Abstract

Effects of incident illumination on phase-contrast images obtained by means of free-space propagation are investigated under the “transport-of-intensity” approximation. Analytical expressions for image intensity distribution are derived in the cases of coherent quasi-plane and quasi-spherical incident waves, as well as for spatially incoherent and quasi-homogeneous sources and some other types of sources. Practical methods for measuring the relevant parameters of the incident radiation are discussed together with formulas allowing one to calculate the effect of these parameters on the image intensity distribution. The results are expected to be useful in quantitative in-line imaging, phase retrieval, and tomography with polychromatic and spatially partially coherent radiation. As an application we present a method for simultaneous “automatic” phase retrieval and spatial deconvolution in in-line imaging of homogeneous objects using extended polychromatic x-ray sources.

© 2006 Optical Society of America

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  2. J. M. Cowley, Diffraction Physics, 3rd ed. (Elsevier, 1995).
  3. D. Paganin and K. A. Nugent, in Advances in Imaging and Electron Physics, P.Hawkes, ed. (Harcourt, 2001), Vol. 118, pp. 85-127.
    [CrossRef]
  4. D. Gabor, "A new microscopic principle, Nature 161, 777-778 (1948).
    [CrossRef] [PubMed]
  5. M. R. Teague, "Deterministic phase retrieval: a Green's function solution," J. Opt. Soc. Am. 73, 1434-1441 (1983).
    [CrossRef]
  6. D. Van Dyck and W. Coene, "A new procedure for wave function restoration in high resolution electron microscopy," Optik (Stuttgart) 77, 125-128 (1987).
  7. N. Streibl, "Phase imaging by the transport of intensity equation," Opt. Commun. 49, 6-10 (1984).
    [CrossRef]
  8. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature 384, 335-338 (1996).
    [CrossRef]
  9. D. Paganin and K. A. Nugent, "Noninterferometric phase imaging with partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
    [CrossRef]
  10. T. E. Gureyev, "Transport of intensity equation for beams in arbitrary state of temporal and spatial coherence," Optik (Stuttgart) 110, 263-266 (1999).
  11. K. A. Nugent, D. Paganin, and T. E. Gureyev, "A phase odyssey," Phys. Today 54 (No. 8), 27-32 (2001).
    [CrossRef]
  12. T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
    [CrossRef] [PubMed]
  13. A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
    [CrossRef]
  14. P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J.-P. Guigay, and M. Schlenker, "Holotomography: quantitative phase tomography with micrometre resolution using hard synchrotron radiation x-rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
    [CrossRef]
  15. R. Fitzgerald, "Phase-sensitive x-ray imaging," Phys. Today 53 (No. 7), 23-26 (2000).
    [CrossRef]
  16. S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
    [CrossRef] [PubMed]
  17. T. E. Gureyev, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, "Image deblurring by means of defocus," Opt. Commun. 240, 81-88 (2004).
    [CrossRef]
  18. D. M. Paganin, T. E. Gureyev, and S. W. Wilkins, "Proximity correction for projection imaging," provisional patent no. 2,004,904,206, priority date July 2004.
  19. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  20. M. V. Fedoryuk, "The stationary phase method and pseudodifferential operators," Russ. Math. Surveys 26(1), 65-115 (1971).
    [CrossRef]
  21. T. E. Gureyev and S. W. Wilkins, "On x-ray phase imaging with a point source," J. Opt. Soc. Am. A 15, 579-585 (1998).
    [CrossRef]
  22. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486-5492 (1995).
    [CrossRef]
  23. H. H. Hopkins, "On the diffraction theory of optical images," Proc. R. Soc. London, Ser. A 217, 408-432 (1953).
    [CrossRef]
  24. 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, 33-40 (2002).
    [CrossRef] [PubMed]
  25. M. Subbarao, T. C. Wei, and G. Surya, "Focused image recovery from 2 defocused images recorded with different camera settings," IEEE Trans. Image Process. 4, 1613-1628 (1995).
    [CrossRef] [PubMed]
  26. D. Paganin, T. E. Gureyev, and S. W. Wilkins, "Deterministic mask design for phase-contrast nanolithography," in preparation (manuscript available from David.Paganin@csiro.au).

