Abstract

We apply the method of phase-space tomography to reconstruct x-ray beams focused using a compound refractive lens. We show that it is possible to decouple the effect of aberrations in the optical system from the field and hence measure both them and the original field. We recover the complex coherence function and find that it is consistent with expectations.

© 2006 Optical Society of America

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  1. I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, 'High-resolution imaging by Fourier-transform x-ray holography,' Science 256, 1009-1012 (1992).
    [Crossref] [PubMed]
  2. J. W. Miao, P. Charalambous, J. Kirz, and D. Sayre, 'Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,' Nature 400, 342-344 (1999).
    [Crossref]
  3. P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, 'Phase objects in synchrotron radiation hard x-ray imaging,' J. Phys. D 29, 133-146 (1996).
    [Crossref]
  4. Z. H. Cai, B. Lai, W. B. Yun, I. McNulty, K. G. Huang, and T. P. Russell, 'Observation of x-ray speckle by coherent scattering at grazing-incidence,' Phys. Rev. Lett. 73, 82-85 (1994).
    [Crossref] [PubMed]
  5. V. Kohn, I. Snigireva, and A. Snigirev, 'Direct measurement of transverse coherence length of hard x rays from interference fringes,' Phys. Rev. Lett. 85, 2745-2748 (2000).
    [Crossref] [PubMed]
  6. D. Paterson, B. E. Allman, P. J. McMahon, J. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, and D. C. Mancini, 'Spatial coherence measurement of x-ray undulator radiation,' Opt. Commun. 195, 79-84 (2001).
    [Crossref]
  7. J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, and I. McNulty, 'Measurement of the spatial coherence function of undulator radiation using a phase mask,' Phys. Rev. Lett. 90, 074801 (2003).
    [Crossref] [PubMed]
  8. M. G. Raymer, M. Beck, and D. F. McAlister, 'Complex wave-field reconstruction using phase-space tomography,' Phys. Rev. Lett. 72, 1137-1140 (1994).
    [Crossref] [PubMed]
  9. C. Q. Tran, A. G. Peele, D. Paterson, A. Roberts, I. McNulty, and K. A. Nugent, 'Synchrotron beam coherence measured using phase-space tomography,' Opt. Lett. 30, 204-206 (2005).
    [Crossref] [PubMed]
  10. C. Q. Tran, A. G. Peele, D. Paterson, A. Roberts, I. McNulty, and K. A. Nugent, 'X-ray imaging: a generalized approach using phase-space tomography,' J. Opt. Soc. Am. A 22, 1691-1700 (2005).
    [Crossref]
  11. D. Paganin and K. A. Nugent, 'Noninterferometric phase imaging with partially coherent light,' Phys. Rev. Lett. 80, 2586-2589 (1998).
    [Crossref]
  12. K. A. Nugent and D. Paganin, 'Matter-wave phase measurement: a noninterferometric approach,' Phys. Rev. A 61, 063614 (2000).
    [Crossref]
  13. A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, 'A compound refractive lens for focusing high-energy x-rays,' Nature 384, 49-51 (1996).
    [Crossref]

2005 (2)

2003 (1)

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, and I. McNulty, 'Measurement of the spatial coherence function of undulator radiation using a phase mask,' Phys. Rev. Lett. 90, 074801 (2003).
[Crossref] [PubMed]

2001 (1)

D. Paterson, B. E. Allman, P. J. McMahon, J. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, and D. C. Mancini, 'Spatial coherence measurement of x-ray undulator radiation,' Opt. Commun. 195, 79-84 (2001).
[Crossref]

2000 (2)

K. A. Nugent and D. Paganin, 'Matter-wave phase measurement: a noninterferometric approach,' Phys. Rev. A 61, 063614 (2000).
[Crossref]

V. Kohn, I. Snigireva, and A. Snigirev, 'Direct measurement of transverse coherence length of hard x rays from interference fringes,' Phys. Rev. Lett. 85, 2745-2748 (2000).
[Crossref] [PubMed]

1999 (1)

