Abstract

What we believe to be a new illumination scheme that achieves space invariance and permits analytical determination of the image intensity distribution is proposed for partially coherent imaging systems. No additional lenses are needed for phase correction in front of the object and no restriction on illumination coherence is required. Conditions on the axial placement of the condenser with respect to the source and the object are specifically derived in terms of the condenser focal length and the distance from the object to the objective lens. By attaining space invariance, this new illumination scheme permits the use of transfer functions without approximation on illumination coherence. In addition, this illumination method establishes an exact Fourier transform relationship between the illumination and source mutual intensities, and thus significantly simplifies the analysis. Comparison with Köhler illumination is also presented. This apparatus is especially favorable for x-ray microscopy where lenses in this spectral region typically have a low efficiency.

© 2010 Optical Society of America

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References

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  1. D. A. Tichenor and J. W. Goodman, “Coherent transfer function,” J. Opt. Soc. Am. 62, 293–295 (1972).
    [CrossRef]
  2. M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge Univ. Press, 1997).
  3. J. W. Goodman, Statistical Optics (Wiley, 1985).
  4. C. Chang, P. Naulleau, and D. Attwood, “Analysis of illumination coherence properties in small-source systems such as synchrotrons,” Appl. Opt. 42, 2506–2512 (2003).
    [CrossRef] [PubMed]
  5. M. Wieland, T. Wilhein, C. Spielmann, and U. Kleineberg, “Zone-plate interferometry at 13 nm wavelength,” Appl. Phys. B 76, 885–889 (2003).
    [CrossRef]
  6. W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft X-ray microscopy at a spatial resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
    [CrossRef] [PubMed]
  7. D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102, 15343–15346 (2005).
    [CrossRef] [PubMed]
  8. S. Rehbein, S. Heim, P. Guttmann, S. Werner, and G. Schneider, “Ultrahigh-resolution soft-x-ray microscopy with zone plates in high orders of diffraction,” Phys. Rev. Lett. 103, 110801 (2009).
    [CrossRef] [PubMed]
  9. K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Raabe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99, 264801 (2007).
    [CrossRef]
  10. D. G. Fischer and T. D. Visser, “Spatial correlation properties of focused partially coherent light,” J. Opt. Soc. Am. A 21, 2097–2102 (2004).
    [CrossRef]
  11. G. Schneider, G. Denbeaux, E. H. Anderson, B. Bates, A. Pearson, M. A. Meyer, E. Zschech, D. Hambach, and E. A. Stach, “Dynamical x-ray microscopy investigation of electromigration in passivated inlaid Cu interconnect structures,” Appl. Phys. Lett. 81, 2535–2537 (2002).
    [CrossRef]
  12. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
    [CrossRef] [PubMed]
  13. 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, 258–261 (2006).
    [CrossRef]
  14. T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, S. Cabrini, and J. Susini, “Differential interference contrast x-ray microscopy with submicron resolution,” Appl. Phys. Lett. 78, 2082–2084 (2001).
    [CrossRef]
  15. C. Chang, A. Sakdinawat, P. Fischer, E. Anderson, and D. Attwood, “Single-element objective lens for soft x-ray differential interference contrast microscopy,” Opt. Lett. 31, 1564–1566 (2006).
    [CrossRef] [PubMed]
  16. O. von Hofsten, M. Bertilson, and U. Vogt, “Theoretical development of a high-resolution differential-interference-contrast optic for x-ray microscopy,” Opt. Express 16, 1132–1141 (2008).
    [CrossRef] [PubMed]
  17. A. Sakdinawat and Y. Liu, “Phase contrast soft x-ray microscopy using Zernike zone plates,” Opt. Express 16, 1559–1564 (2008).
    [CrossRef] [PubMed]

2009 (1)

S. Rehbein, S. Heim, P. Guttmann, S. Werner, and G. Schneider, “Ultrahigh-resolution soft-x-ray microscopy with zone plates in high orders of diffraction,” Phys. Rev. Lett. 103, 110801 (2009).
[CrossRef] [PubMed]

2008 (3)

2007 (1)

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Raabe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99, 264801 (2007).
[CrossRef]

2006 (2)

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, 258–261 (2006).
[CrossRef]

