Abstract

We discuss the role of coherence in x-ray imaging and consider how phase-space tomography can be used to extract information about partial coherence. We describe the application of phase-space tomography to x-ray imaging and recover the spatial coherence properties of a one-dimensional soft (1.5 keV) x-ray beam from a synchrotron undulator source. We present phase-space information from a Young’s experiment and observe negative regions in the quasi-probability distribution. We show that, given knowledge of the coherence of the beam, we can use partially coherent diffraction data to recover fully coherent information, and we present some simple experimental demonstrations of this capability.

© 2005 Optical Society of America

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  1. M. Sutton, S. G. J. Mochrie, T. Greytak, S. E. Nagler, L. E. Berman, G. A. Held, G. B. Stephenson, “Observation of speckle by diffraction with coherent x-rays,” Nature (London) 352, 608–610 (1991).
    [CrossRef]
  2. C. Jacobsen, M. Howells, J. Kirz, S. Rothman, “X-ray holographic microscopy using photoresists,” J. Opt. Soc. Am. A 7, 1847–1861 (1990).
    [CrossRef]
  3. P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D 29, 133–146 (1996).
    [CrossRef]
  4. J. W. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature (London) 400, 342–344 (1999).
    [CrossRef]
  5. K. A. Nugent, T. E. Gureyev., D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
    [CrossRef] [PubMed]
  6. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 522–526.
  7. For example, D. Cernadas, C. Trillo, A. F. Doval, J. C. Lopez, B. V. Dorrio, J. L. Fernandez, M. Perez-Amor, “Non-destructive testing with surface acoustic waves using double-pulse TV holography,” Meas. Sci. Technol. 13, 438–444 (2002).
  8. M. Jezek, Z. Hradil, “Reconstruction of spatial, phase and coherence properties of light,” J. Opt. Soc. Am. A 8, 1407–1416 (2004).
    [CrossRef]
  9. K. A. Nugent, “Wave field determination using 3-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992).
    [CrossRef] [PubMed]
  10. M. G. Raymer, M. Beck, D. F. Mcalister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
    [CrossRef] [PubMed]
  11. C. Q. Tran, A. G. Peele, A. Roberts, K. A. Nugent, D. Paterson, I. McNulty, “Synchrotron beam coherence: a spatially resolved measurement,” Opt. Lett. 30, 204–206 (2005).
    [CrossRef] [PubMed]
  12. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  13. E. Wolf, “Coherent mode new theory of partial coherence in the space-frequency domain. 1. Spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [CrossRef]
  14. M. J. Bastiaans, “Application of the Wigner function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
    [CrossRef]
  15. J. W. Goodman, Statistical Optics (Wiley, 1985), pp. 199–200.
  16. I. McNulty, D. Paterson, J. Arko, M. Erdmann, S. P. Frigo, K. Goetze, P. Ilinski, N. Krapf, T. Mooney, C. C. Retsch, A. P. J. Stampfl, S. Vogt, Y. Wang, S. Xu, “The 2-ID-B intermediate-energy scanning X-ray microscope at the APS,” J. Phys. IV 104, 11–15 (2003).
  17. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), p. 163.
  18. D. Paterson, B. E. Allman, P. J. McMahon, J. J. A. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, D. C. Mancini, “Spatial coherence measurement of x-ray undulator radiation,” Opt. Commun. 195, 79–84 (2001).
    [CrossRef]
  19. K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
    [CrossRef]
  20. C. Kurtsiefer, T. Pfau, J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature (London) 386, 150–153 (1997).
    [CrossRef]
  21. A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002).
    [CrossRef]

2005

2004

M. Jezek, Z. Hradil, “Reconstruction of spatial, phase and coherence properties of light,” J. Opt. Soc. Am. A 8, 1407–1416 (2004).
[CrossRef]

2003

I. McNulty, D. Paterson, J. Arko, M. Erdmann, S. P. Frigo, K. Goetze, P. Ilinski, N. Krapf, T. Mooney, C. C. Retsch, A. P. J. Stampfl, S. Vogt, Y. Wang, S. Xu, “The 2-ID-B intermediate-energy scanning X-ray microscope at the APS,” J. Phys. IV 104, 11–15 (2003).