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

1998 (1)

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

1996 (2)

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

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, 'A compound refractive lens for focusing high-energy x-rays,' Nature 384, 49-51 (1996).
[Crossref]

1994 (2)

Z. H. Cai, B. Lai, W. B. Yun, I. McNulty, K. G. Huang, and T. P. Russell, 'Observation of x-ray speckle by coherent scattering at grazing-incidence,' Phys. Rev. Lett. 73, 82-85 (1994).
[Crossref] [PubMed]

M. G. Raymer, M. Beck, and D. F. McAlister, 'Complex wave-field reconstruction using phase-space tomography,' Phys. Rev. Lett. 72, 1137-1140 (1994).
[Crossref] [PubMed]

1992 (1)

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, 'High-resolution imaging by Fourier-transform x-ray holography,' Science 256, 1009-1012 (1992).
[Crossref] [PubMed]

Allman, B. E.

D. Paterson, B. E. Allman, P. J. McMahon, J. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, and D. C. Mancini, 'Spatial coherence measurement of x-ray undulator radiation,' Opt. Commun. 195, 79-84 (2001).
[Crossref]

Anderson, E. H.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, 'High-resolution imaging by Fourier-transform x-ray holography,' Science 256, 1009-1012 (1992).
[Crossref] [PubMed]

Barrett, R.

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

Baruchel, J.

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

Beck, M.

M. G. Raymer, M. Beck, and D. F. McAlister, 'Complex wave-field reconstruction using phase-space tomography,' Phys. Rev. Lett. 72, 1137-1140 (1994).
[Crossref] [PubMed]

Cai, Z.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, and I. McNulty, 'Measurement of the spatial coherence function of undulator radiation using a phase mask,' Phys. Rev. Lett. 90, 074801 (2003).
[Crossref] [PubMed]

Cai, Z. H.

Z. H. Cai, B. Lai, W. B. Yun, I. McNulty, K. G. Huang, and T. P. Russell, 'Observation of x-ray speckle by coherent scattering at grazing-incidence,' Phys. Rev. Lett. 73, 82-85 (1994).
[Crossref] [PubMed]

Chantler, C. T.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, and I. McNulty, 'Measurement of the spatial coherence function of undulator radiation using a phase mask,' Phys. Rev. Lett. 90, 074801 (2003).
[Crossref] [PubMed]

D. Paterson, B. E. Allman, P. J. McMahon, J. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, and 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, and D. Sayre, 'Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,' Nature 400, 342-344 (1999).
[Crossref]

Cloetens, P.

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

Guigay, J. P.

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

Howells, M. R.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, 'High-resolution imaging by Fourier-transform x-ray holography,' Science 256, 1009-1012 (1992).
[Crossref] [PubMed]

Huang, K. G.

Z. H. Cai, B. Lai, W. B. Yun, I. McNulty, K. G. Huang, and T. P. Russell, 'Observation of x-ray speckle by coherent scattering at grazing-incidence,' Phys. Rev. Lett. 73, 82-85 (1994).
[Crossref] [PubMed]

Irving, T. H. K.

D. Paterson, B. E. Allman, P. J. McMahon, J. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, and D. C. Mancini, 'Spatial coherence measurement of x-ray undulator radiation,' Opt. Commun. 195, 79-84 (2001).
[Crossref]

Jacobsen, C.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, 'High-resolution imaging by Fourier-transform x-ray holography,' Science 256, 1009-1012 (1992).
[Crossref] [PubMed]

Kern, D. P.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, 'High-resolution imaging by Fourier-transform x-ray holography,' Science 256, 1009-1012 (1992).
[Crossref] [PubMed]

Kirz, J.

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

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, 'High-resolution imaging by Fourier-transform x-ray holography,' Science 256, 1009-1012 (1992).
[Crossref] [PubMed]

Kohn, V.

