]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D 29, 133-146 (1996).
[CrossRef]

Cowley, J. M.

J. M. Cowley, Diffraction Physics (North-Holland, 1984). The use of the Fresnel and the small-angle approximations require that parameters of the form (x?X)2/R be small compared with unity; hence, in experiments, the coordinates of the object and the image should be restricted to small distances from the axis of propagation of the x radiation.

DiMichiel, M.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Dyck, D. V.

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]

Gao, D.

A. Pogany, D. Gao, and S. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
[CrossRef]

S. W. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard x-rays," Nature 384, 335-338 (1996).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, 1965).

Guigay, J. P.

S. Zabler, P. Cloetens, J. P. Guigay, and J. Baruchel, "Optimization of phase contrast imaging using hard x-rays," Rev. Sci. Instrum. 76, 073705 (2005).
[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]

Guigay, J.-P.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D 29, 133-146 (1996).
[CrossRef]

Gureyev, T.

S. W. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard x-rays," Nature 384, 335-338 (1996).
[CrossRef]

Klein, M. V.

M. V. Klein, Optics (Wiley, 1970).

Landuyt, J. V.

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]

Leidholdt, E.

J. Bushberg, J. Seibert, and E. Leidholdt, Jr., The Essential Physics of Medical Imaging (Williams and Wilkins, 1994).

Longo, R.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Ludwig, W.

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]

Mancuso, A. P.

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

Nugent, D. A.

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

Olivo, A.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Palma, L. D.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Pani, S.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Peele, A. G.

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

Pogany, A.

A. Pogany, D. Gao, and S. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
[CrossRef]

S. W. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard x-rays," Nature 384, 335-338 (1996).
[CrossRef]

Pontoni, D.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Poropat, P.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Prest, M.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Rashevsky, A.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, 1965).

Schlenker, M.

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]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D 29, 133-146 (1996).
[CrossRef]

Seibert, J.

J. Bushberg, J. Seibert, and E. Leidholdt, Jr., The Essential Physics of Medical Imaging (Williams and Wilkins, 1994).

Stevenson, A. W.

S. W. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard x-rays," Nature 384, 335-338 (1996).
[CrossRef]

Tromba, G.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Vacchi, A.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Vallazza, E.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Wilkins, S.

A. Pogany, D. Gao, and S. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
[CrossRef]

Wilkins, S. W.

S. W. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard x-rays," Nature 384, 335-338 (1996).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Zabler, S.

S. Zabler, P. Cloetens, J. P. Guigay, and J. Baruchel, "Optimization of phase contrast imaging using hard x-rays," Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

Zanconati, F.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Appl. Phys. Lett.

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]

J. Phys. D

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D 29, 133-146 (1996).
[CrossRef]

Nature

S. W. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard x-rays," Nature 384, 335-338 (1996).
[CrossRef]

Phys. Med. Biol.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. DiMichiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998).
[CrossRef] [PubMed]

Rev. Sci. Instrum.

S. Zabler, P. Cloetens, J. P. Guigay, and J. Baruchel, "Optimization of phase contrast imaging using hard x-rays," Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

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

A. Pogany, D. Gao, and S. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
[CrossRef]

Other

J. M. Cowley, Diffraction Physics (North-Holland, 1984). The use of the Fresnel and the small-angle approximations require that parameters of the form (x?X)2/R be small compared with unity; hence, in experiments, the coordinates of the object and the image should be restricted to small distances from the axis of propagation of the x radiation.

M. V. Klein, Optics (Wiley, 1970).

The electric field is taken as proportional to 1/r. The intensity is expressed in units where the dielectric constant and permittivity are taken as unity so that the intensity becomes proportional to 1/r2.

We use the convention used in Ref. where the field is attenuated proportional to exp[??(z)]. The intensity therefore is attenuated as exp[?2?(z)]. The parameter ?0 is taken as an intensity absorption coefficient.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, 1965).

The spectrum of the x-ray tube, which uses only a Be window, can be found in Ref. , p. 99.

J. Bushberg, J. Seibert, and E. Leidholdt, Jr., The Essential Physics of Medical Imaging (Williams and Wilkins, 1994).

For objects where the weak phase approximation is not applicable, Fourier transformation (with respect to ?) of functions of the form exp[iphiv(?+?Df/2)], where D is a length parameter and f is spatial frequency, is required to determine the intensity in the frequency domain; that is, for each value of f in the frequency domain, a Fourier transformation with respect to ? must first be carried out. See Appendix A2 in Ref. .

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

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Figures (5)

Fig. 1
Fig. 1

Schematic drawing of the imaging apparatus and coordinates for phase-contrast imaging with a finite source size. The coordinates on the source, object, and image planes are denoted ( ξ , η ) , ( X , Y ) , and ( x , y ) , respectively.