2002

A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002).
[CrossRef]

For example, D. Cernadas, C. Trillo, A. F. Doval, J. C. Lopez, B. V. Dorrio, J. L. Fernandez, M. Perez-Amor, “Non-destructive testing with surface acoustic waves using double-pulse TV holography,” Meas. Sci. Technol. 13, 438–444 (2002).

2001

D. Paterson, B. E. Allman, P. J. McMahon, J. J. A. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, D. C. Mancini, “Spatial coherence measurement of x-ray undulator radiation,” Opt. Commun. 195, 79–84 (2001).
[CrossRef]

1999

K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
[CrossRef]

J. W. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature (London) 400, 342–344 (1999).
[CrossRef]

1997

C. Kurtsiefer, T. Pfau, J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature (London) 386, 150–153 (1997).
[CrossRef]

1996

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

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

1994

M. G. Raymer, M. Beck, D. F. Mcalister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

1992

K. A. Nugent, “Wave field determination using 3-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992).
[CrossRef] [PubMed]

1991

M. Sutton, S. G. J. Mochrie, T. Greytak, S. E. Nagler, L. E. Berman, G. A. Held, G. B. Stephenson, “Observation of speckle by diffraction with coherent x-rays,” Nature (London) 352, 608–610 (1991).
[CrossRef]

1990

1986

1982

1969

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Allman, B. E.

D. Paterson, B. E. Allman, P. J. McMahon, J. J. A. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, D. C. Mancini, “Spatial coherence measurement of x-ray undulator radiation,” Opt. Commun. 195, 79–84 (2001).
[CrossRef]

Alonso, M. A.

Arko, J.

I. McNulty, D. Paterson, J. Arko, M. Erdmann, S. P. Frigo, K. Goetze, P. Ilinski, N. Krapf, T. Mooney, C. C. Retsch, A. P. J. Stampfl, S. Vogt, Y. Wang, S. Xu, “The 2-ID-B intermediate-energy scanning X-ray microscope at the APS,” J. Phys. IV 104, 11–15 (2003).

Barnea, Z.

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

Barrett, R.

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

Baruchel, J.

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

Bastiaans, M. J.

Beck, M.

M. G. Raymer, M. Beck, D. F. Mcalister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Berman, L. E.

M. Sutton, S. G. J. Mochrie, T. Greytak, S. E. Nagler, L. E. Berman, G. A. Held, G. B. Stephenson, “Observation of speckle by diffraction with coherent x-rays,” Nature (London) 352, 608–610 (1991).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 522–526.

Cernadas, D.

For example, D. Cernadas, C. Trillo, A. F. Doval, J. C. Lopez, B. V. Dorrio, J. L. Fernandez, M. Perez-Amor, “Non-destructive testing with surface acoustic waves using double-pulse TV holography,” Meas. Sci. Technol. 13, 438–444 (2002).

Chantler, C. T.

D. Paterson, B. E. Allman, P. J. McMahon, J. J. A. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, D. C. Mancini, “Spatial coherence measurement of x-ray undulator radiation,” Opt. Commun. 195, 79–84 (2001).
[CrossRef]

Charalambous, P.

J. W. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature (London) 400, 342–344 (1999).
[CrossRef]

Cloetens, P.

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

Cookson, D. F.

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

Dorrio, B. V.

For example, D. Cernadas, C. Trillo, A. F. Doval, J. C. Lopez, B. V. Dorrio, J. L. Fernandez, M. Perez-Amor, “Non-destructive testing with surface acoustic waves using double-pulse TV holography,” Meas. Sci. Technol. 13, 438–444 (2002).

Doval, A. F.

For example, D. Cernadas, C. Trillo, A. F. Doval, J. C. Lopez, B. V. Dorrio, J. L. Fernandez, M. Perez-Amor, “Non-destructive testing with surface acoustic waves using double-pulse TV holography,” Meas. Sci. Technol. 13, 438–444 (2002).

Erdmann, M.

I. McNulty, D. Paterson, J. Arko, M. Erdmann, S. P. Frigo, K. Goetze, P. Ilinski, N. Krapf, T. Mooney, C. C. Retsch, A. P. J. Stampfl, S. Vogt, Y. Wang, S. Xu, “The 2-ID-B intermediate-energy scanning X-ray microscope at the APS,” J. Phys. IV 104, 11–15 (2003).

Fernandez, J. L.

