Abstract

Image magnification via twofold asymmetric Bragg reflection (a setup called the ”Bragg Magnifier”) is a recently established technique allowing to achieve both sub-micrometer spatial resolution and phase contrast in X-ray imaging. The present article extends a previously developed theoretical formalism to account for partially coherent illumination. At a typical synchrotron setup polychromatic illumination is identified as the main influence of partial coherence and the implications on imaging characteristics are analyzed by numerical simulations. We show that contrast decreases by about 50% when compared to the monochromatic case, while sub-micrometer spatial resolution is preserved. The theoretical formalism is experimentally verified by correctly describing the dispersive interaction of the two orthogonal magnifier crystals, an effect that has to be taken into account for precise data evaluation.

© 2009 Optical Society of America

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References

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  3. T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S.W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
    [Crossref]
  4. V. N. Ingal and E. A. Beliaevskaya, “Imaging of biological objects in the plane-wave diffraction scheme,” Nuovo Cimento 19, 553–560 (1997).
    [Crossref]
  5. K. Kobayashi, K. Izumi, H. Kimura, S. Kimura, T. Ibuki, Y. Yokoyama, Y. Tsusaka, Y. Kagoshima, and J. Matsui, “X-ray phase-contrast imaging with submicron resolution by using extremely asymmetric Bragg diffractions,” Appl. Phys. Lett. 78, 132–134 (2001).
    [Crossref]
  6. R. Köhler and P. Schäfer, “Asymmetric Bragg reflection as magnifying optics,” Cryst. Res. Technol. 37, 734–746 (2002).
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  7. D. Korytár, P. Mikulík, C. Ferrari, J. Hrdý, T. Baumbach, A. Freund, and A. Kubena, “Two-dimensional x-ray magnification based on a monolithic beam conditioner,” J. Phys. D: Appl. Phys. 36, A65–A68 (2003).
    [Crossref]
  8. M. Stampanoni, G. Borchert, and R. Abela, “Towards nanotomography with asymmetrically cut crystals,” Nucl. Instrum. Meth. A 551, 119–124 (2005).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  15. A. Bravin, V. Mocella, P. Coan, A. Astolfo, and C. Ferrero, “A numerical wave-optical approach for the simulation of analyzer-based x-ray imaging,” Opt. Express 15, 5641–5648 (2007).
    [Crossref] [PubMed]
  16. Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, “Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging,” J. Phys. D: Appl. Phys. 38, 4259–4271 (2005).
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    [Crossref]
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    [Crossref]
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    [Crossref]
  21. P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–146 (1996).
    [Crossref]
  22. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
    [Crossref] [PubMed]
  23. M. Kuriyama, W. J. Boettinger, and G. G. Cohen, “Synchrotron radiation topography,” Annu. Rev. Mater. Sci. 12, 23–50 (1982).
    [Crossref]
  24. J. Als-Niehlsen and D. McMorrowElements of modern x-ray physics, Wiley & Sons (2001).
  25. P. Modregger, D. Lübbert, P. Schäfer, R. Köhler, T. Weitkamp, M. Hanke, and T. Baumbach, “Fresnel diffraction in the case of an inclined image plane,” Opt. Express 16, 5141–5149 (2008).
    [Crossref] [PubMed]
  26. J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, San Fransisco, pp. 106–110 (1968).
  27. P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Two dimensional diffraction enhanced imaging algorithm,” Appl. Phys. Lett. 90, 193501 (2007).
    [Crossref]
  28. E. WilsonFourier Series and Optical Transform Techniques in Contemporary Optics, Wiley & Sons (1995).
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    [Crossref]
  30. B. Batterman and H. Cole, “Dynamical diffraction of x rays by perfect crystals,” Rev. Mod. Phys. 36, 681–716 (1964).
    [Crossref]

2008 (2)

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Müller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Gräber, R. Heldele, B. Mayzel, J. Goebbels, and T. Baumbach, “The high resolution synchrotron-based imaging stations at the BAMline (BESSY) and TopoTomo (ANKA),” Proc. SPIE 7078, 70780X (2008).
[Crossref]

P. Modregger, D. Lübbert, P. Schäfer, R. Köhler, T. Weitkamp, M. Hanke, and T. Baumbach, “Fresnel diffraction in the case of an inclined image plane,” Opt. Express 16, 5141–5149 (2008).
[Crossref] [PubMed]

2007 (5)

A. Bravin, V. Mocella, P. Coan, A. Astolfo, and C. Ferrero, “A numerical wave-optical approach for the simulation of analyzer-based x-ray imaging,” Opt. Express 15, 5641–5648 (2007).
[Crossref] [PubMed]