Fig. 2
Fig. 2

(left) x-ray phase-contrast images for a 60 μ m radius nylon line with R 1 = 0.2 m , R 2 = 2.4 m , and λ = 41.3 pm for (a) a point x-ray source from Eq. (18) and (b) a nonvanishing source with σ = 6.75 μ m from Eq. (16). The parameters describing the index of refraction of nylon are taken as δ = 1.0 × 10 6 and μ 0 = 0 . (right) Calculated x-ray phase-contrast images for a 50 μ m radius gold wire with R 1 = 0.2 m , R 2 = 2.4 m , and λ = 41.3 pm for (c) a point x-ray source, from Eq. (18), and (d) a nonvanishing source with σ = 6.75 μ m from Eq. (16). The parameters describing the index of refraction of gold are taken as δ = 3.55 × 10 6 and μ 0 = 5.0 × 10 4 m 1 .

Fig. 3
Fig. 3

Calculated x-ray phase-contrast images for a crossed pair of 120 μ m diameter nylon threads with R 1 = 0.2 m , R 2 = 2.4 m , and λ = 41.3 pm (a) for a point x-ray source, from Eq. (18), and (b) with a finite source size with σ = 6.75 μ m calculated using Eq. (16). The parameters δ and μ 0 were taken to be 1.0 × 10 6 and 0, respectively. The maximum and minimum values of the intensity are (a) 6.34 and 0.01 and (b) 1.21 and 0.90. Calculated x-ray phase-contrast images for an 1.0 μ m thick Teflon square taken with R 1 = 1.3 m , R 2 = 1.3 m , and λ = 41.3 pm for (c) a point x-ray source from Eq. (21) evaluated with σ = 0 , and (d) a finite source size with σ = 6.75 μ m calculated using Eq. (21). The parameters δ and μ 0 were taken as 4.9 × 10 7 and 74.8 m 1 , respectively. The maximum and minimum values of the intensity are (c) 1.04 and 0.94 and (d) 1.01 and 0.98. The integration was carried out over a region 100 μ m by 100 μ m for both (c) and (d).

Fig. 4
Fig. 4

(left) (a) Intensity versus distance for a phase-contrast image of a 120 μ m diameter nylon line taken from a CCD image. The imaging parameters are R 1 = 0.18 m and R 2 = 2.42 m , with the x-ray tube operating for 3 min at 90 kV and 100 μ A . (b) Intensity versus distance for a nylon line calculated from Eq. (16) with λ = 41 pm , R 1 = 0.2 m , R 2 = 2.4 m , σ = 6.75 μ m , and μ 0 = 0 . Inset: section of an experimental CCD image used to determine the curve in (a). (right) (c) Intensity versus distance for a phase-contrast image of a 100 μ m diameter gold wire taken from a CCD image taken with R 1 = 0.18 m and R 2 = 2.42 m , with the x-ray tube operating for 3 min at 90 kV and 100 μ A , and (d) as calculated from Eq. (16) with λ = 41 pm , R 1 = 0.2 m , R 2 = 2.4 m , σ = 6.75 μ m , and μ 0 = 50 × 10 3 m 1 . Inset: section of a CCD image used to generate the curve in (c).

Fig. 5
Fig. 5

(a) CCD image of a pair of crossed 100 μ m diameter nylon lines taken with R 1 = 0.18 m and R 2 = 2.42 m and with the x-ray tube operating for 3 min at 90 kV and 100 μ A . (b) A contour plot of the nylon lines calculated from Eq. (16) with λ = 41 pm , R 1 = 0.2 m , R 2 = 2.4 m , σ = 6.75 μ m , and μ 0 = 0 . (c) Intensity versus distance taken from a CCD image of a 1 cm long section of an 80 μ m thick Teflon rectangle with R 1 = 0.2 m and R 2 = 2.4 m and with the x-ray tube operating for 3 min at 90 kV and 100 μ A . (d) A contour plot of the corner calculated from Eq. (16) with λ = 41 pm , R 1 = 0.2 m , R 2 = 2.4 m , σ = 6.75 μ m , δ = 1 × 10 6 , and μ 0 = 0 .