For example, D. Cernadas, C. Trillo, A. F. Doval, J. C. Lopez, B. V. Dorrio, J. L. Fernandez, M. Perez-Amor, “Non-destructive testing with surface acoustic waves using double-pulse TV holography,” Meas. Sci. Technol. 13, 438–444 (2002).

Forbes, G. W.

Frigo, S. P.

I. McNulty, D. Paterson, J. Arko, M. Erdmann, S. P. Frigo, K. Goetze, P. Ilinski, N. Krapf, T. Mooney, C. C. Retsch, A. P. J. Stampfl, S. Vogt, Y. Wang, S. Xu, “The 2-ID-B intermediate-energy scanning X-ray microscope at the APS,” J. Phys. IV 104, 11–15 (2003).

Goetze, K.

I. McNulty, D. Paterson, J. Arko, M. Erdmann, S. P. Frigo, K. Goetze, P. Ilinski, N. Krapf, T. Mooney, C. C. Retsch, A. P. J. Stampfl, S. Vogt, Y. Wang, S. Xu, “The 2-ID-B intermediate-energy scanning X-ray microscope at the APS,” J. Phys. IV 104, 11–15 (2003).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985), pp. 199–200.

Greytak, T.

M. Sutton, S. G. J. Mochrie, T. Greytak, S. E. Nagler, L. E. Berman, G. A. Held, G. B. Stephenson, “Observation of speckle by diffraction with coherent x-rays,” Nature (London) 352, 608–610 (1991).
[CrossRef]

Guigay, J. P.

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

Gureyev., T. E.

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

Harvey, E.

Hayes, J. P.

Held, G. A.

M. Sutton, S. G. J. Mochrie, T. Greytak, S. E. Nagler, L. E. Berman, G. A. Held, G. B. Stephenson, “Observation of speckle by diffraction with coherent x-rays,” Nature (London) 352, 608–610 (1991).
[CrossRef]

Howells, M.

Hradil, Z.

M. Jezek, Z. Hradil, “Reconstruction of spatial, phase and coherence properties of light,” J. Opt. Soc. Am. A 8, 1407–1416 (2004).
[CrossRef]

Ilinski, P.

I. McNulty, D. Paterson, J. Arko, M. Erdmann, S. P. Frigo, K. Goetze, P. Ilinski, N. Krapf, T. Mooney, C. C. Retsch, A. P. J. Stampfl, S. Vogt, Y. Wang, S. Xu, “The 2-ID-B intermediate-energy scanning X-ray microscope at the APS,” J. Phys. IV 104, 11–15 (2003).

Irving, T. H. K.

D. Paterson, B. E. Allman, P. J. McMahon, J. J. A. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, D. C. Mancini, “Spatial coherence measurement of x-ray undulator radiation,” Opt. Commun. 195, 79–84 (2001).
[CrossRef]

Jacobsen, C.

Jezek, M.

M. Jezek, Z. Hradil, “Reconstruction of spatial, phase and coherence properties of light,” J. Opt. Soc. Am. A 8, 1407–1416 (2004).
[CrossRef]

Kirz, J.

J. W. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature (London) 400, 342–344 (1999).
[CrossRef]

C. Jacobsen, M. Howells, J. Kirz, S. Rothman, “X-ray holographic microscopy using photoresists,” J. Opt. Soc. Am. A 7, 1847–1861 (1990).
[CrossRef]

Krapf, N.

I. McNulty, D. Paterson, J. Arko, M. Erdmann, S. P. Frigo, K. Goetze, P. Ilinski, N. Krapf, T. Mooney, C. C. Retsch, A. P. J. Stampfl, S. Vogt, Y. Wang, S. Xu, “The 2-ID-B intermediate-energy scanning X-ray microscope at the APS,” J. Phys. IV 104, 11–15 (2003).

Kurtsiefer, C.

C. Kurtsiefer, T. Pfau, J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature (London) 386, 150–153 (1997).
[CrossRef]

Lai, B.

Lin, J. J. A.

D. Paterson, B. E. Allman, P. J. McMahon, J. J. A. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, D. C. Mancini, “Spatial coherence measurement of x-ray undulator radiation,” Opt. Commun. 195, 79–84 (2001).
[CrossRef]

Lopez, J. C.

For example, D. Cernadas, C. Trillo, A. F. Doval, J. C. Lopez, B. V. Dorrio, J. L. Fernandez, M. Perez-Amor, “Non-destructive testing with surface acoustic waves using double-pulse TV holography,” Meas. Sci. Technol. 13, 438–444 (2002).