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Two dimensional diffraction enhanced imaging algorithm,” Appl. Phys. Lett. 90, 193501 (2007).
[Crossref]

M. G. Hönnicke and C. Cusatis, “Analyzer-based x-ray phase-contrast microscopy combining channel-cut and asymmetrically cut crystals,” Rev. Sci. Instrum. 78, 113708 (2007).
[Crossref] [PubMed]

J.P. Guigay, E. Pagot, and P. Cloetens, “Fourier optics approach to X-ray analyser-based imaging,” Opt. Commun. 270, 180–188 (2007).
[Crossref]

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Spatial resolution in Bragg-magnified X-ray images as determined by Fourier analysis,” Phys. Status Solidi (a) 204, 2746–2752 (2007).
[Crossref]

2006 (1)

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory,” Phys. Rev. B 74054107 (2006).
[Crossref]

2005 (4)

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, “Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging,” J. Phys. D: Appl. Phys. 38, 4259–4271 (2005).
[Crossref]

M. Stampanoni, G. Borchert, and R. Abela, “Towards nanotomography with asymmetrically cut crystals,” Nucl. Instrum. Meth. A 551, 119–124 (2005).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[Crossref] [PubMed]

P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synch. Rad. 12, 241–245 (2005)
[Crossref]

2004 (1)

Ya.I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S.W. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D: Appl. Phys. 371262–1274 (2004).
[Crossref]

2003 (1)

D. Korytár, P. Mikulík, C. Ferrari, J. Hrdý, T. Baumbach, A. Freund, and A. Kubena, “Two-dimensional x-ray magnification based on a monolithic beam conditioner,” J. Phys. D: Appl. Phys. 36, A65–A68 (2003).
[Crossref]

2002 (2)

R. Köhler and P. Schäfer, “Asymmetric Bragg reflection as magnifying optics,” Cryst. Res. Technol. 37, 734–746 (2002).
[Crossref]

J. Keyriläinen, M. Fernandez, and P. Suortti, “Refraction contrast in x-ray imaging,” Nucl. Instrum. Meth. A 488, 419–427 (2002).
[Crossref]

2001 (2)

R. Spal, “Submicrometer resolution hard X-Ray holography with the asymmetric Bragg diffraction microscope,” Phys. Rev. Lett. 86, 3044–3047 (2001).
[Crossref] [PubMed]

K. Kobayashi, K. Izumi, H. Kimura, S. Kimura, T. Ibuki, Y. Yokoyama, Y. Tsusaka, Y. Kagoshima, and J. Matsui, “X-ray phase-contrast imaging with submicron resolution by using extremely asymmetric Bragg diffractions,” Appl. Phys. Lett. 78, 132–134 (2001).
[Crossref]

1997 (1)

V. N. Ingal and E. A. Beliaevskaya, “Imaging of biological objects in the plane-wave diffraction scheme,” Nuovo Cimento 19, 553–560 (1997).
[Crossref]

1996 (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: Appl. Phys. 29, 133–146 (1996).
[Crossref]

1995 (1)

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S.W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[Crossref]

1990 (1)

M. Kuriyama, R. C. Dobbyn, R. D. Spal, H. E. Burdette, and D.R. Black, “Hard x-ray microscope with submicrometer spatial resolution,” J. Res. Natl. Inst. Stand. Technol. 95, 559–574 (1990).

1982 (1)

M. Kuriyama, W. J. Boettinger, and G. G. Cohen, “Synchrotron radiation topography,” Annu. Rev. Mater. Sci. 12, 23–50 (1982).
[Crossref]

1980 (1)

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15, 937–945 (1980).
[Crossref]

1964 (1)

B. Batterman and H. Cole, “Dynamical diffraction of x rays by perfect crystals,” Rev. Mod. Phys. 36, 681–716 (1964).
[Crossref]

1916 (1)

C. M. Sparrow, “On spectroscopic resolving power,” Astrophys. J. 44, 76–86 (1916).
[Crossref]

Abela, R.

M. Stampanoni, G. Borchert, and R. Abela, “Towards nanotomography with asymmetrically cut crystals,” Nucl. Instrum. Meth. A 551, 119–124 (2005).
[Crossref]

Als-Niehlsen, J.

J. Als-Niehlsen and D. McMorrowElements of modern x-ray physics, Wiley & Sons (2001).

Astolfo, A.

Authier, A.

A. AuthierDynamical Theory of X-Ray Diffraction, Vol. 11 of IUCr Monographs on Crystallography, 2nd ed. (Oxford University Press, Oxford2001).

Banhart, J.

