Abstract

Certain phase retrieval methods use knowledge about the free space propagation of a wave to phase a paraxial beam passing through one or more measurement planes. This approach has been widely applied and has been shown to quantitatively retrieve the refractive index profile of a sample. The quality of the phase retrieval will depend on a range of factors including sample feature size, propagation distance, measurement plane separation, wavelength and noise. Here we describe an optimisation study for two-plane phase retrieval using a laboratory-based X-ray source that considers all of these factors. We discuss our results in the context of a three-dimensional reconstruction of a sample refractive index profile.

© 2010 OSA

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    [CrossRef]
  2. P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
    [CrossRef]
  3. J. N. Clark, G. J. Williams, H. M. Quiney, L. Whitehead, M. D. de Jonge, E. Hanssen, M. Altissimo, K. A. Nugent, and A. G. Peele, “Quantitative phase measurement in coherent diffraction imaging,” Opt. Express 16(5), 3342–3348 (2008).
    [CrossRef] [PubMed]
  4. W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, “Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy,” Phys. Rev. Lett. 69(26), 3743–3746 (1992).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. D. Paganin, S. C. 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. 206(1), 33–40 (2002).
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    [CrossRef] [PubMed]
  9. B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
    [CrossRef] [PubMed]
  10. S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83(1-2), 67–73 (2000).
    [CrossRef] [PubMed]
  11. D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
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    [CrossRef]
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    [CrossRef] [PubMed]
  15. T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231(1-6), 53–70 (2004).
    [CrossRef]
  16. L. D. Turner, B. Dhal, J. Hayes, A. P. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, “X-ray phase imaging: Demonstration of extended conditions for homogeneous objects,” Opt. Express 12(13), 2960–2965 (2004).
    [CrossRef] [PubMed]
  17. T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard X-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D Appl. Phys. 32(5), 563–567 (1999).
    [CrossRef]
  18. T. E. Gureyev and S. W. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A 15(3), 579–585 (1998).
    [CrossRef]
  19. D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).
  20. D. Paganin, “Studies in phase retrieval,” PhD thesis (University of Melbourne, 1999).
  21. A. N. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).
  22. B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75(12), 5271–5276 (2004).
    [CrossRef]
  23. K. A. Nugent, “Partially coherent diffraction patterns and coherence measurement,” J. Opt. Soc. Am. A 8(10), 1574–1579 (1991).
    [CrossRef]
  24. T. E. Gureyev, S. C. Mayo, D. E. Myers, Y. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: Propagation-based x-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105(10), 102005 (2009).
    [CrossRef]

2010 (1)

B. D. Arhatari, W. P. Gates, N. Eshtiaghi, and A. G. Peele, “Phase retrieval tomography in the presence of noise,” J. Appl. Phys. 107(034904), 1–7 (2010).
[CrossRef]

2009 (1)

T. E. Gureyev, S. C. Mayo, D. E. Myers, Y. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: Propagation-based x-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105(10), 102005 (2009).
[CrossRef]

2008 (2)

2004 (4)

L. D. Turner, B. Dhal, J. Hayes, A. P. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, “X-ray phase imaging: Demonstration of extended conditions for homogeneous objects,” Opt. Express 12(13), 2960–2965 (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(12), 5271–5276 (2004).
[CrossRef]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231(1-6), 53–70 (2004).
[CrossRef]

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

2002 (1)

D. Paganin, S. C. 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. 206(1), 33–40 (2002).
[CrossRef] [PubMed]

2000 (3)

T. E. Gureyev, A. W. Stevenson, D. Paganin, S. C. Mayo, A. Pogany, D. Gao, and S. W. Wilkins, “Quantitative methods in phase-contrast x-ray imaging,” J. Digit. Imaging 13(S1), 121–126 (2000).
[CrossRef] [PubMed]

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
[CrossRef] [PubMed]

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83(1-2), 67–73 (2000).
[CrossRef] [PubMed]

1999 (2)

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard X-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D Appl. Phys. 32(5), 563–567 (1999).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[CrossRef]

1998 (3)

1996 (1)

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

1992 (1)

W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, “Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy,” Phys. Rev. Lett. 69(26), 3743–3746 (1992).
[CrossRef] [PubMed]

1991 (1)

1983 (1)

1963 (1)

A. N. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).

Allman, B. E.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
[CrossRef] [PubMed]

Altissimo, M.