C. Chang, A. Sakdinawat, P. Fischer, E. Anderson, and D. Attwood, “Single-element objective lens for soft x-ray differential interference contrast microscopy,” Opt. Lett. 31, 1564–1566 (2006).
[CrossRef] [PubMed]

2005 (2)

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft X-ray microscopy at a spatial resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef] [PubMed]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102, 15343–15346 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (2)

C. Chang, P. Naulleau, and D. Attwood, “Analysis of illumination coherence properties in small-source systems such as synchrotrons,” Appl. Opt. 42, 2506–2512 (2003).
[CrossRef] [PubMed]

M. Wieland, T. Wilhein, C. Spielmann, and U. Kleineberg, “Zone-plate interferometry at 13 nm wavelength,” Appl. Phys. B 76, 885–889 (2003).
[CrossRef]

2002 (1)

G. Schneider, G. Denbeaux, E. H. Anderson, B. Bates, A. Pearson, M. A. Meyer, E. Zschech, D. Hambach, and E. A. Stach, “Dynamical x-ray microscopy investigation of electromigration in passivated inlaid Cu interconnect structures,” Appl. Phys. Lett. 81, 2535–2537 (2002).
[CrossRef]

2001 (1)

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, S. Cabrini, and J. Susini, “Differential interference contrast x-ray microscopy with submicron resolution,” Appl. Phys. Lett. 78, 2082–2084 (2001).
[CrossRef]

1972 (1)

Anderson, E.

Anderson, E. H.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft X-ray microscopy at a spatial resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef] [PubMed]

G. Schneider, G. Denbeaux, E. H. Anderson, B. Bates, A. Pearson, M. A. Meyer, E. Zschech, D. Hambach, and E. A. Stach, “Dynamical x-ray microscopy investigation of electromigration in passivated inlaid Cu interconnect structures,” Appl. Phys. Lett. 81, 2535–2537 (2002).
[CrossRef]

Attwood, D.

Attwood, D. T.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft X-ray microscopy at a spatial resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef] [PubMed]

Bates, B.

G. Schneider, G. Denbeaux, E. H. Anderson, B. Bates, A. Pearson, M. A. Meyer, E. Zschech, D. Hambach, and E. A. Stach, “Dynamical x-ray microscopy investigation of electromigration in passivated inlaid Cu interconnect structures,” Appl. Phys. Lett. 81, 2535–2537 (2002).
[CrossRef]

Beetz, T.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102, 15343–15346 (2005).
[CrossRef] [PubMed]

Bertilson, M.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge Univ. Press, 1997).

Bunk, O.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

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, 258–261 (2006).
[CrossRef]

Cabrini, S.

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, S. Cabrini, and J. Susini, “Differential interference contrast x-ray microscopy with submicron resolution,” Appl. Phys. Lett. 78, 2082–2084 (2001).
[CrossRef]

Chang, C.

Chao, W.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft X-ray microscopy at a spatial resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef] [PubMed]

David, C.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Raabe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99, 264801 (2007).
[CrossRef]

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, 258–261 (2006).
[CrossRef]

Denbeaux, G.

G. Schneider, G. Denbeaux, E. H. Anderson, B. Bates, A. Pearson, M. A. Meyer, E. Zschech, D. Hambach, and E. A. Stach, “Dynamical x-ray microscopy investigation of electromigration in passivated inlaid Cu interconnect structures,” Appl. Phys. Lett. 81, 2535–2537 (2002).
[CrossRef]

Di Fabrizio, E.

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, S. Cabrini, and J. Susini, “Differential interference contrast x-ray microscopy with submicron resolution,” Appl. Phys. Lett. 78, 2082–2084 (2001).
[CrossRef]

Dierolf, M.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

Elser, V.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102, 15343–15346 (2005).
[CrossRef] [PubMed]

Fischer, D. G.

Fischer, P.

Goodman, J. W.

Guttmann, P.

S. Rehbein, S. Heim, P. Guttmann, S. Werner, and G. Schneider, “Ultrahigh-resolution soft-x-ray microscopy with zone plates in high orders of diffraction,” Phys. Rev. Lett. 103, 110801 (2009).
[CrossRef] [PubMed]

Hambach, D.