Mancini, D. C.

D. Paterson, B. E. Allman, P. J. McMahon, J. J. A. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, D. C. Mancini, “Spatial coherence measurement of x-ray undulator radiation,” Opt. Commun. 195, 79–84 (2001).
[CrossRef]

Mancuso, A. P.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), p. 163.

Mcalister, D. F.

M. G. Raymer, M. Beck, D. F. Mcalister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

McMahon, P. J.

A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002).
[CrossRef]

D. Paterson, B. E. Allman, P. J. McMahon, J. J. A. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, D. C. Mancini, “Spatial coherence measurement of x-ray undulator radiation,” Opt. Commun. 195, 79–84 (2001).
[CrossRef]

McNulty, I.

C. Q. Tran, A. G. Peele, A. Roberts, K. A. Nugent, D. Paterson, I. McNulty, “Synchrotron beam coherence: a spatially resolved measurement,” Opt. Lett. 30, 204–206 (2005).
[CrossRef] [PubMed]

I. McNulty, D. Paterson, J. Arko, M. Erdmann, S. P. Frigo, K. Goetze, P. Ilinski, N. Krapf, T. Mooney, C. C. Retsch, A. P. J. Stampfl, S. Vogt, Y. Wang, S. Xu, “The 2-ID-B intermediate-energy scanning X-ray microscope at the APS,” J. Phys. IV 104, 11–15 (2003).

A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002).
[CrossRef]

D. Paterson, B. E. Allman, P. J. McMahon, J. J. A. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, D. C. Mancini, “Spatial coherence measurement of x-ray undulator radiation,” Opt. Commun. 195, 79–84 (2001).
[CrossRef]

Miao, J. W.

J. W. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature (London) 400, 342–344 (1999).
[CrossRef]

Mlynek, J.

C. Kurtsiefer, T. Pfau, J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature (London) 386, 150–153 (1997).
[CrossRef]

Mochrie, S. G. J.

M. Sutton, S. G. J. Mochrie, T. Greytak, S. E. Nagler, L. E. Berman, G. A. Held, G. B. Stephenson, “Observation of speckle by diffraction with coherent x-rays,” Nature (London) 352, 608–610 (1991).
[CrossRef]

Moldovan, N.

D. Paterson, B. E. Allman, P. J. McMahon, J. J. A. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, D. C. Mancini, “Spatial coherence measurement of x-ray undulator radiation,” Opt. Commun. 195, 79–84 (2001).
[CrossRef]

Mooney, T.

I. McNulty, D. Paterson, J. Arko, M. Erdmann, S. P. Frigo, K. Goetze, P. Ilinski, N. Krapf, T. Mooney, C. C. Retsch, A. P. J. Stampfl, S. Vogt, Y. Wang, S. Xu, “The 2-ID-B intermediate-energy scanning X-ray microscope at the APS,” J. Phys. IV 104, 11–15 (2003).

Nagler, S. E.

M. Sutton, S. G. J. Mochrie, T. Greytak, S. E. Nagler, L. E. Berman, G. A. Held, G. B. Stephenson, “Observation of speckle by diffraction with coherent x-rays,” Nature (London) 352, 608–610 (1991).
[CrossRef]

Nugent, K. A.

C. Q. Tran, A. G. Peele, A. Roberts, K. A. Nugent, D. Paterson, I. McNulty, “Synchrotron beam coherence: a spatially resolved measurement,” Opt. Lett. 30, 204–206 (2005).
[CrossRef] [PubMed]

A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002).
[CrossRef]

D. Paterson, B. E. Allman, P. J. McMahon, J. J. A. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, D. C. Mancini, “Spatial coherence measurement of x-ray undulator radiation,” Opt. Commun. 195, 79–84 (2001).
[CrossRef]

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

K. A. Nugent, “Wave field determination using 3-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992).
[CrossRef] [PubMed]

Paganin, D.

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

Paterson, D.