2004

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

T. E. Gureyev, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, "Image deblurring by means of defocus," Opt. Commun. 240, 81-88 (2004).
[CrossRef]

2003

2002

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, 33-40 (2002).
[CrossRef] [PubMed]

2001

K. A. Nugent, D. Paganin, and T. E. Gureyev, "A phase odyssey," Phys. Today 54 (No. 8), 27-32 (2001).
[CrossRef]

2000

R. Fitzgerald, "Phase-sensitive x-ray imaging," Phys. Today 53 (No. 7), 23-26 (2000).
[CrossRef]

1999

T. E. Gureyev, "Transport of intensity equation for beams in arbitrary state of temporal and spatial coherence," Optik (Stuttgart) 110, 263-266 (1999).

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

1998

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

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

1997

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

1996

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

1995

M. Subbarao, T. C. Wei, and G. Surya, "Focused image recovery from 2 defocused images recorded with different camera settings," IEEE Trans. Image Process. 4, 1613-1628 (1995).
[CrossRef] [PubMed]

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

1987

D. Van Dyck and W. Coene, "A new procedure for wave function restoration in high resolution electron microscopy," Optik (Stuttgart) 77, 125-128 (1987).

1984

N. Streibl, "Phase imaging by the transport of intensity equation," Opt. Commun. 49, 6-10 (1984).
[CrossRef]

1983

1971

M. V. Fedoryuk, "The stationary phase method and pseudodifferential operators," Russ. Math. Surveys 26(1), 65-115 (1971).
[CrossRef]

1953

H. H. Hopkins, "On the diffraction theory of optical images," Proc. R. Soc. London, Ser. A 217, 408-432 (1953).
[CrossRef]

1948

D. Gabor, "A new microscopic principle, Nature 161, 777-778 (1948).
[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 micrometre resolution using hard synchrotron radiation x-rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

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 micrometre resolution using hard synchrotron radiation x-rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Coene, W.

D. Van Dyck and W. Coene, "A new procedure for wave function restoration in high resolution electron microscopy," Optik (Stuttgart) 77, 125-128 (1987).

Cowley, J. M.

J. M. Cowley, Diffraction Physics, 3rd ed. (Elsevier, 1995).

Davis, T. J.

Fedoryuk, M. V.

M. V. Fedoryuk, "The stationary phase method and pseudodifferential operators," Russ. Math. Surveys 26(1), 65-115 (1971).
[CrossRef]

Fitzgerald, R.

R. Fitzgerald, "Phase-sensitive x-ray imaging," Phys. Today 53 (No. 7), 23-26 (2000).
[CrossRef]

Gabor, D.

D. Gabor, "A new microscopic principle, Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

Gao, D.

A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 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, 335-338 (1996).
[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 micrometre resolution using hard synchrotron radiation x-rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Gureyev, T. E.

T. E. Gureyev, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, "Image deblurring by means of defocus," Opt. Commun. 240, 81-88 (2004).
[CrossRef]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[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, 33-40 (2002).
[CrossRef] [PubMed]

K. A. Nugent, D. Paganin, and T. E. Gureyev, "A phase odyssey," Phys. Today 54 (No. 8), 27-32 (2001).
[CrossRef]

T. E. Gureyev, "Transport of intensity equation for beams in arbitrary state of temporal and spatial coherence," Optik (Stuttgart) 110, 263-266 (1999).

T. E. Gureyev and S. W. Wilkins, "On x-ray phase imaging with a point source," J. Opt. Soc. Am. A 15, 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, 335-338 (1996).
[CrossRef]

D. Paganin, T. E. Gureyev, and S. W. Wilkins, "Deterministic mask design for phase-contrast nanolithography," in preparation (manuscript available from David.Paganin@csiro.au).

D. M. Paganin, T. E. Gureyev, and S. W. Wilkins, "Proximity correction for projection imaging," provisional patent no. 2,004,904,206, priority date July 2004.

Hopkins, H. H.

H. H. Hopkins, "On the diffraction theory of optical images," Proc. R. Soc. London, Ser. A 217, 408-432 (1953).
[CrossRef]

Kohn, V.

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

Kuznetsov, S.

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

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 micrometre resolution using hard synchrotron radiation x-rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Mayo, S. C.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[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, 33-40 (2002).
[CrossRef] [PubMed]

Miller, P. R.