G. Schneider, G. Denbeaux, E. H. Anderson, B. Bates, A. Pearson, M. A. Meyer, E. Zschech, D. Hambach, and E. A. Stach, “Dynamical x-ray microscopy investigation of electromigration in passivated inlaid Cu interconnect structures,” Appl. Phys. Lett. 81, 2535–2537 (2002).
[CrossRef]

Harteneck, B. D.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft X-ray microscopy at a spatial resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef] [PubMed]

Heim, S.

S. Rehbein, S. Heim, P. Guttmann, S. Werner, and G. Schneider, “Ultrahigh-resolution soft-x-ray microscopy with zone plates in high orders of diffraction,” Phys. Rev. Lett. 103, 110801 (2009).
[CrossRef] [PubMed]

Howells, M.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102, 15343–15346 (2005).
[CrossRef] [PubMed]

Jacobsen, C.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102, 15343–15346 (2005).
[CrossRef] [PubMed]

Jefimovs, K.

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Raabe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99, 264801 (2007).
[CrossRef]

Kaulich, B.

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, S. Cabrini, and J. Susini, “Differential interference contrast x-ray microscopy with submicron resolution,” Appl. Phys. Lett. 78, 2082–2084 (2001).
[CrossRef]

Kirz, J.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102, 15343–15346 (2005).
[CrossRef] [PubMed]

Kleineberg, U.

M. Wieland, T. Wilhein, C. Spielmann, and U. Kleineberg, “Zone-plate interferometry at 13 nm wavelength,” Appl. Phys. B 76, 885–889 (2003).
[CrossRef]

Liddle, J. A.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft X-ray microscopy at a spatial resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef] [PubMed]

Lima, E.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102, 15343–15346 (2005).
[CrossRef] [PubMed]

Liu, Y.

Menzel, A.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

Meyer, M. A.

G. Schneider, G. Denbeaux, E. H. Anderson, B. Bates, A. Pearson, M. A. Meyer, E. Zschech, D. Hambach, and E. A. Stach, “Dynamical x-ray microscopy investigation of electromigration in passivated inlaid Cu interconnect structures,” Appl. Phys. Lett. 81, 2535–2537 (2002).
[CrossRef]

Miao, H.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102, 15343–15346 (2005).
[CrossRef] [PubMed]

Naulleau, P.

Neiman, A.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102, 15343–15346 (2005).
[CrossRef] [PubMed]

Pearson, A.

G. Schneider, G. Denbeaux, E. H. Anderson, B. Bates, A. Pearson, M. A. Meyer, E. Zschech, D. Hambach, and E. A. Stach, “Dynamical x-ray microscopy investigation of electromigration in passivated inlaid Cu interconnect structures,” Appl. Phys. Lett. 81, 2535–2537 (2002).
[CrossRef]

Pfeiffer, F.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

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, 258–261 (2006).
[CrossRef]

Pilvi, T.

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Raabe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99, 264801 (2007).
[CrossRef]

Raabe, J.

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Raabe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99, 264801 (2007).
[CrossRef]

Rehbein, S.

S. Rehbein, S. Heim, P. Guttmann, S. Werner, and G. Schneider, “Ultrahigh-resolution soft-x-ray microscopy with zone plates in high orders of diffraction,” Phys. Rev. Lett. 103, 110801 (2009).
[CrossRef] [PubMed]

Ritala, M.

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Raabe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99, 264801 (2007).
[CrossRef]

Romanato, F.

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, S. Cabrini, and J. Susini, “Differential interference contrast x-ray microscopy with submicron resolution,” Appl. Phys. Lett. 78, 2082–2084 (2001).
[CrossRef]

Sakdinawat, A.

Sayre, D.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102, 15343–15346 (2005).
[CrossRef] [PubMed]

Schneider, G.