V. Kohn, I. Snigireva, and A. Snigirev, 'Direct measurement of transverse coherence length of hard x rays from interference fringes,' Phys. Rev. Lett. 85, 2745-2748 (2000).
[Crossref] [PubMed]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, 'A compound refractive lens for focusing high-energy x-rays,' Nature 384, 49-51 (1996).
[Crossref]

Lai, B.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, and I. McNulty, 'Measurement of the spatial coherence function of undulator radiation using a phase mask,' Phys. Rev. Lett. 90, 074801 (2003).
[Crossref] [PubMed]

Z. H. Cai, B. Lai, W. B. Yun, I. McNulty, K. G. Huang, and T. P. Russell, 'Observation of x-ray speckle by coherent scattering at grazing-incidence,' Phys. Rev. Lett. 73, 82-85 (1994).
[Crossref] [PubMed]

Lengeler, B.

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, 'A compound refractive lens for focusing high-energy x-rays,' Nature 384, 49-51 (1996).
[Crossref]

Lin, J.

D. Paterson, B. E. Allman, P. J. McMahon, J. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, and D. C. Mancini, 'Spatial coherence measurement of x-ray undulator radiation,' Opt. Commun. 195, 79-84 (2001).
[Crossref]

Lin, J. J. A.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, and I. McNulty, 'Measurement of the spatial coherence function of undulator radiation using a phase mask,' Phys. Rev. Lett. 90, 074801 (2003).
[Crossref] [PubMed]

Mancini, D. C.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, and I. McNulty, 'Measurement of the spatial coherence function of undulator radiation using a phase mask,' Phys. Rev. Lett. 90, 074801 (2003).
[Crossref] [PubMed]

D. Paterson, B. E. Allman, P. J. McMahon, J. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, and D. C. Mancini, 'Spatial coherence measurement of x-ray undulator radiation,' Opt. Commun. 195, 79-84 (2001).
[Crossref]

McAlister, D. F.

M. G. Raymer, M. Beck, and D. F. McAlister, 'Complex wave-field reconstruction using phase-space tomography,' Phys. Rev. Lett. 72, 1137-1140 (1994).
[Crossref] [PubMed]

McMahon, P. J.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, and I. McNulty, 'Measurement of the spatial coherence function of undulator radiation using a phase mask,' Phys. Rev. Lett. 90, 074801 (2003).
[Crossref] [PubMed]

D. Paterson, B. E. Allman, P. J. McMahon, J. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, and 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, D. Paterson, A. Roberts, I. McNulty, and K. A. Nugent, 'Synchrotron beam coherence measured using phase-space tomography,' Opt. Lett. 30, 204-206 (2005).
[Crossref] [PubMed]

C. Q. Tran, A. G. Peele, D. Paterson, A. Roberts, I. McNulty, and K. A. Nugent, 'X-ray imaging: a generalized approach using phase-space tomography,' J. Opt. Soc. Am. A 22, 1691-1700 (2005).
[Crossref]

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, and I. McNulty, 'Measurement of the spatial coherence function of undulator radiation using a phase mask,' Phys. Rev. Lett. 90, 074801 (2003).
[Crossref] [PubMed]

D. Paterson, B. E. Allman, P. J. McMahon, J. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, and D. C. Mancini, 'Spatial coherence measurement of x-ray undulator radiation,' Opt. Commun. 195, 79-84 (2001).
[Crossref]

Z. H. Cai, B. Lai, W. B. Yun, I. McNulty, K. G. Huang, and T. P. Russell, 'Observation of x-ray speckle by coherent scattering at grazing-incidence,' Phys. Rev. Lett. 73, 82-85 (1994).
[Crossref] [PubMed]

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, 'High-resolution imaging by Fourier-transform x-ray holography,' Science 256, 1009-1012 (1992).
[Crossref] [PubMed]

Miao, J. W.

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

Moldovan, N.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, and I. McNulty, 'Measurement of the spatial coherence function of undulator radiation using a phase mask,' Phys. Rev. Lett. 90, 074801 (2003).
[Crossref] [PubMed]

D. Paterson, B. E. Allman, P. J. McMahon, J. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, and D. C. Mancini, 'Spatial coherence measurement of x-ray undulator radiation,' Opt. Commun. 195, 79-84 (2001).
[Crossref]

Nugent, K. A.