Equations (42)

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f = i λ Σ 0 e i k r r r r d σ ,
f ( x , y ) = i λ e i k ( R 1 + R 2 ) R 1 R 2 e i k ( ξ X ) 2 2 R 1 e i k ( x X ) 2 2 R 2 q x ( X ) d X e i k ( η Y ) 2 2 R 1 e i k ( y Y ) 2 2 R 2 q y ( Y ) d Y ,
q ( X , Y ) = { q x ( X ) q y ( Y ) if X a and Y b 1 if X > a or Y > b } ,
q ( ξ ) = e i ϕ ( ξ ) μ ( ξ ) ,
E q ( X ) d X = E d X + a a E [ q ( X ) 1 ] d X ,
α = e i k ξ 2 2 R 1 i k x 2 2 R 2 e i k X 2 ( 1 R 1 + 1 R 2 ) 2 + i k ( ξ R 1 + x R 2 ) X d X .
α = 2 π R 1 R 2 i k ( R 1 + R 2 ) e i k ( x ξ ) 2 2 ( R 1 + R 2 ) .
s ( ξ , η ) = 1 π σ 2 e ( ξ 2 + η 2 ) σ 2 ,
I ( x , y ) = f ( x , y ) 2 s ( ξ , η ) d ξ d η ,
I ( x , y ) = ( α + β ) ( γ + κ ) 2 s ( ξ , η ) d ξ d η .
s ( ξ , η ) = s ( ξ ) s ( η ) ;
I ( x , y ) = ( 1 λ R 1 R 2 ) 2 1 σ π ( α α * + 2 Re ( α β * ) + β β * ) e ξ 2 σ 2 d ξ × 1 σ π ( γ γ * + 2 Re ( γ κ * ) + κ κ * ) e η 2 σ 2 d η .
I ( x , y ) = ( 1 λ R 1 R 2 ) 2 [ I α α * ( x ; q x ) + 2 Re I α β * ( x ; q x ) + I β β * ( x ; q x ) ] × [ I α α * ( y ; q y ) + 2 Re I α β * ( y ; q y ) + I β β * ( y ; q y ) ] ,
I α α * ( x ; q x ) = 1 σ π α α * e ξ 2 σ 2 d ξ ,
I α β * ( x ; q x ) = 1 σ π α β * e ξ 2 σ 2 d ξ ,
I β β * ( x ; q x ) = 1 σ π β β * e ξ 2 σ 2 d ξ .
I α α * ( x ; q x ) = 2 π R 1 R 2 k ( R 1 + R 2 ) .
I β β * ( x ; q x ) = 1 σ π a a e i k ( x X ) 2 2 R 2 [ q x ( X ) 1 ] d X a a e i k ( x X ) 2 2 R 2 [ q x ( X ) 1 ] * d X × e i k ( ξ X ) 2 2 R 1 e i k ( ξ X ) 2 2 R 1 e ξ 2 σ 2 d ξ .
I β β * ( x ; q x ) = a a e i k ( x X ) 2 2 R 2 i k ( X 2 2 R 1 ) [ q x ( X ) 1 ] d X × a a e i k ( x X ) 2 2 R 2 + i k ( X 2 2 R 1 ) [ q x ( X ) 1 ] * e k 2 σ 2 ( X X ) 2 4 R 1 2 d X .
I α β * ( x ; q x ) = 1 σ 2 R 1 R 2 i k ( R 1 + R 2 ) a a e i k ( x X ) 2 2 R 2 [ q x ( X ) 1 ] * d X e i k ( x ξ ) 2 2 ( R 1 + R 2 ) e i k ( ξ X ) 2 2 R 1 ξ 2 σ 2 d ξ .
I α β * ( x ; q x ) = 2 π R 1 R 2 i β ̂ k ( R 1 + R 2 ) e i k ( x 2 2 ( R 1 + R 2 ) β ̂ ) a a e i k ( x X ) 2 2 R 2 ) [ q x ( X ) 1 ] * e i k ( X 2 2 R 1 β ̂ ) e k 2 σ 2 ( x X ) 2 4 R 1 ( R 1 + R 2 ) β ̂ d X ,
β ̂ = 1 i k σ 2 ( R 2 R 1 ) 2 ( R 1 + R 2 ) .
I ( x , y ) = 1 ( R 1 + R 2 ) 2 [ 1 + 2 Re I ̂ α β * ( x ; q x ) + I ̂ β β * ( x ; q x ) ] [ 1 + 2 Re I ̂ α β * ( y ; q y ) + I ̂ β β * ( y ; q y ) ] ,
I ̂ α β * ( x ; q x ) = k ( R 1 + R 2 ) i 2 π β ̂ R 1 R 2 e i k ( x 2 2 ( R 1 + R 2 ) β ̂ ) a a e i k ( x X ) 2 2 R 2 ) [ q x ( X ) 1 ] * e i k ( X 2 2 R 1 β ̂ ) e k 2 σ 2 ( x X ) 2 4 R 1 ( R 1 + R 2 ) β ̂ d X ,
