Abstract

Edge illumination (EI) has emerged as an X-ray phase-contrast imaging (XPCi) modality which could present significant advantages in terms of translation to clinical and laboratory applications. In this paper, we model its signal through the use of the “transport of intensity” equation. The validity conditions for this approach and its relationship with previous theoretical models for EI XPCi are discussed. The proposed model enables a simple estimation of the different contributions to the signal, which is shown to complement previously obtained results. In particular, it allows taking into account the effect of both slowly and rapidly varying refraction angles, corresponding to large and small object features. The derived framework is then used to investigate the effect on the signal of the smoothness of the mask edges, of the blurring from the source size and of the width of the object edge.

© 2014 Optical Society of America

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Spatial resolution of edge illumination X-ray phase-contrast imaging

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Theory and preliminary experimental verification of quantitative edge illumination x-ray phase contrast tomography

C. K. Hagen, P. C. Diemoz, M. Endrizzi, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, and A. Olivo
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  5. 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(11), 2015–2025 (1997).
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [PubMed]
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    [Crossref] [PubMed]
  25. 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), 51–70 (2004).
  26. A. Peterzol, A. Olivo, L. Rigon, S. Pani, and D. Dreossi, “The effects of the imaging system on the validity limits of the ray-optical approach to phase contrast imaging,” Med. Phys. 32(12), 3617–3627 (2005).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  30. P. C. Diemoz, F. A. Vittoria, and A. Olivo, “Spatial resolution of edge illumination X-ray phase-contrast imaging,” Opt. Express 22(13), 15514–15529 (2014).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  36. F. A. Vittoria, M. Endrizzi, P. C. Diemoz, U. H. Wagner, C. Rau, I. K. Robinson, and A. Olivo, “Virtual edge illumination and one dimensional beam tracking for absorption, refraction, and scattering retrieval,” Appl. Phys. Lett. 104(13), 134102 (2014).
    [Crossref]

2014 (4)

S. W. Wilkins, Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. Pogany, and A. W. Stevenson, “On the evolution and relative merits of hard X-ray phase-contrast imaging methods,” Philos Trans A Math Phys. Eng. Sci. 372(2010), 20130021 (2014).
[Crossref] [PubMed]

M. Endrizzi, P. C. Diemoz, T. P. Millard, L. J. Jones, R. D. Speller, I. K. Robinson, and A. Olivo, “Hard X-ray dark-field imaging with incoherent sample illumination,” Appl. Phys. Lett. 104(2), 024106 (2014).
[Crossref]

P. C. Diemoz, F. A. Vittoria, and A. Olivo, “Spatial resolution of edge illumination X-ray phase-contrast imaging,” Opt. Express 22(13), 15514–15529 (2014).
[Crossref] [PubMed]

F. A. Vittoria, M. Endrizzi, P. C. Diemoz, U. H. Wagner, C. Rau, I. K. Robinson, and A. Olivo, “Virtual edge illumination and one dimensional beam tracking for absorption, refraction, and scattering retrieval,” Appl. Phys. Lett. 104(13), 134102 (2014).
[Crossref]

2013 (8)

P. R. T. Munro and A. Olivo, “X-ray phase-contrast imaging with polychromatic sources and the concept of effective energy,” Phys. Rev. A 87(5), 053838 (2013).
[Crossref]

P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pešić, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-ray phase-contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110(13), 138105 (2013).
[Crossref] [PubMed]

P. R. T. Munro, C. K. Hagen, M. B. Szafraniec, and A. Olivo, “A simplified approach to quantitative coded aperture X-ray phase imaging,” Opt. Express 21(9), 11187–11201 (2013).
[Crossref] [PubMed]

P. R. T. Munro, L. Rigon, K. Ignatyev, F. C. M. Lopez, D. Dreossi, R. D. Speller, and A. Olivo, “A quantitative, non-interferometric X-ray phase contrast imaging technique,” Opt. Express 21(1), 647–661 (2013).
[Crossref] [PubMed]

T. P. Millard, M. Endrizzi, K. Ignatyev, C. K. Hagen, P. R. T. Munro, R. D. Speller, and A. Olivo, “Method for automatization of the alignment of a laboratory based x-ray phase contrast edge illumination system,” Rev. Sci. Instrum. 84(8), 083702 (2013).
[Crossref] [PubMed]

F. A. Vittoria, P. C. Diemoz, M. Endrizzi, L. Rigon, F. C. Lopez, D. Dreossi, P. R. T. Munro, and A. Olivo, “Strategies for efficient and fast wave optics simulation of coded-aperture and other x-ray phase-contrast imaging methods,” Appl. Opt. 52(28), 6940–6947 (2013).
[PubMed]

A. Olivo, S. Gkoumas, M. Endrizzi, C. K. Hagen, M. B. Szafraniec, P. C. Diemoz, P. R. T. Munro, K. Ignatyev, B. Johnson, J. A. Horrocks, S. J. Vinnicombe, J. L. Jones, and R. D. Speller, “Low-dose phase contrast mammography with conventional x-ray sources,” Med. Phys. 40(9), 090701 (2013).
[Crossref] [PubMed]

P. C. Diemoz, C. K. Hagen, M. Endrizzi, and A. Olivo, “Sensitivity of laboratory based implementations of edge illumination X-ray phase-contrast imaging,” Appl. Phys. Lett. 103(24), 244104 (2013).
[Crossref]

2012 (2)

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Phase and absorption retrieval using incoherent X-ray sources,” Proc. Natl. Acad. Sci. U.S.A. 109(35), 13922–13927 (2012).
[Crossref] [PubMed]

M. Marenzana, C. K. Hagen, P. Das Neves Borges, M. Endrizzi, M. B. Szafraniec, K. Ignatyev, and A. Olivo, “Visualization of small lesions in rat cartilage by means of laboratory-based x-ray phase contrast imaging,” Phys. Med. Biol. 57(24), 8173–8184 (2012).
[Crossref] [PubMed]

2011 (2)

K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “Quantitative single-exposure x-ray phase contrast imaging using a single attenuation grid,” Opt. Express 19(20), 19781–19789 (2011).
[Crossref] [PubMed]

K. Ignatyev, P. R. T. Munro, R. D. Speller, and A. Olivo, “Effects of signal diffusion on x-ray phase contrast images,” Rev. Sci. Instrum. 82(7), 073702 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (1)

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation X- ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. 94(4), 044108 (2009).
[Crossref]

2008 (2)

2007 (2)

A. Olivo and R. D. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. 52(22), 6555–6573 (2007).
[Crossref] [PubMed]

A. Olivo and R. D. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91(7), 074106 (2007).
[Crossref]

2006 (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[Crossref]

2005 (1)

A. Peterzol, A. Olivo, L. Rigon, S. Pani, and D. Dreossi, “The effects of the imaging system on the validity limits of the ray-optical approach to phase contrast imaging,” Med. Phys. 32(12), 3617–3627 (2005).
[Crossref] [PubMed]

2004 (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), 51–70 (2004).