C. Q. Tran, A. G. Peele, D. Paterson, A. Roberts, I. McNulty, and K. A. Nugent, 'X-ray imaging: a generalized approach using phase-space tomography,' J. Opt. Soc. Am. A 22, 1691-1700 (2005).
[Crossref]

C. Q. Tran, A. G. Peele, D. Paterson, A. Roberts, I. McNulty, and K. A. Nugent, 'Synchrotron beam coherence measured using phase-space tomography,' Opt. Lett. 30, 204-206 (2005).
[Crossref] [PubMed]

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, and I. McNulty, 'Measurement of the spatial coherence function of undulator radiation using a phase mask,' Phys. Rev. Lett. 90, 074801 (2003).
[Crossref] [PubMed]

D. Paterson, B. E. Allman, P. J. McMahon, J. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, and D. C. Mancini, 'Spatial coherence measurement of x-ray undulator radiation,' Opt. Commun. 195, 79-84 (2001).
[Crossref]

K. A. Nugent and D. Paganin, 'Matter-wave phase measurement: a noninterferometric approach,' Phys. Rev. A 61, 063614 (2000).
[Crossref]

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

Paganin, D.

K. A. Nugent and D. Paganin, 'Matter-wave phase measurement: a noninterferometric approach,' Phys. Rev. A 61, 063614 (2000).
[Crossref]

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

Paterson, D.

C. Q. Tran, A. G. Peele, D. Paterson, A. Roberts, I. McNulty, and K. A. Nugent, 'Synchrotron beam coherence measured using phase-space tomography,' Opt. Lett. 30, 204-206 (2005).
[Crossref] [PubMed]

C. Q. Tran, A. G. Peele, D. Paterson, A. Roberts, I. McNulty, and K. A. Nugent, 'X-ray imaging: a generalized approach using phase-space tomography,' J. Opt. Soc. Am. A 22, 1691-1700 (2005).
[Crossref]

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, and I. McNulty, 'Measurement of the spatial coherence function of undulator radiation using a phase mask,' Phys. Rev. Lett. 90, 074801 (2003).
[Crossref] [PubMed]

D. Paterson, B. E. Allman, P. J. McMahon, J. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, and D. C. Mancini, 'Spatial coherence measurement of x-ray undulator radiation,' Opt. Commun. 195, 79-84 (2001).
[Crossref]

Peele, A. G.

Raymer, M. G.

M. G. Raymer, M. Beck, and D. F. McAlister, 'Complex wave-field reconstruction using phase-space tomography,' Phys. Rev. Lett. 72, 1137-1140 (1994).
[Crossref] [PubMed]

Retsch, C. C.

D. Paterson, B. E. Allman, P. J. McMahon, J. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, and D. C. Mancini, 'Spatial coherence measurement of x-ray undulator radiation,' Opt. Commun. 195, 79-84 (2001).
[Crossref]

Roberts, A.

Russell, T. P.

Z. H. Cai, B. Lai, W. B. Yun, I. McNulty, K. G. Huang, and T. P. Russell, 'Observation of x-ray speckle by coherent scattering at grazing-incidence,' Phys. Rev. Lett. 73, 82-85 (1994).
[Crossref] [PubMed]

Sayre, D.

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

Schlenker, M.

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

Snigirev, A.

V. Kohn, I. Snigireva, and A. Snigirev, 'Direct measurement of transverse coherence length of hard x rays from interference fringes,' Phys. Rev. Lett. 85, 2745-2748 (2000).
[Crossref] [PubMed]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, 'A compound refractive lens for focusing high-energy x-rays,' Nature 384, 49-51 (1996).
[Crossref]

Snigireva, I.

V. Kohn, I. Snigireva, and A. Snigirev, 'Direct measurement of transverse coherence length of hard x rays from interference fringes,' Phys. Rev. Lett. 85, 2745-2748 (2000).
[Crossref] [PubMed]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, 'A compound refractive lens for focusing high-energy x-rays,' Nature 384, 49-51 (1996).
[Crossref]

Tran, C. Q.