Arhatari, B. D.

B. D. Arhatari, W. P. Gates, N. Eshtiaghi, and A. G. Peele, “Phase retrieval tomography in the presence of noise,” J. Appl. Phys. 107(034904), 1–7 (2010).
[CrossRef]

B. D. Arhatari, K. Hannah, E. Balaur, and A. G. Peele, “Phase imaging using a polychromatic x-ray laboratory source,” Opt. Express 16(24), 19950–19956 (2008).
[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(12), 5271–5276 (2004).
[CrossRef]

Arif, M.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
[CrossRef] [PubMed]

Bajt, S.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83(1-2), 67–73 (2000).
[CrossRef] [PubMed]

Balaur, E.

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83(1-2), 67–73 (2000).
[CrossRef] [PubMed]

Baruchel, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[CrossRef]

Clark, J. N.

Cloetens, P.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[CrossRef]

Coene, W.

W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, “Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy,” Phys. Rev. Lett. 69(26), 3743–3746 (1992).
[CrossRef] [PubMed]

Cookson, D. J.

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

de Jonge, M. D.

Dhal, B.

Eshtiaghi, N.

B. D. Arhatari, W. P. Gates, N. Eshtiaghi, and A. G. Peele, “Phase retrieval tomography in the presence of noise,” J. Appl. Phys. 107(034904), 1–7 (2010).
[CrossRef]

Friese, M. E.

Gao, D.

T. E. Gureyev, A. W. Stevenson, D. Paganin, S. C. Mayo, A. Pogany, D. Gao, and S. W. Wilkins, “Quantitative methods in phase-contrast x-ray imaging,” J. Digit. Imaging 13(S1), 121–126 (2000).
[CrossRef] [PubMed]

Gates, W. P.

B. D. Arhatari, W. P. Gates, N. Eshtiaghi, and A. G. Peele, “Phase retrieval tomography in the presence of noise,” J. Appl. Phys. 107(034904), 1–7 (2010).
[CrossRef]

Guigay, J. P.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[CrossRef]

Gureyev, T. E.

T. E. Gureyev, S. C. Mayo, D. E. Myers, Y. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: Propagation-based x-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105(10), 102005 (2009).
[CrossRef]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231(1-6), 53–70 (2004).
[CrossRef]

D. Paganin, S. C. 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. 206(1), 33–40 (2002).
[CrossRef] [PubMed]

T. E. Gureyev, A. W. Stevenson, D. Paganin, S. C. Mayo, A. Pogany, D. Gao, and S. W. Wilkins, “Quantitative methods in phase-contrast x-ray imaging,” J. Digit. Imaging 13(S1), 121–126 (2000).
[CrossRef] [PubMed]

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard X-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D Appl. Phys. 32(5), 563–567 (1999).
[CrossRef]

T. E. Gureyev and S. W. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A 15(3), 579–585 (1998).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Hannah, K.

Hanssen, E.

Hayes, J.

Heckenberg, N. R.

Jacobson, D. L.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
[CrossRef] [PubMed]

Janssen, G.

W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, “Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy,” Phys. Rev. Lett. 69(26), 3743–3746 (1992).
[CrossRef] [PubMed]

Ludwig, W.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[CrossRef]

Mancuso, A. P.

Mayo, S. C.