2003 (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), L866–L868 (2003).
[Crossref]

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]

2001 (1)

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28(8), 1610–1619 (2001).
[Crossref] [PubMed]

1998 (1)

1997 (2)

A. Pogany, D. Gao, and S. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997).
[Crossref]

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(11), 2015–2025 (1997).
[Crossref] [PubMed]

1996 (2)

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[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(14), 2961–2964 (1996).
[Crossref] [PubMed]

1995 (2)

T. 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 373(6515), 595–598 (1995).
[Crossref]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibility of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

1983 (1)

Arfelli, F.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28(8), 1610–1619 (2001).
[Crossref] [PubMed]

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(11), 2015–2025 (1997).
[Crossref] [PubMed]

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(14), 2961–2964 (1996).
[Crossref] [PubMed]

Bohndiek, S. E.

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation X- ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. 94(4), 044108 (2009).
[Crossref]

Bravin, A.

P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pešić, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-ray phase-contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110(13), 138105 (2013).
[Crossref] [PubMed]

Bunk, O.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[Crossref]

Cantatore, G.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28(8), 1610–1619 (2001).
[Crossref] [PubMed]

Castelli, E.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28(8), 1610–1619 (2001).
[Crossref] [PubMed]

Chapman, D.

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(11), 2015–2025 (1997).
[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(14), 2961–2964 (1996).
[Crossref] [PubMed]

Das Neves Borges, P.

M. Marenzana, C. K. Hagen, P. Das Neves Borges, M. Endrizzi, M. B. Szafraniec, K. Ignatyev, and A. Olivo, “Visualization of small lesions in rat cartilage by means of laboratory-based x-ray phase contrast imaging,” Phys. Med. Biol. 57(24), 8173–8184 (2012).
[Crossref] [PubMed]

David, C.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[Crossref]

Davis, T.

T. 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 373(6515), 595–598 (1995).
[Crossref]

Diemoz, P. C.

M. Endrizzi, P. C. Diemoz, T. P. Millard, L. J. Jones, R. D. Speller, I. K. Robinson, and A. Olivo, “Hard X-ray dark-field imaging with incoherent sample illumination,” Appl. Phys. Lett. 104(2), 024106 (2014).
[Crossref]

F. A. Vittoria, M. Endrizzi, P. C. Diemoz, U. H. Wagner, C. Rau, I. K. Robinson, and A. Olivo, “Virtual edge illumination and one dimensional beam tracking for absorption, refraction, and scattering retrieval,” Appl. Phys. Lett. 104(13), 134102 (2014).
[Crossref]

P. C. Diemoz, F. A. Vittoria, and A. Olivo, “Spatial resolution of edge illumination X-ray phase-contrast imaging,” Opt. Express 22(13), 15514–15529 (2014).
[Crossref] [PubMed]

F. A. Vittoria, P. C. Diemoz, M. Endrizzi, L. Rigon, F. C. Lopez, D. Dreossi, P. R. T. Munro, and A. Olivo, “Strategies for efficient and fast wave optics simulation of coded-aperture and other x-ray phase-contrast imaging methods,” Appl. Opt. 52(28), 6940–6947 (2013).
[PubMed]

P. C. Diemoz, C. K. Hagen, M. Endrizzi, and A. Olivo, “Sensitivity of laboratory based implementations of edge illumination X-ray phase-contrast imaging,” Appl. Phys. Lett. 103(24), 244104 (2013).
[Crossref]

P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pešić, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-ray phase-contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110(13), 138105 (2013).
[Crossref] [PubMed]

A. Olivo, S. Gkoumas, M. Endrizzi, C. K. Hagen, M. B. Szafraniec, P. C. Diemoz, P. R. T. Munro, K. Ignatyev, B. Johnson, J. A. Horrocks, S. J. Vinnicombe, J. L. Jones, and R. D. Speller, “Low-dose phase contrast mammography with conventional x-ray sources,” Med. Phys. 40(9), 090701 (2013).
[Crossref] [PubMed]

Dreossi, D.

Endrizzi, M.

M. Endrizzi, P. C. Diemoz, T. P. Millard, L. J. Jones, R. D. Speller, I. K. Robinson, and A. Olivo, “Hard X-ray dark-field imaging with incoherent sample illumination,” Appl. Phys. Lett. 104(2), 024106 (2014).
[Crossref]

F. A. Vittoria, M. Endrizzi, P. C. Diemoz, U. H. Wagner, C. Rau, I. K. Robinson, and A. Olivo, “Virtual edge illumination and one dimensional beam tracking for absorption, refraction, and scattering retrieval,” Appl. Phys. Lett. 104(13), 134102 (2014).
[Crossref]

F. A. Vittoria, P. C. Diemoz, M. Endrizzi, L. Rigon, F. C. Lopez, D. Dreossi, P. R. T. Munro, and A. Olivo, “Strategies for efficient and fast wave optics simulation of coded-aperture and other x-ray phase-contrast imaging methods,” Appl. Opt. 52(28), 6940–6947 (2013).
[PubMed]

P. C. Diemoz, C. K. Hagen, M. Endrizzi, and A. Olivo, “Sensitivity of laboratory based implementations of edge illumination X-ray phase-contrast imaging,” Appl. Phys. Lett. 103(24), 244104 (2013).
[Crossref]

T. P. Millard, M. Endrizzi, K. Ignatyev, C. K. Hagen, P. R. T. Munro, R. D. Speller, and A. Olivo, “Method for automatization of the alignment of a laboratory based x-ray phase contrast edge illumination system,” Rev. Sci. Instrum. 84(8), 083702 (2013).
[Crossref] [PubMed]

P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pešić, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-ray phase-contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110(13), 138105 (2013).
[Crossref] [PubMed]

A. Olivo, S. Gkoumas, M. Endrizzi, C. K. Hagen, M. B. Szafraniec, P. C. Diemoz, P. R. T. Munro, K. Ignatyev, B. Johnson, J. A. Horrocks, S. J. Vinnicombe, J. L. Jones, and R. D. Speller, “Low-dose phase contrast mammography with conventional x-ray sources,” Med. Phys. 40(9), 090701 (2013).
[Crossref] [PubMed]

M. Marenzana, C. K. Hagen, P. Das Neves Borges, M. Endrizzi, M. B. Szafraniec, K. Ignatyev, and A. Olivo, “Visualization of small lesions in rat cartilage by means of laboratory-based x-ray phase contrast imaging,” Phys. Med. Biol. 57(24), 8173–8184 (2012).
[Crossref] [PubMed]

Gao, D.