C. Q. Tran, A. G. Peele, A. Roberts, K. A. Nugent, D. Paterson, I. McNulty, “Synchrotron beam coherence: a spatially resolved measurement,” Opt. Lett. 30, 204–206 (2005).
[CrossRef] [PubMed]

I. McNulty, D. Paterson, J. Arko, M. Erdmann, S. P. Frigo, K. Goetze, P. Ilinski, N. Krapf, T. Mooney, C. C. Retsch, A. P. J. Stampfl, S. Vogt, Y. Wang, S. Xu, “The 2-ID-B intermediate-energy scanning X-ray microscope at the APS,” J. Phys. IV 104, 11–15 (2003).

A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002).
[CrossRef]

D. Paterson, B. E. Allman, P. J. McMahon, J. J. A. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, D. C. Mancini, “Spatial coherence measurement of x-ray undulator radiation,” Opt. Commun. 195, 79–84 (2001).
[CrossRef]

Peele, A. G.

Perez-Amor, M.

For example, D. Cernadas, C. Trillo, A. F. Doval, J. C. Lopez, B. V. Dorrio, J. L. Fernandez, M. Perez-Amor, “Non-destructive testing with surface acoustic waves using double-pulse TV holography,” Meas. Sci. Technol. 13, 438–444 (2002).

Pfau, T.

C. Kurtsiefer, T. Pfau, J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature (London) 386, 150–153 (1997).
[CrossRef]

Raymer, M. G.

M. G. Raymer, M. Beck, D. F. Mcalister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Retsch, C. C.

I. McNulty, D. Paterson, J. Arko, M. Erdmann, S. P. Frigo, K. Goetze, P. Ilinski, N. Krapf, T. Mooney, C. C. Retsch, A. P. J. Stampfl, S. Vogt, Y. Wang, S. Xu, “The 2-ID-B intermediate-energy scanning X-ray microscope at the APS,” J. Phys. IV 104, 11–15 (2003).

D. Paterson, B. E. Allman, P. J. McMahon, J. J. A. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, D. C. Mancini, “Spatial coherence measurement of x-ray undulator radiation,” Opt. Commun. 195, 79–84 (2001).
[CrossRef]

Roberts, A.

Rothman, S.

Sayre, D.

J. W. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature (London) 400, 342–344 (1999).
[CrossRef]

Schlenker, M.

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

Stampfl, A. P. J.

I. McNulty, D. Paterson, J. Arko, M. Erdmann, S. P. Frigo, K. Goetze, P. Ilinski, N. Krapf, T. Mooney, C. C. Retsch, A. P. J. Stampfl, S. Vogt, Y. Wang, S. Xu, “The 2-ID-B intermediate-energy scanning X-ray microscope at the APS,” J. Phys. IV 104, 11–15 (2003).

Stephenson, G. B.

M. Sutton, S. G. J. Mochrie, T. Greytak, S. E. Nagler, L. E. Berman, G. A. Held, G. B. Stephenson, “Observation of speckle by diffraction with coherent x-rays,” Nature (London) 352, 608–610 (1991).
[CrossRef]

Sutton, M.

M. Sutton, S. G. J. Mochrie, T. Greytak, S. E. Nagler, L. E. Berman, G. A. Held, G. B. Stephenson, “Observation of speckle by diffraction with coherent x-rays,” Nature (London) 352, 608–610 (1991).
[CrossRef]

Tran, C. Q.

Trillo, C.

For example, D. Cernadas, C. Trillo, A. F. Doval, J. C. Lopez, B. V. Dorrio, J. L. Fernandez, M. Perez-Amor, “Non-destructive testing with surface acoustic waves using double-pulse TV holography,” Meas. Sci. Technol. 13, 438–444 (2002).

Vogt, S.

I. McNulty, D. Paterson, J. Arko, M. Erdmann, S. P. Frigo, K. Goetze, P. Ilinski, N. Krapf, T. Mooney, C. C. Retsch, A. P. J. Stampfl, S. Vogt, Y. Wang, S. Xu, “The 2-ID-B intermediate-energy scanning X-ray microscope at the APS,” J. Phys. IV 104, 11–15 (2003).

Wang, Y.

I. McNulty, D. Paterson, J. Arko, M. Erdmann, S. P. Frigo, K. Goetze, P. Ilinski, N. Krapf, T. Mooney, C. C. Retsch, A. P. J. Stampfl, S. Vogt, Y. Wang, S. Xu, “The 2-ID-B intermediate-energy scanning X-ray microscope at the APS,” J. Phys. IV 104, 11–15 (2003).

Wolf, E.