S. Rehbein, S. Heim, P. Guttmann, S. Werner, and G. Schneider, “Ultrahigh-resolution soft-x-ray microscopy with zone plates in high orders of diffraction,” Phys. Rev. Lett. 103, 110801 (2009).
[CrossRef] [PubMed]

G. Schneider, G. Denbeaux, E. H. Anderson, B. Bates, A. Pearson, M. A. Meyer, E. Zschech, D. Hambach, and E. A. Stach, “Dynamical x-ray microscopy investigation of electromigration in passivated inlaid Cu interconnect structures,” Appl. Phys. Lett. 81, 2535–2537 (2002).
[CrossRef]

Shapiro, D.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102, 15343–15346 (2005).
[CrossRef] [PubMed]

Spielmann, C.

M. Wieland, T. Wilhein, C. Spielmann, and U. Kleineberg, “Zone-plate interferometry at 13 nm wavelength,” Appl. Phys. B 76, 885–889 (2003).
[CrossRef]

Stach, E. A.

G. Schneider, G. Denbeaux, E. H. Anderson, B. Bates, A. Pearson, M. A. Meyer, E. Zschech, D. Hambach, and E. A. Stach, “Dynamical x-ray microscopy investigation of electromigration in passivated inlaid Cu interconnect structures,” Appl. Phys. Lett. 81, 2535–2537 (2002).
[CrossRef]

Susini, J.

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, S. Cabrini, and J. Susini, “Differential interference contrast x-ray microscopy with submicron resolution,” Appl. Phys. Lett. 78, 2082–2084 (2001).
[CrossRef]

Thibault, P.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102, 15343–15346 (2005).
[CrossRef] [PubMed]

Tichenor, D. A.

Vila-Comamala, J.

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Raabe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99, 264801 (2007).
[CrossRef]

Visser, T. D.

Vogt, U.

von Hofsten, O.

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, 258–261 (2006).
[CrossRef]

Werner, S.

S. Rehbein, S. Heim, P. Guttmann, S. Werner, and G. Schneider, “Ultrahigh-resolution soft-x-ray microscopy with zone plates in high orders of diffraction,” Phys. Rev. Lett. 103, 110801 (2009).
[CrossRef] [PubMed]

Wieland, M.

M. Wieland, T. Wilhein, C. Spielmann, and U. Kleineberg, “Zone-plate interferometry at 13 nm wavelength,” Appl. Phys. B 76, 885–889 (2003).
[CrossRef]

Wilhein, T.

M. Wieland, T. Wilhein, C. Spielmann, and U. Kleineberg, “Zone-plate interferometry at 13 nm wavelength,” Appl. Phys. B 76, 885–889 (2003).
[CrossRef]

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, S. Cabrini, and J. Susini, “Differential interference contrast x-ray microscopy with submicron resolution,” Appl. Phys. Lett. 78, 2082–2084 (2001).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge Univ. Press, 1997).

Zschech, E.

G. Schneider, G. Denbeaux, E. H. Anderson, B. Bates, A. Pearson, M. A. Meyer, E. Zschech, D. Hambach, and E. A. Stach, “Dynamical x-ray microscopy investigation of electromigration in passivated inlaid Cu interconnect structures,” Appl. Phys. Lett. 81, 2535–2537 (2002).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

M. Wieland, T. Wilhein, C. Spielmann, and U. Kleineberg, “Zone-plate interferometry at 13 nm wavelength,” Appl. Phys. B 76, 885–889 (2003).
[CrossRef]

Appl. Phys. Lett. (2)

G. Schneider, G. Denbeaux, E. H. Anderson, B. Bates, A. Pearson, M. A. Meyer, E. Zschech, D. Hambach, and E. A. Stach, “Dynamical x-ray microscopy investigation of electromigration in passivated inlaid Cu interconnect structures,” Appl. Phys. Lett. 81, 2535–2537 (2002).
[CrossRef]

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, S. Cabrini, and J. Susini, “Differential interference contrast x-ray microscopy with submicron resolution,” Appl. Phys. Lett. 78, 2082–2084 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

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

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, 258–261 (2006).
[CrossRef]

Nature (1)

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft X-ray microscopy at a spatial resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. Lett. (2)

S. Rehbein, S. Heim, P. Guttmann, S. Werner, and G. Schneider, “Ultrahigh-resolution soft-x-ray microscopy with zone plates in high orders of diffraction,” Phys. Rev. Lett. 103, 110801 (2009).
[CrossRef] [PubMed]

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Raabe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99, 264801 (2007).
[CrossRef]

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

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102, 15343–15346 (2005).
[CrossRef] [PubMed]

Science (1)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

Other (2)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge Univ. Press, 1997).