A. Pogany, D. Gao, and S. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997).
[Crossref]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

T. 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 373(6515), 595–598 (1995).
[Crossref]

Gkoumas, S.

A. Olivo, S. Gkoumas, M. Endrizzi, C. K. Hagen, M. B. Szafraniec, P. C. Diemoz, P. R. T. Munro, K. Ignatyev, B. Johnson, J. A. Horrocks, S. J. Vinnicombe, J. L. Jones, and R. D. Speller, “Low-dose phase contrast mammography with conventional x-ray sources,” Med. Phys. 40(9), 090701 (2013).
[Crossref] [PubMed]

Gmür, N.

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(11), 2015–2025 (1997).
[Crossref] [PubMed]

Griffiths, J. A.

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation X- ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. 94(4), 044108 (2009).
[Crossref]

Gureyev, T. E.

S. W. Wilkins, Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. Pogany, and A. W. Stevenson, “On the evolution and relative merits of hard X-ray phase-contrast imaging methods,” Philos Trans A Math Phys. Eng. Sci. 372(2010), 20130021 (2014).
[Crossref] [PubMed]

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008).
[Crossref] [PubMed]

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), 51–70 (2004).

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 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. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

T. 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 373(6515), 595–598 (1995).
[Crossref]

Hagen, C. K.

A. Olivo, S. Gkoumas, M. Endrizzi, C. K. Hagen, M. B. Szafraniec, P. C. Diemoz, P. R. T. Munro, K. Ignatyev, B. Johnson, J. A. Horrocks, S. J. Vinnicombe, J. L. Jones, and R. D. Speller, “Low-dose phase contrast mammography with conventional x-ray sources,” Med. Phys. 40(9), 090701 (2013).
[Crossref] [PubMed]

T. P. Millard, M. Endrizzi, K. Ignatyev, C. K. Hagen, P. R. T. Munro, R. D. Speller, and A. Olivo, “Method for automatization of the alignment of a laboratory based x-ray phase contrast edge illumination system,” Rev. Sci. Instrum. 84(8), 083702 (2013).
[Crossref] [PubMed]

P. C. Diemoz, C. K. Hagen, M. Endrizzi, and A. Olivo, “Sensitivity of laboratory based implementations of edge illumination X-ray phase-contrast imaging,” Appl. Phys. Lett. 103(24), 244104 (2013).
[Crossref]

P. R. T. Munro, C. K. Hagen, M. B. Szafraniec, and A. Olivo, “A simplified approach to quantitative coded aperture X-ray phase imaging,” Opt. Express 21(9), 11187–11201 (2013).
[Crossref] [PubMed]

M. Marenzana, C. K. Hagen, P. Das Neves Borges, M. Endrizzi, M. B. Szafraniec, K. Ignatyev, and A. Olivo, “Visualization of small lesions in rat cartilage by means of laboratory-based x-ray phase contrast imaging,” Phys. Med. Biol. 57(24), 8173–8184 (2012).
[Crossref] [PubMed]

Hamaishi, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), L866–L868 (2003).
[Crossref]

Horrocks, J. A.

A. Olivo, S. Gkoumas, M. Endrizzi, C. K. Hagen, M. B. Szafraniec, P. C. Diemoz, P. R. T. Munro, K. Ignatyev, B. Johnson, J. A. Horrocks, S. J. Vinnicombe, J. L. Jones, and R. D. Speller, “Low-dose phase contrast mammography with conventional x-ray sources,” Med. Phys. 40(9), 090701 (2013).
[Crossref] [PubMed]

Ignatyev, K.

A. Olivo, S. Gkoumas, M. Endrizzi, C. K. Hagen, M. B. Szafraniec, P. C. Diemoz, P. R. T. Munro, K. Ignatyev, B. Johnson, J. A. Horrocks, S. J. Vinnicombe, J. L. Jones, and R. D. Speller, “Low-dose phase contrast mammography with conventional x-ray sources,” Med. Phys. 40(9), 090701 (2013).
[Crossref] [PubMed]

T. P. Millard, M. Endrizzi, K. Ignatyev, C. K. Hagen, P. R. T. Munro, R. D. Speller, and A. Olivo, “Method for automatization of the alignment of a laboratory based x-ray phase contrast edge illumination system,” Rev. Sci. Instrum. 84(8), 083702 (2013).
[Crossref] [PubMed]

P. R. T. Munro, L. Rigon, K. Ignatyev, F. C. M. Lopez, D. Dreossi, R. D. Speller, and A. Olivo, “A quantitative, non-interferometric X-ray phase contrast imaging technique,” Opt. Express 21(1), 647–661 (2013).
[Crossref] [PubMed]

M. Marenzana, C. K. Hagen, P. Das Neves Borges, M. Endrizzi, M. B. Szafraniec, K. Ignatyev, and A. Olivo, “Visualization of small lesions in rat cartilage by means of laboratory-based x-ray phase contrast imaging,” Phys. Med. Biol. 57(24), 8173–8184 (2012).
[Crossref] [PubMed]

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Phase and absorption retrieval using incoherent X-ray sources,” Proc. Natl. Acad. Sci. U.S.A. 109(35), 13922–13927 (2012).
[Crossref] [PubMed]

K. Ignatyev, P. R. T. Munro, R. D. Speller, and A. Olivo, “Effects of signal diffusion on x-ray phase contrast images,” Rev. Sci. Instrum. 82(7), 073702 (2011).
[Crossref] [PubMed]

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “The relationship between wave and geometrical optics models of coded aperture type x-ray phase contrast imaging systems,” Opt. Express 18(5), 4103–4117 (2010).
[Crossref] [PubMed]

Johnson, B.

A. Olivo, S. Gkoumas, M. Endrizzi, C. K. Hagen, M. B. Szafraniec, P. C. Diemoz, P. R. T. Munro, K. Ignatyev, B. Johnson, J. A. Horrocks, S. J. Vinnicombe, J. L. Jones, and R. D. Speller, “Low-dose phase contrast mammography with conventional x-ray sources,” Med. Phys. 40(9), 090701 (2013).
[Crossref] [PubMed]

Johnston, R. E.