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Müller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Gräber, R. Heldele, B. Mayzel, J. Goebbels, and T. Baumbach, “The high resolution synchrotron-based imaging stations at the BAMline (BESSY) and TopoTomo (ANKA),” Proc. SPIE 7078, 70780X (2008).
[Crossref]

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: Appl. Phys. 29, 133–146 (1996).
[Crossref]

Baruchel, J.

P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synch. Rad. 12, 241–245 (2005)
[Crossref]

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

Batterman, B.

B. Batterman and H. Cole, “Dynamical diffraction of x rays by perfect crystals,” Rev. Mod. Phys. 36, 681–716 (1964).
[Crossref]

Baumbach, T.

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Müller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Gräber, R. Heldele, B. Mayzel, J. Goebbels, and T. Baumbach, “The high resolution synchrotron-based imaging stations at the BAMline (BESSY) and TopoTomo (ANKA),” Proc. SPIE 7078, 70780X (2008).
[Crossref]

P. Modregger, D. Lübbert, P. Schäfer, R. Köhler, T. Weitkamp, M. Hanke, and T. Baumbach, “Fresnel diffraction in the case of an inclined image plane,” Opt. Express 16, 5141–5149 (2008).
[Crossref] [PubMed]

D. Korytár, P. Mikulík, C. Ferrari, J. Hrdý, T. Baumbach, A. Freund, and A. Kubena, “Two-dimensional x-ray magnification based on a monolithic beam conditioner,” J. Phys. D: Appl. Phys. 36, A65–A68 (2003).
[Crossref]

Beliaevskaya, E. A.

V. N. Ingal and E. A. Beliaevskaya, “Imaging of biological objects in the plane-wave diffraction scheme,” Nuovo Cimento 19, 553–560 (1997).
[Crossref]

Black, D.R.

M. Kuriyama, R. C. Dobbyn, R. D. Spal, H. E. Burdette, and D.R. Black, “Hard x-ray microscope with submicrometer spatial resolution,” J. Res. Natl. Inst. Stand. Technol. 95, 559–574 (1990).

Boettinger, W. J.

M. Kuriyama, W. J. Boettinger, and G. G. Cohen, “Synchrotron radiation topography,” Annu. Rev. Mater. Sci. 12, 23–50 (1982).
[Crossref]

Borchert, G.

M. Stampanoni, G. Borchert, and R. Abela, “Towards nanotomography with asymmetrically cut crystals,” Nucl. Instrum. Meth. A 551, 119–124 (2005).
[Crossref]

Bravin, A.

A. Bravin, V. Mocella, P. Coan, A. Astolfo, and C. Ferrero, “A numerical wave-optical approach for the simulation of analyzer-based x-ray imaging,” Opt. Express 15, 5641–5648 (2007).
[Crossref] [PubMed]

P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synch. Rad. 12, 241–245 (2005)
[Crossref]

Burdette, H. E.

M. Kuriyama, R. C. Dobbyn, R. D. Spal, H. E. Burdette, and D.R. Black, “Hard x-ray microscope with submicrometer spatial resolution,” J. Res. Natl. Inst. Stand. Technol. 95, 559–574 (1990).

Cloetens, P.

J.P. Guigay, E. Pagot, and P. Cloetens, “Fourier optics approach to X-ray analyser-based imaging,” Opt. Commun. 270, 180–188 (2007).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[Crossref] [PubMed]

P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synch. Rad. 12, 241–245 (2005)
[Crossref]

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

Coan, P.

A. Bravin, V. Mocella, P. Coan, A. Astolfo, and C. Ferrero, “A numerical wave-optical approach for the simulation of analyzer-based x-ray imaging,” Opt. Express 15, 5641–5648 (2007).
[Crossref] [PubMed]

P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synch. Rad. 12, 241–245 (2005)
[Crossref]

Cohen, G. G.

M. Kuriyama, W. J. Boettinger, and G. G. Cohen, “Synchrotron radiation topography,” Annu. Rev. Mater. Sci. 12, 23–50 (1982).
[Crossref]

Cole, H.

B. Batterman and H. Cole, “Dynamical diffraction of x rays by perfect crystals,” Rev. Mod. Phys. 36, 681–716 (1964).
[Crossref]

Cusatis, C.

M. G. Hönnicke and C. Cusatis, “Analyzer-based x-ray phase-contrast microscopy combining channel-cut and asymmetrically cut crystals,” Rev. Sci. Instrum. 78, 113708 (2007).
[Crossref] [PubMed]

Danilewsky, A.

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Müller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Gräber, R. Heldele, B. Mayzel, J. Goebbels, and T. Baumbach, “The high resolution synchrotron-based imaging stations at the BAMline (BESSY) and TopoTomo (ANKA),” Proc. SPIE 7078, 70780X (2008).
[Crossref]

David, C.