J. W. Goodman, Statistical Optics (Wiley, 1985).

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Figures (1)

Fig. 1
Fig. 1

Schematic diagram of a partially coherent imaging system.

Equations (47)

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J c ( x ̃ 1 , y ̃ 1 ; x ̃ 2 , y ̃ 2 ) = 1 ( λ z 1 ) 2 exp { j π λ z 1 [ x ̃ 2 2 + y ̃ 2 2 x ̃ 1 2 y ̃ 1 2 ] } + J s ( α 1 , β 1 ; α 2 , β 2 ) exp { j π λ z 1 [ α 2 2 + β 2 2 α 1 2 β 1 2 ] } exp { j 2 π λ z 1 [ x ̃ 2 α 2 + y ̃ 2 β 2 x ̃ 1 α 1 y ̃ 1 β 1 ] } d α 1 d β 1 d α 2 d β 2 ,
J c ( x ̃ 1 , y ̃ 1 ; x ̃ 2 , y ̃ 2 ) = J c ( x ̃ 1 , y ̃ 1 ; x ̃ 2 , y ̃ 2 ) exp { j π λ f c [ x ̃ 2 2 + y ̃ 2 2 x ̃ 1 2 y ̃ 1 2 ] } ,
J o ( ξ 1 , η 1 ; ξ 2 , η 2 ) = 1 ( λ z 2 ) 2 exp { j π λ z 2 [ ξ 2 2 + η 2 2 ξ 1 2 η 1 2 ] } + J c ( x ̃ 1 , y ̃ 1 ; x ̃ 2 , y ̃ 2 ) exp { j π λ z 2 [ x ̃ 2 2 + y ̃ 2 2 x ̃ 1 2 y ̃ 1 2 ] } exp { j 2 π λ z 2 [ ξ 2 x ̃ 2 + η 2 y ̃ 2 ξ 1 x ̃ 1 η 1 y ̃ 1 ] } d x ̃ 1 d y ̃ 1 d x ̃ 2 d y ̃ 2 ,
J o ( ξ 1 , η 1 ; ξ 2 , η 2 ) = 1 ( λ z 1 ) 2 ( λ z 2 ) 2 exp { j π λ z 2 [ ξ 2 2 + η 2 2 ξ 1 2 η 1 2 ] } + d α 1 d β 1 d α 2 d β 2 J s ( α 1 , β 1 ; α 2 , β 2 ) exp { j π λ z 1 [ α 2 2 + β 2 2 α 1 2 β 1 2 ] } + d x ̃ 1 d y ̃ 1 d x ̃ 2 d y ̃ 2   exp { j π λ [ 1 z 1 + 1 z 2 1 f c ] [ x ̃ 2 2 + y ̃ 2 2 x ̃ 1 2 y ̃ 1 2 ] } exp { j 2 π λ [ x ̃ 2 ( α 2 z 1 + ξ 2 z 2 ) + y ̃ 2 ( β 2 z 1 + η 2 z 2 ) x ̃ 1 ( α 1 z 1 + ξ 1 z 2 ) y ̃ 1 ( β 1 z 1 + η 1 z 2 ) ] } .
+ exp ( j π λ f x 2 ) exp ( ± j 2 π ν x ) d x = j λ f   exp ( ± j π λ f ν 2 ) ,
J o ( ξ 1 , η 1 ; ξ 2 , η 2 ) = 1 λ 2 z 1 2 z 2 2 ( 1 1 / z 1 + 1 / z 2 1 / f c ) 2 exp { j π λ z 2 [ ξ 2 2 + η 2 2 ξ 1 2 η 1 2 ] } exp { j π λ [ 1 1 z 1 + 1 z 2 1 f c ] [ 1 z 2 2 ( ξ 2 2 + η 2 2 ξ 1 2 η 1 2 ) ] } + d α 1 d β 1 d α 2 d β 2 J s ( α 1 , β 1 ; α 2 , β 2 ) exp { j π λ z 1 [ α 2 2 + β 2 2 α 1 2 β 1 2 ] } exp { j π λ [ 1 1 z 1 + 1 z 2 1 f c ] [ 1 z 1 2 ( α 2 2 + β 2 2 α 1 2 β 1 2 ) ] } exp { j π λ [ 1 1 z 1 + 1 z 2 1 f c ] [ 2 z 1 z 2 ( α 2 ξ 2 + β 2 η 2 α 1 ξ 1 β 1 η 1 ) ] } .