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(11), 2015–2025 (1997).
[Crossref] [PubMed]

Jones, J. L.

A. Olivo, S. Gkoumas, M. Endrizzi, C. K. Hagen, M. B. Szafraniec, P. C. Diemoz, P. R. T. Munro, K. Ignatyev, B. Johnson, J. A. Horrocks, S. J. Vinnicombe, J. L. Jones, and R. D. Speller, “Low-dose phase contrast mammography with conventional x-ray sources,” Med. Phys. 40(9), 090701 (2013).
[Crossref] [PubMed]

Jones, L. J.

M. Endrizzi, P. C. Diemoz, T. P. Millard, L. J. Jones, R. D. Speller, I. K. Robinson, and A. Olivo, “Hard X-ray dark-field imaging with incoherent sample illumination,” Appl. Phys. Lett. 104(2), 024106 (2014).
[Crossref]

Kawamoto, S.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), L866–L868 (2003).
[Crossref]

Kohn, V.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibility of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

Konstantinidis, A.

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation X- ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. 94(4), 044108 (2009).
[Crossref]

Koyama, I.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), L866–L868 (2003).
[Crossref]

Kuznetsov, S.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibility of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

Longo, R.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28(8), 1610–1619 (2001).
[Crossref] [PubMed]

Lopez, F. C.

Lopez, F. C. M.

Marenzana, M.

M. Marenzana, C. K. Hagen, P. Das Neves Borges, M. Endrizzi, M. B. Szafraniec, K. Ignatyev, and A. Olivo, “Visualization of small lesions in rat cartilage by means of laboratory-based x-ray phase contrast imaging,” Phys. Med. Biol. 57(24), 8173–8184 (2012).
[Crossref] [PubMed]

Mayo, S. C.

S. W. Wilkins, Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. Pogany, and A. W. Stevenson, “On the evolution and relative merits of hard X-ray phase-contrast imaging methods,” Philos Trans A Math Phys. Eng. Sci. 372(2010), 20130021 (2014).
[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]

Menk, R.

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(11), 2015–2025 (1997).
[Crossref] [PubMed]

Menk, R. H.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28(8), 1610–1619 (2001).
[Crossref] [PubMed]

Millard, T. P.

M. Endrizzi, P. C. Diemoz, T. P. Millard, L. J. Jones, R. D. Speller, I. K. Robinson, and A. Olivo, “Hard X-ray dark-field imaging with incoherent sample illumination,” Appl. Phys. Lett. 104(2), 024106 (2014).
[Crossref]

T. P. Millard, M. Endrizzi, K. Ignatyev, C. K. Hagen, P. R. T. Munro, R. D. Speller, and A. Olivo, “Method for automatization of the alignment of a laboratory based x-ray phase contrast edge illumination system,” Rev. Sci. Instrum. 84(8), 083702 (2013).
[Crossref] [PubMed]

Miller, P. R.

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008).
[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]

Momose, A.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), L866–L868 (2003).
[Crossref]

Morgan, K. S.

Munro, P. R. T.

P. R. T. Munro, L. Rigon, K. Ignatyev, F. C. M. Lopez, D. Dreossi, R. D. Speller, and A. Olivo, “A quantitative, non-interferometric X-ray phase contrast imaging technique,” Opt. Express 21(1), 647–661 (2013).
[Crossref] [PubMed]

P. R. T. Munro, C. K. Hagen, M. B. Szafraniec, and A. Olivo, “A simplified approach to quantitative coded aperture X-ray phase imaging,” Opt. Express 21(9), 11187–11201 (2013).
[Crossref] [PubMed]

F. A. Vittoria, P. C. Diemoz, M. Endrizzi, L. Rigon, F. C. Lopez, D. Dreossi, P. R. T. Munro, and A. Olivo, “Strategies for efficient and fast wave optics simulation of coded-aperture and other x-ray phase-contrast imaging methods,” Appl. Opt. 52(28), 6940–6947 (2013).
[PubMed]

A. Olivo, S. Gkoumas, M. Endrizzi, C. K. Hagen, M. B. Szafraniec, P. C. Diemoz, P. R. T. Munro, K. Ignatyev, B. Johnson, J. A. Horrocks, S. J. Vinnicombe, J. L. Jones, and R. D. Speller, “Low-dose phase contrast mammography with conventional x-ray sources,” Med. Phys. 40(9), 090701 (2013).
[Crossref] [PubMed]

T. P. Millard, M. Endrizzi, K. Ignatyev, C. K. Hagen, P. R. T. Munro, R. D. Speller, and A. Olivo, “Method for automatization of the alignment of a laboratory based x-ray phase contrast edge illumination system,” Rev. Sci. Instrum. 84(8), 083702 (2013).
[Crossref] [PubMed]

P. R. T. Munro and A. Olivo, “X-ray phase-contrast imaging with polychromatic sources and the concept of effective energy,” Phys. Rev. A 87(5), 053838 (2013).
[Crossref]

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Phase and absorption retrieval using incoherent X-ray sources,” Proc. Natl. Acad. Sci. U.S.A. 109(35), 13922–13927 (2012).
[Crossref] [PubMed]

K. Ignatyev, P. R. T. Munro, R. D. Speller, and A. Olivo, “Effects of signal diffusion on x-ray phase contrast images,” Rev. Sci. Instrum. 82(7), 073702 (2011).
[Crossref] [PubMed]

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “The relationship between wave and geometrical optics models of coded aperture type x-ray phase contrast imaging systems,” Opt. Express 18(5), 4103–4117 (2010).
[Crossref] [PubMed]

Nesterets, Y. I.

S. W. Wilkins, Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. Pogany, and A. W. Stevenson, “On the evolution and relative merits of hard X-ray phase-contrast imaging methods,” Philos Trans A Math Phys. Eng. Sci. 372(2010), 20130021 (2014).
[Crossref] [PubMed]

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008).
[Crossref] [PubMed]

Nugent, K. A.

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(14), 2961–2964 (1996).
[Crossref] [PubMed]

Olivo, A.