Davis, T. J.

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S.W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[Crossref]

Diaz, A.

Dobbyn, R. C.

M. Kuriyama, R. C. Dobbyn, R. D. Spal, H. E. Burdette, and D.R. Black, “Hard x-ray microscope with submicrometer spatial resolution,” J. Res. Natl. Inst. Stand. Technol. 95, 559–574 (1990).

Fernandez, M.

J. Keyriläinen, M. Fernandez, and P. Suortti, “Refraction contrast in x-ray imaging,” Nucl. Instrum. Meth. A 488, 419–427 (2002).
[Crossref]

Ferrari, C.

D. Korytár, P. Mikulík, C. Ferrari, J. Hrdý, T. Baumbach, A. Freund, and A. Kubena, “Two-dimensional x-ray magnification based on a monolithic beam conditioner,” J. Phys. D: Appl. Phys. 36, A65–A68 (2003).
[Crossref]

Ferrero, C.

Fiedler, S.

P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synch. Rad. 12, 241–245 (2005)
[Crossref]

Förster, E.

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15, 937–945 (1980).
[Crossref]

Freund, A.

D. Korytár, P. Mikulík, C. Ferrari, J. Hrdý, T. Baumbach, A. Freund, and A. Kubena, “Two-dimensional x-ray magnification based on a monolithic beam conditioner,” J. Phys. D: Appl. Phys. 36, A65–A68 (2003).
[Crossref]

Gao, D.

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S.W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[Crossref]

Goebbels, J.

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Müller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Gräber, R. Heldele, B. Mayzel, J. Goebbels, and T. Baumbach, “The high resolution synchrotron-based imaging stations at the BAMline (BESSY) and TopoTomo (ANKA),” Proc. SPIE 7078, 70780X (2008).
[Crossref]

Goetz, K.

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15, 937–945 (1980).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, San Fransisco, pp. 106–110 (1968).

Gräber, H.

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Müller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Gräber, R. Heldele, B. Mayzel, J. Goebbels, and T. Baumbach, “The high resolution synchrotron-based imaging stations at the BAMline (BESSY) and TopoTomo (ANKA),” Proc. SPIE 7078, 70780X (2008).
[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: Appl. Phys. 29, 133–146 (1996).
[Crossref]

Guigay, J.P.

J.P. Guigay, E. Pagot, and P. Cloetens, “Fourier optics approach to X-ray analyser-based imaging,” Opt. Commun. 270, 180–188 (2007).
[Crossref]

Gureyev, T. E.

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, “Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging,” J. Phys. D: Appl. Phys. 38, 4259–4271 (2005).
[Crossref]

Ya.I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S.W. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D: Appl. Phys. 371262–1274 (2004).
[Crossref]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S.W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[Crossref]

Hanke, M.

Heldele, R.

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Müller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Gräber, R. Heldele, B. Mayzel, J. Goebbels, and T. Baumbach, “The high resolution synchrotron-based imaging stations at the BAMline (BESSY) and TopoTomo (ANKA),” Proc. SPIE 7078, 70780X (2008).
[Crossref]

Helfen, L.

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Müller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Gräber, R. Heldele, B. Mayzel, J. Goebbels, and T. Baumbach, “The high resolution synchrotron-based imaging stations at the BAMline (BESSY) and TopoTomo (ANKA),” Proc. SPIE 7078, 70780X (2008).
[Crossref]

Hönnicke, M. G.

M. G. Hönnicke and C. Cusatis, “Analyzer-based x-ray phase-contrast microscopy combining channel-cut and asymmetrically cut crystals,” Rev. Sci. Instrum. 78, 113708 (2007).
[Crossref] [PubMed]

Hrdý, J.

D. Korytár, P. Mikulík, C. Ferrari, J. Hrdý, T. Baumbach, A. Freund, and A. Kubena, “Two-dimensional x-ray magnification based on a monolithic beam conditioner,” J. Phys. D: Appl. Phys. 36, A65–A68 (2003).
[Crossref]

Ibuki, T.

K. Kobayashi, K. Izumi, H. Kimura, S. Kimura, T. Ibuki, Y. Yokoyama, Y. Tsusaka, Y. Kagoshima, and J. Matsui, “X-ray phase-contrast imaging with submicron resolution by using extremely asymmetric Bragg diffractions,” Appl. Phys. Lett. 78, 132–134 (2001).
[Crossref]

Ingal, V. N.

V. N. Ingal and E. A. Beliaevskaya, “Imaging of biological objects in the plane-wave diffraction scheme,” Nuovo Cimento 19, 553–560 (1997).
[Crossref]

Izumi, K.