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[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, 33-40 (2002).
[CrossRef] [PubMed]

Nesterets, Ya. I.

T. E. Gureyev, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, "Image deblurring by means of defocus," Opt. Commun. 240, 81-88 (2004).
[CrossRef]

Nugent, K. A.

K. A. Nugent, D. Paganin, and T. E. Gureyev, "A phase odyssey," Phys. Today 54 (No. 8), 27-32 (2001).
[CrossRef]

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

D. Paganin and K. A. Nugent, in Advances in Imaging and Electron Physics, P.Hawkes, ed. (Harcourt, 2001), Vol. 118, pp. 85-127.
[CrossRef]

Paganin, D.

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[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, 33-40 (2002).
[CrossRef] [PubMed]

K. A. Nugent, D. Paganin, and T. E. Gureyev, "A phase odyssey," Phys. Today 54 (No. 8), 27-32 (2001).
[CrossRef]

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

D. Paganin and K. A. Nugent, in Advances in Imaging and Electron Physics, P.Hawkes, ed. (Harcourt, 2001), Vol. 118, pp. 85-127.
[CrossRef]

D. Paganin, T. E. Gureyev, and S. W. Wilkins, "Deterministic mask design for phase-contrast nanolithography," in preparation (manuscript available from David.Paganin@csiro.au).

Paganin, D. M.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

D. M. Paganin, T. E. Gureyev, and S. W. Wilkins, "Proximity correction for projection imaging," provisional patent no. 2,004,904,206, priority date July 2004.

Pogany, A.

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 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, 335-338 (1996).
[CrossRef]

Schelokov, I.

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

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 micrometre resolution using hard synchrotron radiation x-rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Snigirev, A.

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

Snigireva, I.

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

Stevenson, A. W.

T. E. Gureyev, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, "Image deblurring by means of defocus," Opt. Commun. 240, 81-88 (2004).
[CrossRef]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[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, 335-338 (1996).
[CrossRef]

Streibl, N.

N. Streibl, "Phase imaging by the transport of intensity equation," Opt. Commun. 49, 6-10 (1984).
[CrossRef]

Subbarao, M.

M. Subbarao, T. C. Wei, and G. Surya, "Focused image recovery from 2 defocused images recorded with different camera settings," IEEE Trans. Image Process. 4, 1613-1628 (1995).
[CrossRef] [PubMed]

Surya, G.

M. Subbarao, T. C. Wei, and G. Surya, "Focused image recovery from 2 defocused images recorded with different camera settings," IEEE Trans. Image Process. 4, 1613-1628 (1995).
[CrossRef] [PubMed]

Teague, M. R.

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 micrometre resolution using hard synchrotron radiation x-rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

D. Van Dyck and W. Coene, "A new procedure for wave function restoration in high resolution electron microscopy," Optik (Stuttgart) 77, 125-128 (1987).

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 micrometre resolution using hard synchrotron radiation x-rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Wei, T. C.

M. Subbarao, T. C. Wei, and G. Surya, "Focused image recovery from 2 defocused images recorded with different camera settings," IEEE Trans. Image Process. 4, 1613-1628 (1995).
[CrossRef] [PubMed]

Wilkins, S. W.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

T. E. Gureyev, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, "Image deblurring by means of defocus," Opt. Commun. 240, 81-88 (2004).
[CrossRef]

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[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, 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, 579-585 (1998).
[CrossRef]

A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 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, 335-338 (1996).
[CrossRef]

D. Paganin, T. E. Gureyev, and S. W. Wilkins, "Deterministic mask design for phase-contrast nanolithography," in preparation (manuscript available from David.Paganin@csiro.au).

D. M. Paganin, T. E. Gureyev, and S. W. Wilkins, "Proximity correction for projection imaging," provisional patent no. 2,004,904,206, priority date July 2004.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Appl. Phys. Lett.

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

IEEE Trans. Image Process.

M. Subbarao, T. C. Wei, and G. Surya, "Focused image recovery from 2 defocused images recorded with different camera settings," IEEE Trans. Image Process. 4, 1613-1628 (1995).
[CrossRef] [PubMed]

J. Microsc.

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, 33-40 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nature

D. Gabor, "A new microscopic principle, Nature 161, 777-778 (1948).
[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, 335-338 (1996).
[CrossRef]

Opt. Commun.