Yun, W. B.

Z. H. Cai, B. Lai, W. B. Yun, I. McNulty, K. G. Huang, and T. P. Russell, 'Observation of x-ray speckle by coherent scattering at grazing-incidence,' Phys. Rev. Lett. 73, 82-85 (1994).
[Crossref] [PubMed]

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

J. Phys. D (1)

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

Nature (2)

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

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, 'A compound refractive lens for focusing high-energy x-rays,' Nature 384, 49-51 (1996).
[Crossref]

Opt. Commun. (1)

D. Paterson, B. E. Allman, P. J. McMahon, J. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, and D. C. Mancini, 'Spatial coherence measurement of x-ray undulator radiation,' Opt. Commun. 195, 79-84 (2001).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (1)

K. A. Nugent and D. Paganin, 'Matter-wave phase measurement: a noninterferometric approach,' Phys. Rev. A 61, 063614 (2000).
[Crossref]

Phys. Rev. Lett. (5)

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, and I. McNulty, 'Measurement of the spatial coherence function of undulator radiation using a phase mask,' Phys. Rev. Lett. 90, 074801 (2003).
[Crossref] [PubMed]

M. G. Raymer, M. Beck, and D. F. McAlister, 'Complex wave-field reconstruction using phase-space tomography,' Phys. Rev. Lett. 72, 1137-1140 (1994).
[Crossref] [PubMed]

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

Z. H. Cai, B. Lai, W. B. Yun, I. McNulty, K. G. Huang, and T. P. Russell, 'Observation of x-ray speckle by coherent scattering at grazing-incidence,' Phys. Rev. Lett. 73, 82-85 (1994).
[Crossref] [PubMed]

V. Kohn, I. Snigireva, and A. Snigirev, 'Direct measurement of transverse coherence length of hard x rays from interference fringes,' Phys. Rev. Lett. 85, 2745-2748 (2000).
[Crossref] [PubMed]

Science (1)

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, 'High-resolution imaging by Fourier-transform x-ray holography,' Science 256, 1009-1012 (1992).
[Crossref] [PubMed]

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Figures (12)

Fig. 1
Fig. 1

Geometry of a converging beam that was used to justify the coordinate change in Eq. (15).

Fig. 2
Fig. 2

Propagation of a PSD function through a focusing lens. The PSD functions of the incident beam (a) before and (b) after a perfect refractive lens and (c) after an aberrated lens. The propagation of the PSD function follows Eqs. (4, 12).

Fig. 3
Fig. 3

Schematic of the experimental arrangement.

Fig. 4
Fig. 4

(a) Logarithmic plot of the measured intensity as a function of lateral position ( x ) and longitudinal position ( z ) (a) for the higher-coherence data and (b) with the entrance slits set to fully open to reduce the coherence of the beam.

Fig. 5
Fig. 5

Phase-space distribution reconstructed at the focal plane from (a) high-coherence data and (b) lower coherence. Note the wider distribution along the u axis in (b) due to the lower coherence.

Fig. 6
Fig. 6

Correction for the effect of spherical aberration (a) Phase-space distribution just after the lens. (b) The effect of spherical aberration is modeled by a cubic function (solid curve) fitted to the u centroid of the data in (a) (squares). The deviation of the solid curve from the dashed line [the linear term in Eq. (32)] is due to the effect of spherical aberration.

Fig. 7
Fig. 7

PSD reconstructed at the focal plane after correction for lens aberrations for (a) the higher-coherence case and (b) the lower-coherence case.

Fig. 8
Fig. 8

Aberration-corrected intensity distributions at the focal plane of the 30 μ m entrance-slit width (solid curve) and of the 3 mm entrance-slit width (dashed curve).