I ̂ β β * ( x ; q x ) = 1 2 π k ( R 1 + R 2 ) R 1 R 2 a a e i k ( x X ) 2 2 R 2 ) e i k ( X 2 2 R 1 ) [ q x ( X ) 1 ] d X a a e i k ( x X ) 2 2 R 2 ) e i k X 2 2 R 1 [ q x ( X ) 1 ] * e k 2 σ 2 ( X X ) 2 4 R 1 2 d X .
I P ( x , y ) = 1 ( R 1 + R 2 ) 2 [ 1 + 2 Re I ̂ α β * P ( x ; q x ) + I ̂ β β * P ( x ; q x ) ] [ 1 + 2 Re I ̂ α β * P ( y ; q y ) + I ̂ β β * P ( y ; q y ) ] ,
I ̂ α β * P ( x ; q x ) = k ( R 1 + R 2 ) i 2 π R 1 R 2 e i k ( x 2 2 ( R 1 + R 2 ) ) a a e i k ( x X ) 2 2 R 2 ) [ q x ( X ) 1 ] * e i k ( X 2 2 R 1 ) d X ,
I ̂ β β * P ( x ; q x ) = k ( R 1 + R 2 ) 2 π R 1 R 2 a a e i k ( x X ) 2 2 R 2 e i k ( X 2 2 R 1 ) [ q x ( X ) 1 ] d X 2 .
I ̂ P ( x , y ) = 1 ( R 1 + R 2 ) 2 1 + I ̂ α β * P ( x ; q x ) 2 1 + I ̂ α β * P ( y ; q y ) 2 .
β ̂ 1 i k σ 2 2 R 1 ( R 2 R 1 ) ,
I P W ( x , y ) = 1 R 1 2 [ 1 + 2 Re I ̂ α β * P W ( x ; q x ) + I ̂ β β * P W ( x ; q x ) ] [ 1 + 2 Re I ̂ α β * P W ( y ; q y ) + I ̂ β β * P W ( y ; q y ) ] ,
I ̂ α β * P W ( x ; q x ) = k i 2 π β ̂ R 2 e i k ( x 2 2 R 1 β ̂ ) a a e i k ( x X ) 2 2 R 2 ) [ q x ( X ) 1 ] * e i k ( X 2 2 R 1 β ̂ ) e k 2 σ 2 ( x X ) 2 4 R 1 2 β ̂ d X ,
I ̂ β β * P W ( x ; q x ) = 1 2 π k R 2 a a e i k ( x X ) 2 2 R 2 ) e i k ( X 2 2 R 1 ) [ q x ( X ) 1 ] d X a a e i k ( x X ) 2 2 R 2 ) e i k ( X 2 2 R 1 ) [ q x ( X ) 1 ] * e k 2 σ 2 ( X X ) 2 4 R 1 d X .
q ( X , Y ) = e i ϕ x ( X ) ϕ y ( Y ) μ x ( X ) μ y ( Y ) ,
q ( X , Y ) = 1 + f ̂ x ( X ) f ̂ y ( Y ) ( i δ k t μ 0 t 2 ) ,
I W ( x , y ) = 1 ( R 1 + R 2 ) 2 [ 1 + 2 Re ( k t δ + i μ 0 t 2 ) I ̂ α β * W ( x ; f ̂ x ) I ̂ α β * W ( y ; f ̂ y ) + ( k 2 δ 2 t 2 + μ 0 2 t 2 4 ) I ̂ β β * W ( x ; f ̂ x ) I ̂ β β * W ( y ; f ̂ y ) ] ,
I ̂ α β * W ( x ; f ̂ x ) = k ( R 1 + R 2 ) 2 π β ̂ R 1 R 2 e i k x 2 2 ( R 1 + R 2 ) β ̂ a a f ̂ x * ( X ) e i k ( X 2 2 R 1 β ̂ ) + i k ( x X ) 2 2 R 2 e k 2 σ 2 ( x X ) 2 4 R 1 ( R 1 + R 2 ) β ̂ d X ,
I ̂ β β * W ( x ; f ̂ x )
= 1 2 π k ( R 1 + R 2 ) R 1 R 2 a a a a f ̂ x ( X ) f ̂ x * ( X ) × e i k [ X 2 2 R 1 + ( x X ) 2 2 R 2 ] + i k [ X 2 2 R 1 + ( x X ) 2 2 R 2 ] k 2 σ 2 ( X X ) 2 4 R 1 2 d X d X .
q x ( X ) = e 2 i k δ a 1 ( X a ) 2 μ 0 a 1 ( X a ) 2 ,
q x ( X ) = e ( 2 i k δ μ 0 ) 2 a 1 ( X a ) 2 for X a ,
q ( X , Y ) = e i ϕ x ( X ) ϕ y ( Y ) μ x ( X ) μ y ( Y ) ,

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