J i ( u 1 , v 1 ; u 2 , v 2 ) = + J o ( ξ 1 , η 1 ; ξ 2 , η 2 ) t o ( ξ 1 , η 1 ) t o ( ξ 2 , η 2 ) K ( u 1 , v 1 ; ξ 1 , η 1 ) K ( u 2 , v 2 ; ξ 2 , η 2 ) d ξ 1 d η 1 d ξ 2 d η 2 ,
K ( u , v ; ξ , η ) = exp [ j π λ z i ( u 2 + v 2 ) ] exp [ j π λ z o ( ξ 2 + η 2 ) ] λ 2 z i z o + P ( x , y ) exp { j 2 π λ z i [ ( u + z i z o ξ ) x + ( v + z i z o η ) y ] } d x d y ,
1 z 1 + 1 z 2 + z o = 1 f c .
J i ( u 1 , v 1 ; u 2 , v 2 ) = exp { j π λ z i ( u 1 2 + v 1 2 ) } exp { j π λ z i ( u 2 2 + v 2 2 ) } + d ξ 1 d η 1 d ξ 2 d η 2 J o , eff ( ξ 1 , η 1 ; ξ 2 , η 2 ) t o ( ξ 1 , η 1 ) t o ( ξ 2 , η 2 ) K eff ( u 1 , v 1 ; ξ 1 , η 1 ) K eff ( u 2 , v 2 ; ξ 2 , η 2 ) ,
J o , eff ( ξ 1 , η 1 ; ξ 2 , η 2 ) = 1 λ 2 z 1 2 z 2 2 ( 1 1 / z 2 1 / ( z 2 + z o ) ) 2 + d α 1 d β 1 d α 2 d β 2 J s ( α 1 , β 1 ; α 2 , β 2 ) exp { j π λ z 1 [ α 2 2 + β 2 2 α 1 2 β 1 2 ] } exp { j π λ [ 1 1 z 2 1 z 2 + z o ] [ 1 z 1 2 ( α 2 2 + β 2 2 α 1 2 β 1 2 ) ] } exp { j π λ [ 1 1 z 2 1 z 2 + z o ] [ 2 z 1 z 2 ( α 2 ξ 2 + β 2 η 2 α 1 ξ 1 β 1 η 1 ) ] } ,
K eff ( u , v ; ξ , η ) = 1 λ 2 z i z o + P ( x , y ) exp { j 2 π λ z i [ ( u + z i z o ξ ) x + ( v + z i z o η ) y ] } d x d y ,
K eff ( u , v ; ξ , η ) K ( u ξ , v η ) = 1 λ 2 z i z o + P ( x , y ) exp { j 2 π λ z i [ ( u ξ ) x + ( v η ) y ] } d x d y .
J i ( u 1 , v 1 ; u 2 , v 2 ) = + d ξ 1 d η 1 d ξ 2 d η 2 M 4 J o ( ξ 1 , η 1 ; ξ 2 , η 2 ) K ( u 1 ξ 1 , v 1 η 1 ) K ( u 2 ξ 2 , v 2 η 2 ) ,
J i ( ν 1 , ν 2 , ν 3 , ν 4 ) = J o ( ν 1 , ν 2 , ν 3 , ν 4 ) K ( ν 1 , ν 2 ) K ( ν 3 , ν 4 ) ,
K ( p , q ) = M P ( λ z o M p , λ z o M q )
z 2 = f c ,
z 1 = f c ( 1 + f c z o ) .
J o , eff ( ξ 1 , η 1 ; ξ 2 , η 2 ) = 1 ( λ f c ) 2 + J s ( α 1 , β 1 ; α 2 , β 2 ) exp { j 2 π λ f c [ α 2 ξ 2 + β 2 η 2 α 1 ξ 1 β 1 η 1 ] } d α 1 d β 1 d α 2 d β 2 .
J o ( p , q , r , s ) = ( M 2 λ f c ) 2 J s ( λ f c M p , λ f c M q , λ f c M r , λ f c M s ) .
J o ( ν 1 , ν 2 , ν 3 , ν 4 ) = + J o ( p , q , r , s ) T o ( ν 1 p , ν 2 q ) T o ( r ν 3 , s ν 4 ) d p d q d r d s ,