M. Endrizzi, P. C. Diemoz, T. P. Millard, L. J. Jones, R. D. Speller, I. K. Robinson, and A. Olivo, “Hard X-ray dark-field imaging with incoherent sample illumination,” Appl. Phys. Lett. 104(2), 024106 (2014).
[Crossref]

F. A. Vittoria, M. Endrizzi, P. C. Diemoz, U. H. Wagner, C. Rau, I. K. Robinson, and A. Olivo, “Virtual edge illumination and one dimensional beam tracking for absorption, refraction, and scattering retrieval,” Appl. Phys. Lett. 104(13), 134102 (2014).
[Crossref]

P. C. Diemoz, F. A. Vittoria, and A. Olivo, “Spatial resolution of edge illumination X-ray phase-contrast imaging,” Opt. Express 22(13), 15514–15529 (2014).
[Crossref] [PubMed]

F. A. Vittoria, P. C. Diemoz, M. Endrizzi, L. Rigon, F. C. Lopez, D. Dreossi, P. R. T. Munro, and A. Olivo, “Strategies for efficient and fast wave optics simulation of coded-aperture and other x-ray phase-contrast imaging methods,” Appl. Opt. 52(28), 6940–6947 (2013).
[PubMed]

P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pešić, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-ray phase-contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110(13), 138105 (2013).
[Crossref] [PubMed]

P. R. T. Munro, C. K. Hagen, M. B. Szafraniec, and A. Olivo, “A simplified approach to quantitative coded aperture X-ray phase imaging,” Opt. Express 21(9), 11187–11201 (2013).
[Crossref] [PubMed]

P. R. T. Munro, L. Rigon, K. Ignatyev, F. C. M. Lopez, D. Dreossi, R. D. Speller, and A. Olivo, “A quantitative, non-interferometric X-ray phase contrast imaging technique,” Opt. Express 21(1), 647–661 (2013).
[Crossref] [PubMed]

T. P. Millard, M. Endrizzi, K. Ignatyev, C. K. Hagen, P. R. T. Munro, R. D. Speller, and A. Olivo, “Method for automatization of the alignment of a laboratory based x-ray phase contrast edge illumination system,” Rev. Sci. Instrum. 84(8), 083702 (2013).
[Crossref] [PubMed]

P. C. Diemoz, C. K. Hagen, M. Endrizzi, and A. Olivo, “Sensitivity of laboratory based implementations of edge illumination X-ray phase-contrast imaging,” Appl. Phys. Lett. 103(24), 244104 (2013).
[Crossref]

P. R. T. Munro and A. Olivo, “X-ray phase-contrast imaging with polychromatic sources and the concept of effective energy,” Phys. Rev. A 87(5), 053838 (2013).
[Crossref]

A. Olivo, S. Gkoumas, M. Endrizzi, C. K. Hagen, M. B. Szafraniec, P. C. Diemoz, P. R. T. Munro, K. Ignatyev, B. Johnson, J. A. Horrocks, S. J. Vinnicombe, J. L. Jones, and R. D. Speller, “Low-dose phase contrast mammography with conventional x-ray sources,” Med. Phys. 40(9), 090701 (2013).
[Crossref] [PubMed]

M. Marenzana, C. K. Hagen, P. Das Neves Borges, M. Endrizzi, M. B. Szafraniec, K. Ignatyev, and A. Olivo, “Visualization of small lesions in rat cartilage by means of laboratory-based x-ray phase contrast imaging,” Phys. Med. Biol. 57(24), 8173–8184 (2012).
[Crossref] [PubMed]

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Phase and absorption retrieval using incoherent X-ray sources,” Proc. Natl. Acad. Sci. U.S.A. 109(35), 13922–13927 (2012).
[Crossref] [PubMed]

K. Ignatyev, P. R. T. Munro, R. D. Speller, and A. Olivo, “Effects of signal diffusion on x-ray phase contrast images,” Rev. Sci. Instrum. 82(7), 073702 (2011).
[Crossref] [PubMed]

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “The relationship between wave and geometrical optics models of coded aperture type x-ray phase contrast imaging systems,” Opt. Express 18(5), 4103–4117 (2010).
[Crossref] [PubMed]

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation X- ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. 94(4), 044108 (2009).
[Crossref]

A. Olivo and R. D. Speller, “Image formation principles in coded-aperture based x-ray phase contrast imaging,” Phys. Med. Biol. 53(22), 6461–6474 (2008).
[Crossref] [PubMed]

A. Olivo and R. D. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. 52(22), 6555–6573 (2007).
[Crossref] [PubMed]

A. Olivo and R. D. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91(7), 074106 (2007).
[Crossref]

A. Peterzol, A. Olivo, L. Rigon, S. Pani, and D. Dreossi, “The effects of the imaging system on the validity limits of the ray-optical approach to phase contrast imaging,” Med. Phys. 32(12), 3617–3627 (2005).
[Crossref] [PubMed]

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28(8), 1610–1619 (2001).
[Crossref] [PubMed]

Paganin, D.

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]

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(14), 2961–2964 (1996).
[Crossref] [PubMed]

Paganin, D. M.

K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “Quantitative single-exposure x-ray phase contrast imaging using a single attenuation grid,” Opt. Express 19(20), 19781–19789 (2011).
[Crossref] [PubMed]

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), 51–70 (2004).

Pani, S.

A. Peterzol, A. Olivo, L. Rigon, S. Pani, and D. Dreossi, “The effects of the imaging system on the validity limits of the ray-optical approach to phase contrast imaging,” Med. Phys. 32(12), 3617–3627 (2005).
[Crossref] [PubMed]

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28(8), 1610–1619 (2001).
[Crossref] [PubMed]

Pešic, Z. D.

P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pešić, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-ray phase-contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110(13), 138105 (2013).
[Crossref] [PubMed]

Peterzol, A.

A. Peterzol, A. Olivo, L. Rigon, S. Pani, and D. Dreossi, “The effects of the imaging system on the validity limits of the ray-optical approach to phase contrast imaging,” Med. Phys. 32(12), 3617–3627 (2005).
[Crossref] [PubMed]

Pfeiffer, F.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[Crossref]

Pisano, E.

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(11), 2015–2025 (1997).
[Crossref] [PubMed]

Pogany, A.

S. W. Wilkins, Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. Pogany, and A. W. Stevenson, “On the evolution and relative merits of hard X-ray phase-contrast imaging methods,” Philos Trans A Math Phys. Eng. Sci. 372(2010), 20130021 (2014).
[Crossref] [PubMed]

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008).
[Crossref] [PubMed]

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), 51–70 (2004).

A. Pogany, D. Gao, and S. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997).
[Crossref]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

Poropat, P.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28(8), 1610–1619 (2001).
[Crossref] [PubMed]

Prest, M.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28(8), 1610–1619 (2001).
[Crossref] [PubMed]

Rau, C.