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[CrossRef]

T. E. Gureyev, A. W. Stevenson, Ya. I. Nesterets, and S. W. Wilkins, "Image deblurring by means of defocus," Opt. Commun. 240, 81-88 (2004).
[CrossRef]

Opt. Express

Optik (Stuttgart)

D. Van Dyck and W. Coene, "A new procedure for wave function restoration in high resolution electron microscopy," Optik (Stuttgart) 77, 125-128 (1987).

T. E. Gureyev, "Transport of intensity equation for beams in arbitrary state of temporal and spatial coherence," Optik (Stuttgart) 110, 263-266 (1999).

Phys. Rev. Lett.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

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

Phys. Today

K. A. Nugent, D. Paganin, and T. E. Gureyev, "A phase odyssey," Phys. Today 54 (No. 8), 27-32 (2001).
[CrossRef]

R. Fitzgerald, "Phase-sensitive x-ray imaging," Phys. Today 53 (No. 7), 23-26 (2000).
[CrossRef]

Proc. R. Soc. London, Ser. A

H. H. Hopkins, "On the diffraction theory of optical images," Proc. R. Soc. London, Ser. A 217, 408-432 (1953).
[CrossRef]

Rev. Sci. Instrum.

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

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

Russ. Math. Surveys

M. V. Fedoryuk, "The stationary phase method and pseudodifferential operators," Russ. Math. Surveys 26(1), 65-115 (1971).
[CrossRef]

Other

D. Paganin, T. E. Gureyev, and S. W. Wilkins, "Deterministic mask design for phase-contrast nanolithography," in preparation (manuscript available from David.Paganin@csiro.au).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

J. M. Cowley, Diffraction Physics, 3rd ed. (Elsevier, 1995).

D. Paganin and K. A. Nugent, in Advances in Imaging and Electron Physics, P.Hawkes, ed. (Harcourt, 2001), Vol. 118, pp. 85-127.
[CrossRef]

D. M. Paganin, T. E. Gureyev, and S. W. Wilkins, "Proximity correction for projection imaging," provisional patent no. 2,004,904,206, priority date July 2004.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

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Figures (1)

Fig. 1
Fig. 1

Schematics of in-line imaging.

Equations (42)