T. E. Gureyev, S. C. Mayo, D. E. Myers, Y. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: Propagation-based x-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105(10), 102005 (2009).
[CrossRef]

D. Paganin, S. C. 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. 206(1), 33–40 (2002).
[CrossRef] [PubMed]

T. E. Gureyev, A. W. Stevenson, D. Paganin, S. C. Mayo, A. Pogany, D. Gao, and S. W. Wilkins, “Quantitative methods in phase-contrast x-ray imaging,” J. Digit. Imaging 13(S1), 121–126 (2000).
[CrossRef] [PubMed]

McCartney, M.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83(1-2), 67–73 (2000).
[CrossRef] [PubMed]

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
[CrossRef] [PubMed]

Miller, P. R.

D. Paganin, S. C. 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. 206(1), 33–40 (2002).
[CrossRef] [PubMed]

Myers, D. E.

T. E. Gureyev, S. C. Mayo, D. E. Myers, Y. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: Propagation-based x-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105(10), 102005 (2009).
[CrossRef]

Nesterets, Y.

T. E. Gureyev, S. C. Mayo, D. E. Myers, Y. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: Propagation-based x-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105(10), 102005 (2009).
[CrossRef]

Nieminen, T. A.

Nugent, K.

Nugent, K. A.

J. N. Clark, G. J. Williams, H. M. Quiney, L. Whitehead, M. D. de Jonge, E. Hanssen, M. Altissimo, K. A. Nugent, and A. G. Peele, “Quantitative phase measurement in coherent diffraction imaging,” Opt. Express 16(5), 3342–3348 (2008).
[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(12), 5271–5276 (2004).
[CrossRef]

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
[CrossRef] [PubMed]

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83(1-2), 67–73 (2000).
[CrossRef] [PubMed]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

K. A. Nugent, “Partially coherent diffraction patterns and coherence measurement,” J. Opt. Soc. Am. A 8(10), 1574–1579 (1991).
[CrossRef]

Op de Beeck, M.

W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, “Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy,” Phys. Rev. Lett. 69(26), 3743–3746 (1992).
[CrossRef] [PubMed]

Paganin, D.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

D. Paganin, S. C. 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. 206(1), 33–40 (2002).
[CrossRef] [PubMed]

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
[CrossRef] [PubMed]

T. E. Gureyev, A. W. Stevenson, D. Paganin, S. C. Mayo, A. Pogany, D. Gao, and S. W. Wilkins, “Quantitative methods in phase-contrast x-ray imaging,” J. Digit. Imaging 13(S1), 121–126 (2000).
[CrossRef] [PubMed]

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83(1-2), 67–73 (2000).
[CrossRef] [PubMed]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Paganin, D. M.

T. E. Gureyev, S. C. Mayo, D. E. Myers, Y. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: Propagation-based x-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105(10), 102005 (2009).
[CrossRef]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231(1-6), 53–70 (2004).
[CrossRef]

Paterson, D.

Peele, A.

Peele, A. G.

B. D. Arhatari, W. P. Gates, N. Eshtiaghi, and A. G. Peele, “Phase retrieval tomography in the presence of noise,” J. Appl. Phys. 107(034904), 1–7 (2010).
[CrossRef]

J. N. Clark, G. J. Williams, H. M. Quiney, L. Whitehead, M. D. de Jonge, E. Hanssen, M. Altissimo, K. A. Nugent, and A. G. Peele, “Quantitative phase measurement in coherent diffraction imaging,” Opt. Express 16(5), 3342–3348 (2008).
[CrossRef] [PubMed]

B. D. Arhatari, K. Hannah, E. Balaur, and A. G. Peele, “Phase imaging using a polychromatic x-ray laboratory source,” Opt. Express 16(24), 19950–19956 (2008).
[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(12), 5271–5276 (2004).
[CrossRef]

Pogany, A.

T. E. Gureyev, S. C. Mayo, D. E. Myers, Y. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: Propagation-based x-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105(10), 102005 (2009).
[CrossRef]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231(1-6), 53–70 (2004).
[CrossRef]

T. E. Gureyev, A. W. Stevenson, D. Paganin, S. C. Mayo, A. Pogany, D. Gao, and S. W. Wilkins, “Quantitative methods in phase-contrast x-ray imaging,” J. Digit. Imaging 13(S1), 121–126 (2000).
[CrossRef] [PubMed]

Quiney, H. M.