F. A. Vittoria, M. Endrizzi, P. C. Diemoz, U. H. Wagner, C. Rau, I. K. Robinson, and A. Olivo, “Virtual edge illumination and one dimensional beam tracking for absorption, refraction, and scattering retrieval,” Appl. Phys. Lett. 104(13), 134102 (2014).
[Crossref]

P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pešić, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-ray phase-contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110(13), 138105 (2013).
[Crossref] [PubMed]

Rigon, L.

P. R. T. Munro, L. Rigon, K. Ignatyev, F. C. M. Lopez, D. Dreossi, R. D. Speller, and A. Olivo, “A quantitative, non-interferometric X-ray phase contrast imaging technique,” Opt. Express 21(1), 647–661 (2013).
[Crossref] [PubMed]

F. A. Vittoria, P. C. Diemoz, M. Endrizzi, L. Rigon, F. C. Lopez, D. Dreossi, P. R. T. Munro, and A. Olivo, “Strategies for efficient and fast wave optics simulation of coded-aperture and other x-ray phase-contrast imaging methods,” Appl. Opt. 52(28), 6940–6947 (2013).
[PubMed]

A. Peterzol, A. Olivo, L. Rigon, S. Pani, and D. Dreossi, “The effects of the imaging system on the validity limits of the ray-optical approach to phase contrast imaging,” Med. Phys. 32(12), 3617–3627 (2005).
[Crossref] [PubMed]

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28(8), 1610–1619 (2001).
[Crossref] [PubMed]

Robinson, I. K.

M. Endrizzi, P. C. Diemoz, T. P. Millard, L. J. Jones, R. D. Speller, I. K. Robinson, and A. Olivo, “Hard X-ray dark-field imaging with incoherent sample illumination,” Appl. Phys. Lett. 104(2), 024106 (2014).
[Crossref]

F. A. Vittoria, M. Endrizzi, P. C. Diemoz, U. H. Wagner, C. Rau, I. K. Robinson, and A. Olivo, “Virtual edge illumination and one dimensional beam tracking for absorption, refraction, and scattering retrieval,” Appl. Phys. Lett. 104(13), 134102 (2014).
[Crossref]

P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pešić, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-ray phase-contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110(13), 138105 (2013).
[Crossref] [PubMed]

Sayers, D.

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(11), 2015–2025 (1997).
[Crossref] [PubMed]

Schelokov, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibility of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

Siu, K. K. W.

Snigirev, A.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibility of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

Snigireva, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibility of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

Speller, R. D.

M. Endrizzi, P. C. Diemoz, T. P. Millard, L. J. Jones, R. D. Speller, I. K. Robinson, and A. Olivo, “Hard X-ray dark-field imaging with incoherent sample illumination,” Appl. Phys. Lett. 104(2), 024106 (2014).
[Crossref]

A. Olivo, S. Gkoumas, M. Endrizzi, C. K. Hagen, M. B. Szafraniec, P. C. Diemoz, P. R. T. Munro, K. Ignatyev, B. Johnson, J. A. Horrocks, S. J. Vinnicombe, J. L. Jones, and R. D. Speller, “Low-dose phase contrast mammography with conventional x-ray sources,” Med. Phys. 40(9), 090701 (2013).
[Crossref] [PubMed]

T. P. Millard, M. Endrizzi, K. Ignatyev, C. K. Hagen, P. R. T. Munro, R. D. Speller, and A. Olivo, “Method for automatization of the alignment of a laboratory based x-ray phase contrast edge illumination system,” Rev. Sci. Instrum. 84(8), 083702 (2013).
[Crossref] [PubMed]

P. R. T. Munro, L. Rigon, K. Ignatyev, F. C. M. Lopez, D. Dreossi, R. D. Speller, and A. Olivo, “A quantitative, non-interferometric X-ray phase contrast imaging technique,” Opt. Express 21(1), 647–661 (2013).
[Crossref] [PubMed]

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Phase and absorption retrieval using incoherent X-ray sources,” Proc. Natl. Acad. Sci. U.S.A. 109(35), 13922–13927 (2012).
[Crossref] [PubMed]

K. Ignatyev, P. R. T. Munro, R. D. Speller, and A. Olivo, “Effects of signal diffusion on x-ray phase contrast images,” Rev. Sci. Instrum. 82(7), 073702 (2011).
[Crossref] [PubMed]

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “The relationship between wave and geometrical optics models of coded aperture type x-ray phase contrast imaging systems,” Opt. Express 18(5), 4103–4117 (2010).
[Crossref] [PubMed]

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation X- ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. 94(4), 044108 (2009).
[Crossref]

A. Olivo and R. D. Speller, “Image formation principles in coded-aperture based x-ray phase contrast imaging,” Phys. Med. Biol. 53(22), 6461–6474 (2008).
[Crossref] [PubMed]

A. Olivo and R. D. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. 52(22), 6555–6573 (2007).
[Crossref] [PubMed]

A. Olivo and R. D. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91(7), 074106 (2007).
[Crossref]

Stevenson, A. W.

S. W. Wilkins, Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. Pogany, and A. W. Stevenson, “On the evolution and relative merits of hard X-ray phase-contrast imaging methods,” Philos Trans A Math Phys. Eng. Sci. 372(2010), 20130021 (2014).
[Crossref] [PubMed]

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008).
[Crossref] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

T. 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 373(6515), 595–598 (1995).
[Crossref]

Suzuki, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), L866–L868 (2003).
[Crossref]

Szafraniec, M. B.

A. Olivo, S. Gkoumas, M. Endrizzi, C. K. Hagen, M. B. Szafraniec, P. C. Diemoz, P. R. T. Munro, K. Ignatyev, B. Johnson, J. A. Horrocks, S. J. Vinnicombe, J. L. Jones, and R. D. Speller, “Low-dose phase contrast mammography with conventional x-ray sources,” Med. Phys. 40(9), 090701 (2013).
[Crossref] [PubMed]

P. R. T. Munro, C. K. Hagen, M. B. Szafraniec, and A. Olivo, “A simplified approach to quantitative coded aperture X-ray phase imaging,” Opt. Express 21(9), 11187–11201 (2013).
[Crossref] [PubMed]

M. Marenzana, C. K. Hagen, P. Das Neves Borges, M. Endrizzi, M. B. Szafraniec, K. Ignatyev, and A. Olivo, “Visualization of small lesions in rat cartilage by means of laboratory-based x-ray phase contrast imaging,” Phys. Med. Biol. 57(24), 8173–8184 (2012).
[Crossref] [PubMed]

Takai, K.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), L866–L868 (2003).
[Crossref]

Teague, M. R.