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V ( x , y , z , t ) = exp ( i 2 π ν t ) U ( x , y , z , ν ) d ν .
S ( x , y , z , ν ) δ ( ν ν ) = U * ( x , y , z , ν ) U ( x , y , z , ν ) ,
I ( x , y , z ) = S ( x , y , z , ν ) d ν ,
V ( x 1 , y 1 , R 1 , t ) = exp ( i 2 π ν t ) U src ( x 1 , y 1 , ν ) d ν
V ( x , y , z , t ) = 1 i λ z exp ( i 2 π ν t ) exp ( i k z ) exp { i k ( 2 z ) [ ( x x 1 ) 2 + ( y y 1 ) 2 ] } U src ( x 1 , y 1 , ν ) d x 1 d y 1 d ν ,
U out ( x , y , 0 , ν ) = Q ( x , y , ν ) U in ( x , y , 0 , ν ) .
F ( x , y , ν ) = i λ R 1 exp [ i k ( x x 1 + y y 1 ) R 1 ] exp [ i k ( x 1 2 + y 1 2 ) 2 R 1 ] U src ( x 1 , y 1 , ν ) d x 1 d y 1 .
U in ( x , y , 0 , ν ) = exp ( i k R 1 ) exp [ i k ( x 2 + y 2 ) ( 2 R 1 ) ] F ( x , y , ν ) .
V ( x 2 , y 2 , R 2 , t ) = exp ( i 2 π ν t ) U ( x 2 , y 2 , R 2 , ν ) d ν ,
U ( x 2 , y 2 , R 2 , ν ) = i exp ( i k R ) λ R 2 Q ( x , y , ν ) F ( x , y , ν ) × exp { i k [ x 2 + y 2 2 R 1 + ( x 2 x ) 2 + ( y 2 y ) 2 2 R 2 ] } d x d y ,
S ( x 2 , y 2 , R 2 , ν ) = 1 λ 2 R 2 2 W in ( x , y , x , y , ν ) Q * ( x , y , ν ) Q ( x , y , ν ) exp { i k [ x 2 x 2 + y 2 y 2 2 R + x 2 ( x x ) + y 2 ( y y ) R 2 ] } d x d y d x d y ,
R 2 ξ ( x 1 , y 1 , ν ) 1 , R ξ ( x 1 , y 1 , ν ) a ( x 1 , y 1 , ν ) 1 ,
F ( x , y , ν ) [ a ( x , y , ν ) ( R 1 2 ) ( a 2 ξ ) ( x , y , ν ) R 1 a ξ ( x , y , ν ) ] exp [ i k ( 2 R 1 ) ( x 2 + y 2 ) + i k ξ ( x , y , ν ) ] ;
I in ( x , y , 0 , ν ) = a 2 ( x , y , ν ) R 1 ( a 2 ξ ) ( x , y , ν ) .
R 2 2 ψ ( x , y , ν ) 1 , R 2 ψ ( x , y , ν ) q ( x , y , ν ) 1 .
U ( x 2 , y 2 , R 2 , ν ) = exp ( i k Ψ ) { ( a q ) ( x 2 , y 2 , ν ) [ 1 ( R 2 2 ) 2 ( ψ + ξ ) ( x 2 , y 2 , ν ) ] R 2 ( a q ) ( ψ + ξ ) ( x 2 , y 2 , ν ) R 1 q [ a ξ + ( a 2 ) 2 ξ ] ( x 2 , y 2 , ν ) } ,
I ( x 2 , y 2 , R 2 , ν ) = ( a 2 q 2 ) ( x 2 , y 2 , ν ) R 1 [ q 2 ( a 2 ξ ) ] ( x 2 , y 2 , ν ) R 2 [ a 2 q 2 ( ξ + ψ ) ] ( x 2 , y 2 , ν ) .
I ( x 2 , y 2 , R 2 , ν ) = ( I in q 2 ) ( x 2 , y 2 , ν ) R 2 [ I in q 2 ( ψ in + ψ ) ] ( x 2 , y 2 , ν ) ,
I ( x 2 , y 2 , R 2 ) = I out ( x 2 , y 2 ) R 2 ( S out ψ out ) ( x 2 , y 2 , ν ) d ν ,
I flat ( x 2 , y 2 , R 2 , ν ) = I in ( x 2 , y 2 , ν ) R 2 ( I in ψ in ) ( x 2 , y 2 , ν ) .
U ( x 2 , y 2 , R 2 , ν ) = exp ( i k Ψ ) M { ( f q ) ( x 2 M , y 2 M , ν ) [ 1 R 2 2 ( ψ + η ) ( x 2 M , y 2 M , ν ) ] R 2 M ( f q ) ( ψ + η ) ( x 2 M , y 2 M , ν ) } ,
I ( x 2 , y 2 , R 2 , ν ) = M 2 { f 2 q 2 ( R 2 M ) [ f 2 q 2 ( η + ψ ) ] } ( x 2 M , y 2 M , ν ) .