E. Wolf, “Coherent mode new theory of partial coherence in the space-frequency domain. 1. Spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[CrossRef]

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), p. 163.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 522–526.

Wolf, K. B.

Xu, S.

I. McNulty, D. Paterson, J. Arko, M. Erdmann, S. P. Frigo, K. Goetze, P. Ilinski, N. Krapf, T. Mooney, C. C. Retsch, A. P. J. Stampfl, S. Vogt, Y. Wang, S. Xu, “The 2-ID-B intermediate-energy scanning X-ray microscope at the APS,” J. Phys. IV 104, 11–15 (2003).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D

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

J. Phys. IV

I. McNulty, D. Paterson, J. Arko, M. Erdmann, S. P. Frigo, K. Goetze, P. Ilinski, N. Krapf, T. Mooney, C. C. Retsch, A. P. J. Stampfl, S. Vogt, Y. Wang, S. Xu, “The 2-ID-B intermediate-energy scanning X-ray microscope at the APS,” J. Phys. IV 104, 11–15 (2003).

Meas. Sci. Technol.

For example, D. Cernadas, C. Trillo, A. F. Doval, J. C. Lopez, B. V. Dorrio, J. L. Fernandez, M. Perez-Amor, “Non-destructive testing with surface acoustic waves using double-pulse TV holography,” Meas. Sci. Technol. 13, 438–444 (2002).

Nature (London)

J. W. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature (London) 400, 342–344 (1999).
[CrossRef]

M. Sutton, S. G. J. Mochrie, T. Greytak, S. E. Nagler, L. E. Berman, G. A. Held, G. B. Stephenson, “Observation of speckle by diffraction with coherent x-rays,” Nature (London) 352, 608–610 (1991).
[CrossRef]

C. Kurtsiefer, T. Pfau, J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature (London) 386, 150–153 (1997).
[CrossRef]

Opt. Commun.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

D. Paterson, B. E. Allman, P. J. McMahon, J. J. A. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, D. C. Mancini, “Spatial coherence measurement of x-ray undulator radiation,” Opt. Commun. 195, 79–84 (2001).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

K. A. Nugent, “Wave field determination using 3-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992).
[CrossRef] [PubMed]

M. G. Raymer, M. Beck, D. F. Mcalister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

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

Other

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 522–526.

J. W. Goodman, Statistical Optics (Wiley, 1985), pp. 199–200.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), p. 163.

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Figures (13)

Fig. 1
Fig. 1

Diagram illustrating the phase-space tomography concept. The numbers in {} indicate the quadrant number referred to in the text. For a measurement with only positive z information, only quadrants {1} and {3} are measured.

Fig. 2
Fig. 2

Measured diffracted intensity of a beam defined by a single-slit aperture of 22 μm width. The intensity is plotted on a logarithmic scale.

Fig. 3
Fig. 3

PSD function reconstructed from the data in Fig. 2, describing a quasi-probability distribution for the light in terms of position and propagation direction. The propagation direction may be interpreted in terms of the transverse momentum of the photons in the beam.

Fig. 4
Fig. 4

(a) Reconstructed intensity distribution at the aperture, (b) reconstructed far-field intensity. Both reconstructions are indistinguishable from the measured data.

Fig. 5
Fig. 5

Isometric plot of the reconstructed MOI function for the beam. This plot is a transformation of the data in Fig. 3.

Fig. 6
Fig. 6

(a) Magnitude and (b) phase of the recovered complex degree of coherence. The nonzero phase arises from the asymmetric far-field distribution.

Fig. 7
Fig. 7

(a) Measured diffracted intensity of a beam defined by a double-slit aperture of 2.0 μm width and 10 μm separation. The measurements were carried out at more than 100 positions ranging from z 0 m to z = 1.688 m ; the intensity is plotted on a logarithmic scale. (b) Example of the Young’s interference fringes used to produce the data point in Fig. 11 below.

Fig. 8
Fig. 8

(a) PSD function reconstructed from the data in Fig. 7a. (b) Reconstruction of simulated data incorporating the finite-detector resolution. The agreement is very good.

Fig. 9
Fig. 9

Isometric plot of the reconstructed phase-space distribution. The negative quasi-probability regions are seen clearly.

Fig. 10
Fig. 10

Correlation information recovered from different points in the field. The fact that the data points lie on top of one another indicates that the form of the correlation information is independent of position.