K. Kobayashi, K. Izumi, H. Kimura, S. Kimura, T. Ibuki, Y. Yokoyama, Y. Tsusaka, Y. Kagoshima, and J. Matsui, “X-ray phase-contrast imaging with submicron resolution by using extremely asymmetric Bragg diffractions,” Appl. Phys. Lett. 78, 132–134 (2001).
[Crossref]

Kagoshima, Y.

K. Kobayashi, K. Izumi, H. Kimura, S. Kimura, T. Ibuki, Y. Yokoyama, Y. Tsusaka, Y. Kagoshima, and J. Matsui, “X-ray phase-contrast imaging with submicron resolution by using extremely asymmetric Bragg diffractions,” Appl. Phys. Lett. 78, 132–134 (2001).
[Crossref]

Keyriläinen, J.

J. Keyriläinen, M. Fernandez, and P. Suortti, “Refraction contrast in x-ray imaging,” Nucl. Instrum. Meth. A 488, 419–427 (2002).
[Crossref]

Kimura, H.

K. Kobayashi, K. Izumi, H. Kimura, S. Kimura, T. Ibuki, Y. Yokoyama, Y. Tsusaka, Y. Kagoshima, and J. Matsui, “X-ray phase-contrast imaging with submicron resolution by using extremely asymmetric Bragg diffractions,” Appl. Phys. Lett. 78, 132–134 (2001).
[Crossref]

Kimura, S.

K. Kobayashi, K. Izumi, H. Kimura, S. Kimura, T. Ibuki, Y. Yokoyama, Y. Tsusaka, Y. Kagoshima, and J. Matsui, “X-ray phase-contrast imaging with submicron resolution by using extremely asymmetric Bragg diffractions,” Appl. Phys. Lett. 78, 132–134 (2001).
[Crossref]

Kobayashi, K.

K. Kobayashi, K. Izumi, H. Kimura, S. Kimura, T. Ibuki, Y. Yokoyama, Y. Tsusaka, Y. Kagoshima, and J. Matsui, “X-ray phase-contrast imaging with submicron resolution by using extremely asymmetric Bragg diffractions,” Appl. Phys. Lett. 78, 132–134 (2001).
[Crossref]

Köhler, R.

P. Modregger, D. Lübbert, P. Schäfer, R. Köhler, T. Weitkamp, M. Hanke, and T. Baumbach, “Fresnel diffraction in the case of an inclined image plane,” Opt. Express 16, 5141–5149 (2008).
[Crossref] [PubMed]

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Two dimensional diffraction enhanced imaging algorithm,” Appl. Phys. Lett. 90, 193501 (2007).
[Crossref]

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Spatial resolution in Bragg-magnified X-ray images as determined by Fourier analysis,” Phys. Status Solidi (a) 204, 2746–2752 (2007).
[Crossref]

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory,” Phys. Rev. B 74054107 (2006).
[Crossref]

R. Köhler and P. Schäfer, “Asymmetric Bragg reflection as magnifying optics,” Cryst. Res. Technol. 37, 734–746 (2002).
[Crossref]

Korytár, D.

D. Korytár, P. Mikulík, C. Ferrari, J. Hrdý, T. Baumbach, A. Freund, and A. Kubena, “Two-dimensional x-ray magnification based on a monolithic beam conditioner,” J. Phys. D: Appl. Phys. 36, A65–A68 (2003).
[Crossref]

Kubena, A.

D. Korytár, P. Mikulík, C. Ferrari, J. Hrdý, T. Baumbach, A. Freund, and A. Kubena, “Two-dimensional x-ray magnification based on a monolithic beam conditioner,” J. Phys. D: Appl. Phys. 36, A65–A68 (2003).
[Crossref]

Kuriyama, M.

M. Kuriyama, R. C. Dobbyn, R. D. Spal, H. E. Burdette, and D.R. Black, “Hard x-ray microscope with submicrometer spatial resolution,” J. Res. Natl. Inst. Stand. Technol. 95, 559–574 (1990).

M. Kuriyama, W. J. Boettinger, and G. G. Cohen, “Synchrotron radiation topography,” Annu. Rev. Mater. Sci. 12, 23–50 (1982).
[Crossref]

Lübbert, D.

P. Modregger, D. Lübbert, P. Schäfer, R. Köhler, T. Weitkamp, M. Hanke, and T. Baumbach, “Fresnel diffraction in the case of an inclined image plane,” Opt. Express 16, 5141–5149 (2008).
[Crossref] [PubMed]

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Two dimensional diffraction enhanced imaging algorithm,” Appl. Phys. Lett. 90, 193501 (2007).
[Crossref]

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Spatial resolution in Bragg-magnified X-ray images as determined by Fourier analysis,” Phys. Status Solidi (a) 204, 2746–2752 (2007).
[Crossref]

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory,” Phys. Rev. B 74054107 (2006).
[Crossref]

Matsui, J.