I ( x 2 , y 2 , R 2 , ν ) = M 2 { ( I in q 2 ) ( x 2 M + x ¯ 1 ( M 1 ) M , y 2 M + y ¯ 1 ( M 1 ) M , ν ) R 2 M [ I in q 2 ( ψ in + ψ ) ] ( x 2 M + x ¯ 1 ( M 1 ) M , y 2 M + y ¯ 1 ( M 1 ) M , ν ) } ,
I ( x 2 , y 2 , R 2 ) = M 2 S ( ν ) [ q ( x 2 M + x ¯ 1 ( M 1 ) M , y 2 M + y ¯ 1 ( M 1 ) M , ν ) R 2 M ( q 2 ψ ) ( x 2 M + x ¯ 1 ( M 1 ) M , y 2 M + y ¯ 1 ( M 1 ) M , ν ) ] d ν .
W in ( x , y , x , y , ν ) = g in ( x x , y y , ν ) S in ( x + x 2 , y + y 2 , ν ) × exp { i k [ ψ in ( x , y , ν ) ψ in ( x , y , ν ) ] } ,
W ( x 1 , y 1 , x 1 , y 1 , ν ) = S src ( ( x 1 + x 1 ) 2 , ( y 1 + y 1 ) 2 , ν ) g src ( x 1 x 1 , y 1 y 1 , ν ) .
R 2 2 ψ in ( x , y , ν ) 1 , R 2 ψ in ( x , y , ν ) S in ( x , y , ν ) 1 .
S ( x 2 , y 2 , R 2 , ν ) = 1 λ 2 R 2 2 exp [ i k ( x + x + y + y R x 2 x + y 2 y R 2 ) ] g in ( x , y , ν ) S in ( x + , y + , ν ) × Q ̃ * ( x + x 2 , y + y 2 , ν ) Q ̃ ( x + + x 2 , y + + y 2 , ν ) d x + d y + d x d y ,
g in ( x , y , ν ) = exp [ i 2 π ( x 1 x λ R 1 + y 1 y λ R 1 ) ] S n src ( x 1 , y 1 , ν ) d x 1 d y 1 ,
S ps ( x 2 , y 2 , ν ) = 1 λ 2 R 2 2 exp [ i 2 π ( x + x + y + y λ R x 2 x + y 2 y λ R 2 ) ] S in ( x + , y + , ν ) Q ̃ * ( x + x 2 , y + y 2 , ν ) Q ̃ ( x + + x 2 , y + + y 2 , ν ) d x + d y + d x d y .
S ps ( x 2 , y 2 , R 2 , ν ) = M 2 { ( S in q 2 ) ( x 2 M , y 2 M , ν ) R [ S in q 2 ( ψ in + ψ ) ] ( x 2 M , y 2 M , ν ) } .
S ( x 2 , y 2 , R 2 , ν ) = M 2 S n src ( x 1 , y 1 , ν ) { S in q 2 R 2 M [ S in q 2 ( ψ in + ψ ) ] } ( x 2 M + x 1 ( M 1 ) M , y 2 M + y 1 ( M 1 ) M , ν ) d x 1 d y 1 .
I ( x 2 , y 2 , R 2 ) = M 2 S n src ( x 1 , y 1 , ν ) { S in q 2 R 2 M [ S in q 2 ( ψ in + ψ ) ] } ( x 2 M + x 1 ( M 1 ) M , y 2 M + y 1 ( M 1 ) M , ν ) d x 1 d y 1 d ν .
S ( x 2 , y 2 , R 2 , ν ) = M 2 S src ( x 1 , y 1 , ν ) { q 2 R 2 M [ ( q 2 ψ ) ] } ( x 2 M + x 1 ( M 1 ) M , y 2 M + y 1 ( M 1 ) M , ν ) d x 1 d y 1 ,
a m n ( ν ) = ( 1 ) m + n m ! n ! x 1 m y 1 n S src ( x 1 , y 1 , ν ) d x 1 d y 1 ,
S ( x 2 , y 2 , R 2 , ν ) M 2 a 0 ( ν ) { q 2 ( x 2 M , y 2 M , ν ) + 1 2 2 q 2 ( x 2 M , y 2 M , ν ) [ ( M 1 ) 2 M 2 σ 2 ( ν ) R 2 δ ( ν ) M k β ( ν ) ] } ,
R opt = ( M 1 ) 2 M 2 β ( ν ) δ ( ν ) k σ 2 ( ν ) .
S ( x 2 , y 2 , R opt M , ν ) M 2 a 0 ( ν ) q 2 ( x 2 M , y 2 M , ν ) ,
I ( x 2 , y 2 , R 2 ) I in M 2 2 M 2 T ( x 2 M , y 2 M ) a 0 ( ν ) k β ( ν ) d ν 2 T ( x 2 M , y 2 M ) [ 2 ( M 1 ) 2 M 2 k β ( ν ) a 2 ( ν ) R 2 δ ( ν ) a 0 ( ν ) M ] d ν .
R opt = ( M 1 ) 2 M 2 2 a 2 ( ν ) k β ( ν ) d ν a 0 ( ν ) δ ( ν ) d ν .
R opt = ( M 1 ) 2 M 2 σ 2 k β ( ν ) S ( ν ) d ν δ ( ν ) S ( ν ) d ν ,
σ max = M M 1 [ R δ ( ν ) S ( ν ) d ν k β ( ν ) S ( ν ) d ν ] 1 2 .

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