Raven, C.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard X-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D Appl. Phys. 32(5), 563–567 (1999).
[CrossRef]

Rubinsztein-Dunlop, H.

Schlenker, M.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[CrossRef]

Scholten, R.

Snigirev, A.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard X-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D Appl. Phys. 32(5), 563–567 (1999).
[CrossRef]

Snigireva, I.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard X-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D Appl. Phys. 32(5), 563–567 (1999).
[CrossRef]

Stevenson, A. W.

T. E. Gureyev, S. C. Mayo, D. E. Myers, Y. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: Propagation-based x-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105(10), 102005 (2009).
[CrossRef]

T. E. Gureyev, A. W. Stevenson, D. Paganin, S. C. Mayo, A. Pogany, D. Gao, and S. W. Wilkins, “Quantitative methods in phase-contrast x-ray imaging,” J. Digit. Imaging 13(S1), 121–126 (2000).
[CrossRef] [PubMed]

Teague, M. R.

Tikhonov, A. N.

A. N. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).

Tran, C.

Turner, L. D.

Van Dyck, D.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[CrossRef]

W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, “Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy,” Phys. Rev. Lett. 69(26), 3743–3746 (1992).
[CrossRef] [PubMed]

Van Landuyt, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[CrossRef]

Wall, M.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83(1-2), 67–73 (2000).
[CrossRef] [PubMed]

Werner, S. A.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
[CrossRef] [PubMed]

Whitehead, L.

Wilkins, S. W.

T. E. Gureyev, S. C. Mayo, D. E. Myers, Y. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: Propagation-based x-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105(10), 102005 (2009).
[CrossRef]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231(1-6), 53–70 (2004).
[CrossRef]

D. Paganin, S. C. 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. 206(1), 33–40 (2002).
[CrossRef] [PubMed]

T. E. Gureyev, A. W. Stevenson, D. Paganin, S. C. Mayo, A. Pogany, D. Gao, and S. W. Wilkins, “Quantitative methods in phase-contrast x-ray imaging,” J. Digit. Imaging 13(S1), 121–126 (2000).
[CrossRef] [PubMed]

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard X-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D Appl. Phys. 32(5), 563–567 (1999).
[CrossRef]

T. E. Gureyev and S. W. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A 15(3), 579–585 (1998).
[CrossRef]

Williams, G. J.

Appl. Phys. Lett. (1)

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[CrossRef]

J. Appl. Phys. (2)

B. D. Arhatari, W. P. Gates, N. Eshtiaghi, and A. G. Peele, “Phase retrieval tomography in the presence of noise,” J. Appl. Phys. 107(034904), 1–7 (2010).
[CrossRef]

T. E. Gureyev, S. C. Mayo, D. E. Myers, Y. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: Propagation-based x-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105(10), 102005 (2009).
[CrossRef]

J. Digit. Imaging (1)

T. E. Gureyev, A. W. Stevenson, D. Paganin, S. C. Mayo, A. Pogany, D. Gao, and S. W. Wilkins, “Quantitative methods in phase-contrast x-ray imaging,” J. Digit. Imaging 13(S1), 121–126 (2000).
[CrossRef] [PubMed]

J. Microsc. (2)

D. Paganin, S. C. 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. 206(1), 33–40 (2002).
[CrossRef] [PubMed]

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Phys. D Appl. Phys. (1)

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard X-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D Appl. Phys. 32(5), 563–567 (1999).
[CrossRef]

Nature (1)

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
[CrossRef] [PubMed]

Opt. Commun. (1)

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231(1-6), 53–70 (2004).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. Lett. (3)

W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, “Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy,” Phys. Rev. Lett. 69(26), 3743–3746 (1992).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[CrossRef]

Rev. Sci. Instrum. (1)

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75(12), 5271–5276 (2004).
[CrossRef]

Sov. Math. Dokl. (1)

A. N. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).

Ultramicroscopy (1)

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83(1-2), 67–73 (2000).
[CrossRef] [PubMed]

Other (2)

D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).