Thomlinson, W.

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(11), 2015–2025 (1997).
[Crossref] [PubMed]

Tromba, G.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28(8), 1610–1619 (2001).
[Crossref] [PubMed]

Vallazza, E.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28(8), 1610–1619 (2001).
[Crossref] [PubMed]

Vinnicombe, S. J.

A. Olivo, S. Gkoumas, M. Endrizzi, C. K. Hagen, M. B. Szafraniec, P. C. Diemoz, P. R. T. Munro, K. Ignatyev, B. Johnson, J. A. Horrocks, S. J. Vinnicombe, J. L. Jones, and R. D. Speller, “Low-dose phase contrast mammography with conventional x-ray sources,” Med. Phys. 40(9), 090701 (2013).
[Crossref] [PubMed]

Vittoria, F. A.

Wagner, U. H.

F. A. Vittoria, M. Endrizzi, P. C. Diemoz, U. H. Wagner, C. Rau, I. K. Robinson, and A. Olivo, “Virtual edge illumination and one dimensional beam tracking for absorption, refraction, and scattering retrieval,” Appl. Phys. Lett. 104(13), 134102 (2014).
[Crossref]

Washburn, D.

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(11), 2015–2025 (1997).
[Crossref] [PubMed]

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[Crossref]

Wilkins, S.

A. Pogany, D. Gao, and S. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997).
[Crossref]

Wilkins, S. W.

S. W. Wilkins, Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. Pogany, and A. W. Stevenson, “On the evolution and relative merits of hard X-ray phase-contrast imaging methods,” Philos Trans A Math Phys. Eng. Sci. 372(2010), 20130021 (2014).
[Crossref] [PubMed]

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008).
[Crossref] [PubMed]

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), 51–70 (2004).

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

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

T. 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 373(6515), 595–598 (1995).
[Crossref]

Zapata, C. E.

P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pešić, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-ray phase-contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110(13), 138105 (2013).
[Crossref] [PubMed]

Zhong, Z.

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(11), 2015–2025 (1997).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (5)

P. C. Diemoz, C. K. Hagen, M. Endrizzi, and A. Olivo, “Sensitivity of laboratory based implementations of edge illumination X-ray phase-contrast imaging,” Appl. Phys. Lett. 103(24), 244104 (2013).
[Crossref]

M. Endrizzi, P. C. Diemoz, T. P. Millard, L. J. Jones, R. D. Speller, I. K. Robinson, and A. Olivo, “Hard X-ray dark-field imaging with incoherent sample illumination,” Appl. Phys. Lett. 104(2), 024106 (2014).
[Crossref]

A. Olivo and R. D. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91(7), 074106 (2007).
[Crossref]

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation X- ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. 94(4), 044108 (2009).
[Crossref]

F. A. Vittoria, M. Endrizzi, P. C. Diemoz, U. H. Wagner, C. Rau, I. K. Robinson, and A. Olivo, “Virtual edge illumination and one dimensional beam tracking for absorption, refraction, and scattering retrieval,” Appl. Phys. Lett. 104(13), 134102 (2014).
[Crossref]

J. Microsc. (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]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), L866–L868 (2003).
[Crossref]

Med. Phys. (3)

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28(8), 1610–1619 (2001).
[Crossref] [PubMed]

A. Olivo, S. Gkoumas, M. Endrizzi, C. K. Hagen, M. B. Szafraniec, P. C. Diemoz, P. R. T. Munro, K. Ignatyev, B. Johnson, J. A. Horrocks, S. J. Vinnicombe, J. L. Jones, and R. D. Speller, “Low-dose phase contrast mammography with conventional x-ray sources,” Med. Phys. 40(9), 090701 (2013).
[Crossref] [PubMed]

A. Peterzol, A. Olivo, L. Rigon, S. Pani, and D. Dreossi, “The effects of the imaging system on the validity limits of the ray-optical approach to phase contrast imaging,” Med. Phys. 32(12), 3617–3627 (2005).
[Crossref] [PubMed]

Nat. Phys. (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[Crossref]

Nature (2)

T. 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 373(6515), 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 384(6607), 335–338 (1996).
[Crossref]

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), 51–70 (2004).

Opt. Express (6)

Philos Trans A Math Phys. Eng. Sci. (1)

S. W. Wilkins, Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. Pogany, and A. W. Stevenson, “On the evolution and relative merits of hard X-ray phase-contrast imaging methods,” Philos Trans A Math Phys. Eng. Sci. 372(2010), 20130021 (2014).
[Crossref] [PubMed]

Phys. Med. Biol. (4)

M. Marenzana, C. K. Hagen, P. Das Neves Borges, M. Endrizzi, M. B. Szafraniec, K. Ignatyev, and A. Olivo, “Visualization of small lesions in rat cartilage by means of laboratory-based x-ray phase contrast imaging,” Phys. Med. Biol. 57(24), 8173–8184 (2012).
[Crossref] [PubMed]

A. Olivo and R. D. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. 52(22), 6555–6573 (2007).
[Crossref] [PubMed]

A. Olivo and R. D. Speller, “Image formation principles in coded-aperture based x-ray phase contrast imaging,” Phys. Med. Biol. 53(22), 6461–6474 (2008).
[Crossref] [PubMed]

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(11), 2015–2025 (1997).
[Crossref] [PubMed]

Phys. Rev. A (1)

P. R. T. Munro and A. Olivo, “X-ray phase-contrast imaging with polychromatic sources and the concept of effective energy,” Phys. Rev. A 87(5), 053838 (2013).
[Crossref]

Phys. Rev. Lett. (2)

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Proc. Natl. Acad. Sci. U.S.A. (1)

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Figures (6)