K. Kobayashi, K. Izumi, H. Kimura, S. Kimura, T. Ibuki, Y. Yokoyama, Y. Tsusaka, Y. Kagoshima, and J. Matsui, “X-ray phase-contrast imaging with submicron resolution by using extremely asymmetric Bragg diffractions,” Appl. Phys. Lett. 78, 132–134 (2001).
[Crossref]

Mayzel, B.

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Müller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Gräber, R. Heldele, B. Mayzel, J. Goebbels, and T. Baumbach, “The high resolution synchrotron-based imaging stations at the BAMline (BESSY) and TopoTomo (ANKA),” Proc. SPIE 7078, 70780X (2008).
[Crossref]

McMorrow, D.

J. Als-Niehlsen and D. McMorrowElements of modern x-ray physics, Wiley & Sons (2001).

Mikulík, P.

D. Korytár, P. Mikulík, C. Ferrari, J. Hrdý, T. Baumbach, A. Freund, and A. Kubena, “Two-dimensional x-ray magnification based on a monolithic beam conditioner,” J. Phys. D: Appl. Phys. 36, A65–A68 (2003).
[Crossref]

Mocella, V.

Modregger, P.

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Müller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Gräber, R. Heldele, B. Mayzel, J. Goebbels, and T. Baumbach, “The high resolution synchrotron-based imaging stations at the BAMline (BESSY) and TopoTomo (ANKA),” Proc. SPIE 7078, 70780X (2008).
[Crossref]

P. Modregger, D. Lübbert, P. Schäfer, R. Köhler, T. Weitkamp, M. Hanke, and T. Baumbach, “Fresnel diffraction in the case of an inclined image plane,” Opt. Express 16, 5141–5149 (2008).
[Crossref] [PubMed]

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Two dimensional diffraction enhanced imaging algorithm,” Appl. Phys. Lett. 90, 193501 (2007).
[Crossref]

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Spatial resolution in Bragg-magnified X-ray images as determined by Fourier analysis,” Phys. Status Solidi (a) 204, 2746–2752 (2007).
[Crossref]

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory,” Phys. Rev. B 74054107 (2006).
[Crossref]

Müller, B.

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Müller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Gräber, R. Heldele, B. Mayzel, J. Goebbels, and T. Baumbach, “The high resolution synchrotron-based imaging stations at the BAMline (BESSY) and TopoTomo (ANKA),” Proc. SPIE 7078, 70780X (2008).
[Crossref]

Nesterets, Ya. I.

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, “Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging,” J. Phys. D: Appl. Phys. 38, 4259–4271 (2005).
[Crossref]

Nesterets, Ya.I.

Ya.I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S.W. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D: Appl. Phys. 371262–1274 (2004).
[Crossref]

Paganin, D.

Ya.I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S.W. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D: Appl. Phys. 371262–1274 (2004).
[Crossref]

Pagot, E.

J.P. Guigay, E. Pagot, and P. Cloetens, “Fourier optics approach to X-ray analyser-based imaging,” Opt. Commun. 270, 180–188 (2007).
[Crossref]

P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synch. Rad. 12, 241–245 (2005)
[Crossref]

Pavlov, K. M.

Ya.I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S.W. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D: Appl. Phys. 371262–1274 (2004).
[Crossref]

Pfeiffer, F.

Rack, A.

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Müller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Gräber, R. Heldele, B. Mayzel, J. Goebbels, and T. Baumbach, “The high resolution synchrotron-based imaging stations at the BAMline (BESSY) and TopoTomo (ANKA),” Proc. SPIE 7078, 70780X (2008).
[Crossref]

Riesemeier, H.

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Müller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Gräber, R. Heldele, B. Mayzel, J. Goebbels, and T. Baumbach, “The high resolution synchrotron-based imaging stations at the BAMline (BESSY) and TopoTomo (ANKA),” Proc. SPIE 7078, 70780X (2008).
[Crossref]

Schäfer, P.

P. Modregger, D. Lübbert, P. Schäfer, R. Köhler, T. Weitkamp, M. Hanke, and T. Baumbach, “Fresnel diffraction in the case of an inclined image plane,” Opt. Express 16, 5141–5149 (2008).
[Crossref] [PubMed]

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Two dimensional diffraction enhanced imaging algorithm,” Appl. Phys. Lett. 90, 193501 (2007).
[Crossref]

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Spatial resolution in Bragg-magnified X-ray images as determined by Fourier analysis,” Phys. Status Solidi (a) 204, 2746–2752 (2007).
[Crossref]

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory,” Phys. Rev. B 74054107 (2006).
[Crossref]

R. Köhler and P. Schäfer, “Asymmetric Bragg reflection as magnifying optics,” Cryst. Res. Technol. 37, 734–746 (2002).
[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: Appl. Phys. 29, 133–146 (1996).
[Crossref]

Spal, R.