D. Paganin, “Studies in phase retrieval,” PhD thesis (University of Melbourne, 1999).

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Figures (12)

Fig. 1
Fig. 1

The set up of the system used in this study: z12 is the source to sample distance, z23 is the sample to detector1 distance, z24 is the sample to detector2 distance and z34 is the inter plane separation between the two detector positions. Nominal images recorded at the detector positions are shown in the insets.

Fig. 2
Fig. 2

Simulation result for regularization parameter evaluation for different noise levels for the set up of z12 = 0.1m, z23 = 0.02m, z24 = 0.54m.

Fig. 3
Fig. 3

Simulation result for (a) fixed z12 = 0.12 m and z24 = 0.54 m with varied distance z23 . (b) fixed z12 = 0.12 m and z23 = 0.02 m with varied distance z24 . (c) fixed z12 = 0.12 m and z34 = 0.25 m with co-varied distance z23 and z24 . Note the result for 0% noise is multiplied by 10x to make it visible.

Fig. 4
Fig. 4

Simulation results with varied source to sample distance z12, for (a) fixed z23 = 0.02 m and z24 = 0.125 m, (b) fixed z23 = 0.05 m and z24 = 0.125 m, (c) fixed z23 = 0.02 m and z24 = 0.54 m (d) fixed z23 = 0.3 m and z24 = 0.54 m.

Fig. 5
Fig. 5

Plot of z34eff as a function of source sample distance variation, z12 , for the set-ups in Fig. 4.

Fig. 6
Fig. 6

System resolution for (a) small geometric magnification (b). large geometric magnification. Nominal images recorded by the detector are shown in the insets.

Fig. 7
Fig. 7

χ2 values are plotted as a function of object periods for various set-ups at 0% noise (left column) and 0.4% noise (middle column). The corresponding scaled Gaussian sources are plotted in the right column.

Fig. 8
Fig. 8

(a). The spectral response function of the system taken at 40kV tube voltage. (b). Retrieved δpolyt vs its real value for kapton sinusoidal sample.

Fig. 9
Fig. 9

(a). intensity image at z23 = 0.02m, (b). intensity image at z24 = 0.125m, (c). Phase retrieved δpolyt.

Fig. 10
Fig. 10

Comparison between the retrieved δpolyt from experimental data and the real value using δpoly of 2.36x10−6 .

Fig. 11
Fig. 11

Simulation vs experimental result for varied distances as explained in the text.

Fig. 12
Fig. 12

(a). A reconstruction slice (b). Three dimensional view of phase retrieval tomography reconstruction of polystyrene spheres (45 μm diameter) mixing with plastic fibre from the data set taken at z12 = 0.1 m, z23 = 0.02 m, z24 = 0.125 m.

Equations (8)

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I ( r ) z + 1 k . ( I ( r ) φ ( r ) ) = 0
φ ( r ) = k 2 ( . { 1 I ( r ) [ 2 I ( r ) z ] } )
z 23 e f f = z 12 z 23 z 12 + z 23 , z 24 e f f = z 12 z 24 z 12 + z 24
z 34 e f f = z 24 e f f z 23 e f f = z 12 z 24 z 12 + z 24 z 12 z 23 z 12 + z 23 = z 12 2 z 34 ( z 12 + z 24 ) ( z 12 + z 23 )
1 u u u 2 + α 2
χ 2 = i ( φ T I E i φ o b j i ) 2 i ( φ o b j i ) 2
δ λ t ( r ) = 2 ( . { 1 I λ ( r ) [ 2 I λ ( r ) z ] } )
δ p o l y t ( r ) = δ λ t ( r ) S ( λ ) d λ S ( λ ) d λ = 2 ( . { 1 I λ ( r ) [ 2 I λ ( r ) z ] } ) S ( λ ) d λ S ( λ ) d λ

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