Fig. 1
Fig. 1 Diagram of the EI experimental setup in its laboratory implementation (not to scale).
Fig. 2
Fig. 2 Example profiles for function K, in the case of point and extended sources. Considered parameters of the experimental setup: z1 = 1.6 m, z2 = 0.4 m, ye = 7.5 µm (which corresponds to a 50% illumination for an aperture a = 12 µm), d = 20 µm, extended source of 70 µm full width at half maximum (parameters matching one of the experimental setups installed at University College London (UCL)).
Fig. 3
Fig. 3 Example profiles for the illumination curve C, in the case of both point and extended sources. Considered parameters of the experimental setup: z1 = 1.6 m, z2 = 0.4 m, a = 12 µm, d = 20 µm, X-ray energy = 20 keV, detector mask period = 85 µm, extended source of 70 µm full width at half maximum (parameters matching one of the experimental setups installed at UCL). The peak of the curves is reached at ye = −2.5 µm, corresponding to perfectly aligned slits.
Fig. 4
Fig. 4 Plots of (a) fT and (b) fR for different source sizes and illumination fractions. Considered experimental parameters: z1 = 1.6 m, z2 = 0.4 m, a = 12 µm, d = 20 µm, detector mask period = 85 µm. ye/M = 6 µm corresponds to + 50% slits misalignment, ye/M = 9 µm to + 75% slits misalignment and ye/M = −2 µm to perfectly aligned slits.
Fig. 5
Fig. 5 Plots of the refraction integral sensitivity function FR (a) as a function of the source FWHM for two different positions of the detector edge, and (b) as a function of the detector edge position for two source FWHMs. The values for the other experimental parameters are: z1 = 1.6 m, z2 = 0.4 m, a = 12 µm, d = 20 µm, detector mask period = 85 µm.
Fig. 6
Fig. 6 Normalized signal profiles obtained using wave optics and TIE-based models for (a) a 200 µm diameter cylinder made of PET, and (b) a 30 µm thick PET slab, with an edge shape expressed as a step function convolved with a Gaussian function of 1 µm standard deviation.

Equations (27)

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E o u t ( x , y ; p ; λ ) = q ( x , y ; λ ) m ( y ) E i n ( x , y ; λ )
I p o i n t ( x ˜ , y ˜ ; p ; λ ) = M 1 I 0 | [ q ( x , y ; λ ) m ( y ) ] H z d e f ( x , y ; λ ) | 2
S ( n x ; p ; y e ; λ ) = ( n x 1 ) Δ x n x Δ x d x ˜ y e y e + d d y ˜ [ I p o i n t ( x ˜ , y ˜ ; p ; λ ) G ( x ˜ , y ˜ ) ]
S ( p ; y e ; λ ) = y e y e + d d y ˜ + d y ¯ g ( y ˜ y ¯ ) I p o i n t ( y ¯ ; p ; λ )
S ( p ; y e ; λ ) = + d y ¯ I p o i n t ( y ¯ ; p ; λ ) K ( y ¯ ; y e )
K ( y ¯ ; y e ) y e y e + d d y ˜ g ( y ˜ y ¯ ) = [ g r e c t d ] ( y e + d / 2 y ¯ )
S ( p ) = T ( p ) y e z 2 Δ θ y ( p ) y e + d z 2 Δ θ y ( p ) d y ˜ I r e f ( y ˜ ) = a I 0 T ( p ) C ( y e z 2 Δ θ y ( p ) )
I F S P ( y ˜ ) = [ I F S P , p o i n t g ] ( y ˜ )
I F S P , p o i n t ( y ˜ ) = M 1 I T ( y ) M 1 k 1 z d e f [ I T ( y ) 2 ϕ y 2 ( y ) + I T y ( y ) ϕ y ( y ) ]
N F 2 π σ M σ o b j / ( λ z d e f ) max { 1 , ϕ m a x }
S ( p ) = + d y K ( M ( y + p ) ) [ I T ( y ) r e c t a ( y + p a / 2 ) k 1 z d e f ( I T ( y ) r e c t a ( y + p a / 2 ) 2 ϕ ( y ) y 2 + y ( I T ( y ) r e c t a ( y + p a / 2 ) ) ϕ ( y ) y ) ]
k 1 z d e f + d y y ( I T ( y ) r e c t a ( y + p a / 2 ) ) ϕ ( y ) y K ( M ( y + p ) ) = k 1 z d e f + d y I T ( y ) r e c t a ( y + p a / 2 ) y [ ϕ ( y ) y K ( M ( y + p ) ) ]
S ( p ) = p p + a d y I T ( y ) K ( M ( y + p ) ) + k 1 z d e f p p + a d y I T ( y ) ϕ ( y ) y K y ( M ( y + p ) ) = [ I T × f T ] ( p ) + k 1 [ ( I T ϕ y ) f R ] ( p )
S ( p ) = p p + a I T ( y ) K ( M ( y + p ) ) + k 1 z d e f [ I T ( y ) ϕ ( y ) y K ( M ( y + p ) ) ] p p + a k 1 z d e f p p + a d y [ I T ( y ) 2 ϕ ( y ) y 2 + I T ( y ) y ϕ ( y ) y ] K ( M ( y + p ) )
S ( p ) = p p + a d y K ( M ( y + p ) ) [ I T ( y ) k 1 z d e f I T ( y ) 2 ϕ ( y ) y 2 k 1 z d e f I T ( y ) y ϕ ( y ) y ] = [ ( I T k 1 z d e f I T 2 ϕ y 2 k 1 z d e f I T ( y ) y ϕ ( y ) y ) f T ] ( p )
S ( p ) = I T ( p ) 0 a d y K ( M y ) + k 1 z d e f I T ( p ) ϕ y ( p ) [ K ( M a ) K ( 0 ) ]
K ( M a ) K ( 0 ) = y e M a y e d y g ( y ) y e + d M a y e + d d y g ( y ) = M a [ I r e f ( y e ) I r e f ( y e + d ) ] = M a C y e ( y e )
S ( p ) = a I T ( p ) C ( y e ) k 1 z 2 a I T ( p ) C y e ( y e ) ϕ y ( p )
K d ( y ) [ g r e c t s m o o t h , d ] ( y e + d / 2 y )
K d ( y ) = [ g 2 r e c t d ] ( y e + d / 2 y )
f T ( y ) = [ f a r e c t a ] ( y a / 2 ) [ g r , 2 r e c t d / M ] ( y e / M + d / 2 M y )
f R ( y ) = z d e f [ f a r e c t a ] ( y a / 2 ) [ g r , 2 ( y e / M y ) g r , 2 ( y e / M + d / M y ) ]
S ( p ) = I T F T I T k 1 z d e f [ 2 ϕ y 2 f T ] ( p )
F T ( y e ) = [ g r f d , r f a r e c t d / M r e c t a ] ( y e / M + d / 2 M a / 2 )
F R ( y e ) = z d e f [ r e c t d / M g r f d , r f a ] ( y e / M + d / 2 M a ) z d e f [ r e c t d / M g r f d , r f a ] ( y e / M + d / 2 M )
K d , a ( y ) = [ g f d f a , M r e c t d ] ( y e + d / 2 y )
S ( p ) = I 0 F T + k 1 I 0 ϕ s t e p [ g o b j f R ] ( p )

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