R. Spal, “Submicrometer resolution hard X-Ray holography with the asymmetric Bragg diffraction microscope,” Phys. Rev. Lett. 86, 3044–3047 (2001).
[Crossref] [PubMed]

Spal, R. D.

M. Kuriyama, R. C. Dobbyn, R. D. Spal, H. E. Burdette, and D.R. Black, “Hard x-ray microscope with submicrometer spatial resolution,” J. Res. Natl. Inst. Stand. Technol. 95, 559–574 (1990).

Sparrow, C. M.

C. M. Sparrow, “On spectroscopic resolving power,” Astrophys. J. 44, 76–86 (1916).
[Crossref]

Stampanoni, M.

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[Crossref] [PubMed]

M. Stampanoni, G. Borchert, and R. Abela, “Towards nanotomography with asymmetrically cut crystals,” Nucl. Instrum. Meth. A 551, 119–124 (2005).
[Crossref]

Stevenson, A. W.

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S.W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[Crossref]

Suortti, P.

J. Keyriläinen, M. Fernandez, and P. Suortti, “Refraction contrast in x-ray imaging,” Nucl. Instrum. Meth. A 488, 419–427 (2002).
[Crossref]

Tsusaka, Y.

K. Kobayashi, K. Izumi, H. Kimura, S. Kimura, T. Ibuki, Y. Yokoyama, Y. Tsusaka, Y. Kagoshima, and J. Matsui, “X-ray phase-contrast imaging with submicron resolution by using extremely asymmetric Bragg diffractions,” Appl. Phys. Lett. 78, 132–134 (2001).
[Crossref]

Weidemann, G.

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Müller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Gräber, R. Heldele, B. Mayzel, J. Goebbels, and T. Baumbach, “The high resolution synchrotron-based imaging stations at the BAMline (BESSY) and TopoTomo (ANKA),” Proc. SPIE 7078, 70780X (2008).
[Crossref]

Weitkamp, T.

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Müller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Gräber, R. Heldele, B. Mayzel, J. Goebbels, and T. Baumbach, “The high resolution synchrotron-based imaging stations at the BAMline (BESSY) and TopoTomo (ANKA),” Proc. SPIE 7078, 70780X (2008).
[Crossref]

P. Modregger, D. Lübbert, P. Schäfer, R. Köhler, T. Weitkamp, M. Hanke, and T. Baumbach, “Fresnel diffraction in the case of an inclined image plane,” Opt. Express 16, 5141–5149 (2008).
[Crossref] [PubMed]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[Crossref] [PubMed]

Wilkins, S. W.

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, “Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging,” J. Phys. D: Appl. Phys. 38, 4259–4271 (2005).
[Crossref]

Wilkins, S.W.

Ya.I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S.W. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D: Appl. Phys. 371262–1274 (2004).
[Crossref]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S.W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[Crossref]

Wilson, E.

E. WilsonFourier Series and Optical Transform Techniques in Contemporary Optics, Wiley & Sons (1995).

Yokoyama, Y.

K. Kobayashi, K. Izumi, H. Kimura, S. Kimura, T. Ibuki, Y. Yokoyama, Y. Tsusaka, Y. Kagoshima, and J. Matsui, “X-ray phase-contrast imaging with submicron resolution by using extremely asymmetric Bragg diffractions,” Appl. Phys. Lett. 78, 132–134 (2001).
[Crossref]

Zabler, S.

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Müller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Gräber, R. Heldele, B. Mayzel, J. Goebbels, and T. Baumbach, “The high resolution synchrotron-based imaging stations at the BAMline (BESSY) and TopoTomo (ANKA),” Proc. SPIE 7078, 70780X (2008).
[Crossref]

Zaumseil, P.

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15, 937–945 (1980).
[Crossref]

Ziegler, E.