I i ( u , v ) = + d p d q d r d s J o ( p , q , r , s ) + d w 1 d w 2 T o ( w 1 , w 2 ) K ( w 1 + p , w 2 + q ) + d ν U d ν V   exp { j 2 π ( u ν U + v ν V ) } T o ( w 1 ν U + p + r , w 2 ν V + q + s ) K ( w 1 ν U + p , w 2 ν V + q ) ,
J s ( α 1 , β 1 ; α 2 , β 2 ) = κ I s ( α 1 , β 1 ) δ ( α 1 α 2 , β 1 β 2 ) ,
J o , eff ( Δ ξ , Δ η ) = κ ( λ f c ) 2 + I s ( α , β ) exp { j 2 π λ f c ( α Δ ξ + β Δ η ) } d α d β ,
I i ( u , v ) = + d p d q J o ( p , q ) + d w 1 d w 2 T o ( w 1 , w 2 ) K ( w 1 p , w 2 q ) + d ν U d ν V   exp { j 2 π ( u ν U + v ν V ) } T o ( w 1 ν U , w 2 ν V ) K ( w 1 p ν U , w 2 q ν V ) ,
J o ( p , q ) = κ M 2 I s ( λ f c M p , λ f c M q )
J o , eff ( ξ 1 , η 1 ; ξ 2 , η 2 ) = 1 ( λ f c ) 2 + J s ( α 1 , β 1 ; α 2 , β 2 ) exp { j 2 π λ f c [ α 2 ξ 2 + β 2 η 2 α 1 ξ 1 β 1 η 1 ] } d α 1 d β 1 d α 2 d β 2 .
J o , eff ( ξ 1 M , η 1 M ; ξ 2 M , η 2 M ) J o ( ξ 1 , η 1 ; ξ 2 , η 2 ) = 1 ( λ f c ) 2 + d α 1 d β 1 d α 2 d β 2 J s ( α 1 , β 1 ; α 2 , β 2 ) exp { j 1 M 2 π λ f c [ α 2 ξ 2 + β 2 η 2 α 1 ξ 1 β 1 η 1 ] } .
J o ( p , q , r , s ) = + d ξ 1 d η 1 d ξ 2 d η 2 J o ( ξ 1 , η 1 ; ξ 2 , η 2 ) exp { j 2 π ( ξ 1 p + η 1 q + ξ 2 r + η 2 s ) } = + d α 1 d β 1 d α 2 d β 2 J s ( α 1 , β 1 ; α 2 , β 2 ) + d ξ 1 d η 1 d ξ 2 d η 2   exp { j 2 π [ ξ 1 ( p + α 1 λ M f c ) + η 1 ( q + β 1 λ M f c ) + ξ 2 ( r α 2 λ M f c ) + η 2 ( s β 2 λ M f c ) ] } = ( M 2 λ f c ) 2 J s ( λ f c M p , λ f c M q , λ f c M r , λ f c M s ) .
J o ( ξ 1 , η 1 ; ξ 2 , η 2 ) = J o ( ξ 1 , η 1 ; ξ 2 , η 2 ) t o ( ξ 1 , η 1 ) t o ( ξ 2 , η 2 ) ,
J o ( ν 1 , ν 2 , ν 3 , ν 4 ) = + d ξ 1 d η 1 d ξ 2 d η 2 J o ( ξ 1 , η 1 ; ξ 2 , η 2 ) t o ( ξ 1 , η 1 ) t o ( ξ 2 , η 2 ) exp { j 2 π ( ξ 1 ν 1 + η 1 ν 2 + ξ 2 ν 3 + η 2 ν 4 ) } ,
J o ( ξ 1 , η 1 ; ξ 2 , η 2 ) = + d p d q d r d s J o ( p , q , r , s ) exp { j 2 π ( p ξ 1 + q η 1 + r ξ 2 + s η 2 ) } ,
t o ( ξ 1 , η 1 ) = + d α d β T o ( α , β ) exp { j 2 π ( α ξ 1 + β η 1 ) } ,
t o ( ξ 2 , η 2 ) = [ + d x d y T o ( x , y ) exp { j 2 π ( x ξ 2 + y η 2 ) } ] ,