Fig. 9
Fig. 9

Correction for the lens response function. (a) The PSD function of the undulator source obtained by deconvoluting the reconstructed PSD from the lens response function, shown as the solid curve in Fig. 8. (b) The intensity distributions at the focal plane before (dashed curve) and after (solid curve) the correction for the lens response function (dotted–dashed curve).

Fig. 10
Fig. 10

Reconstructed PSD function of the incident beam of a wide open entrance slit. This allows for the characterization of the entire undulator source.

Fig. 11
Fig. 11

(a) Real and (b) imaginary parts of the recovered MOI distribution of the incident beam.

Fig. 12
Fig. 12

One-dimensional coherence function of the incident beam. The HWHM of the Gaussian profile fit is ( 4.3 ± 0.7 ) μ m .

Equations (33)

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B ( r , u ) = ( k 2 π ) 2 J ( r + x 2 , r x 2 ) exp [ i k x u ] d x .
J z ( r 1 , r 2 ) = J 0 ( r 1 , r 2 ) exp { i k 2 z [ ( r 1 r 1 ) 2 ( r 2 r 2 ) 2 ] } d r 1 d r 2 ,
J z ( r + x 2 , r x 2 ) = exp ( i k 2 r x ) J 0 ( r + x 2 , r x 2 ) × exp ( i k z r x ) exp [ i k z ( r x + r x ) ] d r d x ,
r 1 r + x 2 ,
r 2 r x 2 ,
B z ( r , u ) = B 0 ( r z u , u ) .
I ( r ) = B ( r , u ) d u ,
I z ( r ) = B 0 ( r z u , u ) d u ,
r z u = const .
J z ( z s + x 2 , z s x 2 ) exp ( i k s x ) J 0 ( r + x 2 , r x 2 ) exp [ i k ( s x 1 z r x ) ] d r d x ,
B ( z s , u ) B 0 ( z ( s + u ) , s ) .
L ( r ) = P ( r ) exp ( i k r 2 2 f ) ,
J out ( r 1 , r 2 ) = J in ( r 1 , r 2 ) exp [ i k 2 f ( r 1 2 r 2 2 ) ] .
B out ( r , u ) = B in ( r , u + r f ) .
B z out ( r , u ) = B in 0 ( ( f z f ) r z u , u + r f ) .
r = f z f r ,
B z out ( r , u ) = B in 0 ( r z u , u + z f z r f ) .
B z out ( r , u ) B in 0 ( r f u , z f z r f ) .
u = 1 z ( f z z ) r .
z eff = z 2 f z .
J out ( r 1 , r 2 ) = J in ( r 1 , r 2 ) P ( r 1 ) P * ( r 2 ) ,
B out ( r , u ) = B in ( r , u ) G ( r , u u ) d u ,
G ( r , u ) ( k 2 π ) 2 P ( r + x 2 ) P * ( r x 2 ) exp ( i k x u ) d u .
L ( r ) = P ( r ) exp { i [ k r 2 2 f ϕ a b ( r ) ] } ,
J out ( r 1 , r 2 ) = J in ( r 1 , r 2 ) exp ( i k { 1 2 f ( r 1 2 r 2 2 ) [ ϕ a b ( r 1 ) ϕ a b ( r 2 ) ] } ) ,
B out ( r , u ) = ( k 2 π ) 2 J in ( r + x 2 , r x 2 ) exp ( i k { x ( u + r f ) [ ϕ a b ( r + x 2 ) ϕ a b ( r x 2 ) ] } ) d x .
ϕ a b ( r + x 2 ) ϕ a b ( r x 2 ) x ϕ a b ( r ) ,
B a b out ( r , u ) ( k 2 π ) 2 J in ( r + x 2 , r x 2 ) exp { i k [ x ( u ϕ a b ( r ) + r f ) ] } d x .
B a b out ( r , u ) B out ( r , u ϕ a b ( r ) ) .
ϕ ( r ) = 1 I ( r ) u B ( r , u ) d u .
B ( r , u ) B ( r , u u ave ) ,
u ave u B ( r , u ) d u ,
u = [ x f + b ( x f ) 3 ] .

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