Annu. Rev. Mater. Sci. (1)

M. Kuriyama, W. J. Boettinger, and G. G. Cohen, “Synchrotron radiation topography,” Annu. Rev. Mater. Sci. 12, 23–50 (1982).
[Crossref]

Appl. Phys. Lett. (2)

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Two dimensional diffraction enhanced imaging algorithm,” Appl. Phys. Lett. 90, 193501 (2007).
[Crossref]

K. Kobayashi, K. Izumi, H. Kimura, S. Kimura, T. Ibuki, Y. Yokoyama, Y. Tsusaka, Y. Kagoshima, and J. Matsui, “X-ray phase-contrast imaging with submicron resolution by using extremely asymmetric Bragg diffractions,” Appl. Phys. Lett. 78, 132–134 (2001).
[Crossref]

Astrophys. J. (1)

C. M. Sparrow, “On spectroscopic resolving power,” Astrophys. J. 44, 76–86 (1916).
[Crossref]

Cryst. Res. Technol. (1)

R. Köhler and P. Schäfer, “Asymmetric Bragg reflection as magnifying optics,” Cryst. Res. Technol. 37, 734–746 (2002).
[Crossref]

J. Phys. D: Appl. Phys. (4)

D. Korytár, P. Mikulík, C. Ferrari, J. Hrdý, T. Baumbach, A. Freund, and A. Kubena, “Two-dimensional x-ray magnification based on a monolithic beam conditioner,” J. Phys. D: Appl. Phys. 36, A65–A68 (2003).
[Crossref]

Ya.I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S.W. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D: Appl. Phys. 371262–1274 (2004).
[Crossref]

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, “Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging,” J. Phys. D: Appl. Phys. 38, 4259–4271 (2005).
[Crossref]

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

J. Res. Natl. Inst. Stand. Technol. (1)

M. Kuriyama, R. C. Dobbyn, R. D. Spal, H. E. Burdette, and D.R. Black, “Hard x-ray microscope with submicrometer spatial resolution,” J. Res. Natl. Inst. Stand. Technol. 95, 559–574 (1990).

J. Synch. Rad. (1)

P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synch. Rad. 12, 241–245 (2005)
[Crossref]

Krist. Tech. (1)

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15, 937–945 (1980).
[Crossref]

Nature (1)

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S.W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[Crossref]

Nucl. Instrum. Meth. A (2)

M. Stampanoni, G. Borchert, and R. Abela, “Towards nanotomography with asymmetrically cut crystals,” Nucl. Instrum. Meth. A 551, 119–124 (2005).
[Crossref]

J. Keyriläinen, M. Fernandez, and P. Suortti, “Refraction contrast in x-ray imaging,” Nucl. Instrum. Meth. A 488, 419–427 (2002).
[Crossref]

Nuovo Cimento (1)

V. N. Ingal and E. A. Beliaevskaya, “Imaging of biological objects in the plane-wave diffraction scheme,” Nuovo Cimento 19, 553–560 (1997).
[Crossref]

Opt. Commun. (1)

J.P. Guigay, E. Pagot, and P. Cloetens, “Fourier optics approach to X-ray analyser-based imaging,” Opt. Commun. 270, 180–188 (2007).
[Crossref]

Opt. Express (3)

Phys. Rev. B (1)

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory,” Phys. Rev. B 74054107 (2006).
[Crossref]

Phys. Rev. Lett. (1)

R. Spal, “Submicrometer resolution hard X-Ray holography with the asymmetric Bragg diffraction microscope,” Phys. Rev. Lett. 86, 3044–3047 (2001).
[Crossref] [PubMed]

Phys. Status Solidi (a) (1)

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Spatial resolution in Bragg-magnified X-ray images as determined by Fourier analysis,” Phys. Status Solidi (a) 204, 2746–2752 (2007).
[Crossref]

Proc. SPIE (1)

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Müller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Gräber, R. Heldele, B. Mayzel, J. Goebbels, and T. Baumbach, “The high resolution synchrotron-based imaging stations at the BAMline (BESSY) and TopoTomo (ANKA),” Proc. SPIE 7078, 70780X (2008).
[Crossref]

Rev. Mod. Phys. (1)

B. Batterman and H. Cole, “Dynamical diffraction of x rays by perfect crystals,” Rev. Mod. Phys. 36, 681–716 (1964).
[Crossref]

Rev. Sci. Instrum. (1)

M. G. Hönnicke and C. Cusatis, “Analyzer-based x-ray phase-contrast microscopy combining channel-cut and asymmetrically cut crystals,” Rev. Sci. Instrum. 78, 113708 (2007).
[Crossref] [PubMed]

Other (4)

A. AuthierDynamical Theory of X-Ray Diffraction, Vol. 11 of IUCr Monographs on Crystallography, 2nd ed. (Oxford University Press, Oxford2001).

J. Als-Niehlsen and D. McMorrowElements of modern x-ray physics, Wiley & Sons (2001).

J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, San Fransisco, pp. 106–110 (1968).

E. WilsonFourier Series and Optical Transform Techniques in Contemporary Optics, Wiley & Sons (1995).

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