J o ( ν 1 , ν 2 , ν 3 , ν 4 ) = + d p d q d r d s J o ( p , q , r , s ) + d α d β T o ( α , β ) + d x d y T o ( x , y ) + d ξ 1 d η 1 d ξ 2 d η 2   exp { j 2 π [ ξ 1 ( ν 1 p α ) + η 1 ( ν 2 q β ) + ξ 2 ( ν 3 r + x ) + η 2 ( ν 4 s + y ) ] } = + J o ( p , q , r , s ) T o ( ν 1 p , ν 2 q ) T o ( r ν 3 , s ν 4 ) d p d q d r d s .
I i ( u , v ) = + d ν 1 d ν 2 d ν 3 d ν 4   exp { j 2 π ( ν 1 u + ν 2 v + ν 3 u + ν 4 v ) } J o ( ν 1 , ν 2 , ν 3 , ν 4 ) K ( ν 1 , ν 2 ) K ( ν 3 , ν 4 ) = + d p d q d r d s J o ( p , q , r , s ) + d ν 1 d ν 2 d ν 3 d ν 4   exp { j 2 π [ u ( ν 1 + ν 3 ) + v ( ν 2 + ν 4 ) ] } K ( ν 1 , ν 2 ) T o ( ν 1 p , ν 2 q ) K ( ν 3 , ν 4 ) T o ( r ν 3 , s ν 4 ) ,
w 1 = ν 1 p ,
w 2 = ν 2 q ,
ν U = ν 1 + ν 3 ,
ν V = ν 2 + ν 4 ,
I i ( u , v ) = + d p d q d r d s J o ( p , q , r , s ) + d w 1 d w 2 T o ( w 1 , w 2 ) K ( w 1 + p , w 2 + q ) + d ν U d ν V   exp { j 2 π ( u ν U + v ν V ) } T o ( w 1 ν U + p + r , w 2 ν V + q + s ) K ( w 1 ν U + p , w 2 ν V + q ) .
J s ( α 1 , β 1 ; α 2 , β 2 ) = κ I s ( α 1 , β 1 ) δ ( α 1 α 2 , β 1 β 2 ) ,
J o ( p , q , r , s ) = κ ( M 2 λ f c ) 2 I s ( λ f c M p , λ f c M q ) δ ( λ f c M ( r + p ) , λ f c M ( s + q ) ) .
I i ( u , v ) = κ ( M 2 λ f c ) 2 + d p d q I s ( λ f c M p , λ f c M q ) + d w 1 d w 2 T o ( w 1 , w 2 ) K ( w 1 + p , w 2 + q ) + d ν U d ν V   exp { j 2 π ( u ν U + v ν V ) } K ( w 1 ν U + p , w 2 ν V + q ) + d r d s T o ( w 1 ν U + p + r , w 2 ν V + q + s ) δ ( λ f c M ( r + p ) , λ f c M ( s + q ) ) .
I i ( u , v ) = κ M 2 + d p d q I s ( λ f c M p , λ f c M q ) + d w 1 d w 2 T o ( w 1 , w 2 ) K ( w 1 + p , w 2 + q ) + d ν U d ν V   exp { j 2 π ( u ν U + v ν V ) } T o ( w 1 ν U , w 2 ν V ) K ( w 1 + p ν U , w 2 + q ν V ) .
I i ( u , v ) = + d p d q J o ( p , q ) + d w 1 d w 2 T o ( w 1 , w 2 ) K ( w 1 p , w 2 q ) + d ν U d ν V   exp { j 2 π ( u ν U + v ν V ) } T o ( w 1 ν U , w 2 ν V ) K ( w 1 p ν U , w 2 q ν V ) ,
J o ( p , q ) = κ M 2 I s ( λ f c M p , λ f c M q ) .

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