Fig. 11
Fig. 11

Average of the results in Fig. 10 along with a Gaussian fit. The fit is clearly very good, indicating a source with Gaussian statistics. The solid circle indicates the degree of correlation from an independent Young’s interference experiment (Fig. 8).

Fig. 12
Fig. 12

Overlay of the measured Young’s fringes (diamonds) with the prediction using the known properties of the aperture and the measured coherence function. The bright central fringe arises from transmission through the bulk of the slit substrate. The only fitting parameter was the thickness of the substrate, and the recovered result is within the uncertainty of the known thickness.

Fig. 13
Fig. 13

(a) MOI of the Young’s fringe beam shown in Fig. 7. (b) Predicted fully coherent fringes obtained from the measured beam MOI (solid curve) along with the partially coherent experimental data (diamonds). The increase in fringe visibility is obvious.

Equations (31)

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Γ ( r 1 , r 2 , τ ) = E ( r 1 , t ) E * ( r 2 , t + τ ) ,
Γ ( r 1 , r 2 , τ ) J ( r 1 , r 2 ) exp ( 2 π i ν ¯ τ ) ,
B ( r , u ) = ( k 2 π ) 2 J ( r + x 2 , r x 2 ) exp ( i k u x ) d x ,
B ( r , u ) z = z 0 = B ( r z 0 u , u ) z = 0 ,
I ( r ) = B ( r , u ) d u .
J ( r 1 , r 2 ) = n = a n ψ n ( r 1 ) ψ n * ( r 2 ) ,
ψ n ( 1 ) ( x ) ψ n ( x , y ) d y ,
J ( 1 ) ( x 1 , x 2 ) J ( x 1 , y 1 , x 2 , y 2 ) d y 1 d y 2 .
J ( 1 ) ( x 1 , x 2 ) = n = a n ψ n ( x 1 , y 1 ) d y 1 ψ n * ( x 2 , y 2 ) d y 2 ,
J ( 1 ) ( x 1 , x 2 ) = n = a n ψ n ( 1 ) ( x 1 ) ψ n ( 1 ) * ( x 2 ) .
B ̂ ( p , q ) = ( k 2 π ) 2 B ( x , u ) exp [ i k ( p x + q u ) ] d x d u .
I ̂ ( p , z ) = B ̂ ( p , z p ) .
J out ( r 1 , r 2 ) = J in ( r 1 , r 2 ) T ( r 1 ) T * ( r 2 ) .
B out ( r , u ) = ( k 2 π ) 2 J in ( r + x 2 , r x 2 ) T ( r + x 2 ) T * ( r x 2 ) exp { i k u x } d x .
G ( r , u ) ( k 2 π ) 2 T ( r + x 2 ) T * ( r x 2 ) exp ( i k u x ) d x ,
B out ( r , u ) = B in ( r , u u ) G ( r , u ) d u .
d 2 λ z 1 ,
r 1 2 ( r 1 + r 2 ) ,
y r 1 r 2 ,
J ( r , y ) = B ( r , u ) exp ( i k u y ) d u ,
B ( x , u ) = B ̂ ( p , q ) exp { i k [ p x + q u ] } d p d q ,
J ( x , y ) = B ̂ ( p , y ) exp ( i k x p ) d p .
B ̂ meas ( p , y ) = { B ̂ ( p , y ) H ( + y ) , y 0 B ̂ ( p , y ) H ( y ) , y < 0 ,
H ( y ) = { 1 y 0 0 y < 0 .
J meas ( x , y ) = { 1 2 J ( x , y ) i J ( x , y ) x x d x y 0 1 2 J ( x , y ) + i J ( x , y ) x x d x y < 0 .
J ( r 1 , r 2 , z ) = J ( r 1 , r 2 , z ) exp [ i Φ 12 ( r 1 , r 2 , z ) ] ,
Φ 12 ( r 1 , r 2 , 0 ) = 0
I ( x , z > z max ) = z max z I ( z max z x , z max ) .
I ( x , z ) = z hi z z hi z lo I ( x , z lo ) + z z lo z hi z lo I ( x , z hi ) ,
I ( x , z ) = z hi z z hi z lo I ( z lo z x , z lo ) + z z lo z hi z lo I ( z hi z x , z hi ) .
γ ( x 1 , x 2 ) J ( x 1 , x 2 ) I ( x 1 ) I ( x 2 ) .

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