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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    [CrossRef]
  2. 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).
  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).
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  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. Kohler and P. Schafer, "Asymmetric Bragg reflection as magnifying optics," Cryst. Res. Technol. 37, 734-746 (2002).
    [CrossRef]
  7. D. Korytar, P. Mikulık, C. Ferrari, J. Hrdy, 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).
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  9. M. G. Honnicke 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]
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    [CrossRef] [PubMed]
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  12. 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. 37, 1262-1274 (2004).
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  14. J. P. Guigay, E. Pagot and P. Cloetens, "Fourier optics approach to X-ray analyser-based imaging," Opt. Commun. 270, 180-188 (2007).
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  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).
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  25. P. Modregger, D. Lubbert, P. Schafer, R. Kohler, 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]
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    [CrossRef]
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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

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Muller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Graber, 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. Lubbert, P. Schafer, R. Kohler, 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

P. Modregger, D. Lubbert, P. Schafer and R. Kohler, "Two dimensional diffraction enhanced imaging algorithm," Appl. Phys. Lett. 90, 193501 (2007).
[CrossRef]

M. G. Honnicke 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]

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. Lubbert, P. Schafer and R. Kohler, "Spatial resolution in Bragg-magnified X-ray images as determined by Fourier analysis," Phys. Status Solidi(a) 204, 2746-2752 (2007).
[CrossRef]

2006

P. Modregger, D. Lubbert, P. Schafer and R. Kohler, "Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory," Phys. Rev. B 74, 054107 (2006).
[CrossRef]

2005

Ya. I. Nesterets, T. E. Gureyev and S. W. Wilkins, "Polychromaticity in the combined propagationbased/ 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]

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]

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]

2004

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. 37, 1262-1274 (2004).
[CrossRef]

2003

D. Korytar, P. Mikulık, C. Ferrari, J. Hrdy, 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

R. Kohler and P. Schafer, "Asymmetric Bragg reflection as magnifying optics," Cryst. Res. Technol. 37, 734-746 (2002).
[CrossRef]

J. Keyrilainen, M. Fernandez and P. Suortti, "Refraction contrast in x-ray imaging," Nucl. Instrum. Meth. A 488, 419-427 (2002).
[CrossRef]

2001

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

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

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

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

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

M. Kuriyama,W. J. Boettinger and G. G. Cohen, "Synchrotron radiation topography," Annu. Rev. Mater. Sci. 12, 23-50 (1982).
[CrossRef]

1980

E. Forster, 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

B. Batterman and H. Cole, "Dynamical diffraction of x rays by perfect crystals," Rev. Mod. Phys. 36, 681-716 (1964).
[CrossRef]

1916

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]

Astolfo, A.

Banhart, J.

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Muller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Graber, 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. Muller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Graber, 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. Lubbert, P. Schafer, R. Kohler, 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]

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. Honnicke 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. Muller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Graber, 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. Keyrilainen, M. Fernandez and P. Suortti, "Refraction contrast in x-ray imaging," Nucl. Instrum. Meth. A 488, 419-427 (2002).
[CrossRef]

Ferrari, C.

D. Korytar, P. Mikulık, C. Ferrari, J. Hrdy, 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]

Forster, E.

E. Forster, K. Goetz and P. Zaumseil, "Double crystal diffractometry for the characterization of targets for laser fusion experiments," Krist. Tech. 15, 937-945 (1980).
[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. Muller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Graber, 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. Forster, K. Goetz and P. Zaumseil, "Double crystal diffractometry for the characterization of targets for laser fusion experiments," Krist. Tech. 15, 937-945 (1980).
[CrossRef]

Graber, H.

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Muller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Graber, 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.

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. 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]

Gureyev, T. E.

Ya. I. Nesterets, T. E. Gureyev and S. W. Wilkins, "Polychromaticity in the combined propagationbased/ 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. 37, 1262-1274 (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. Muller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Graber, 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. Muller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Graber, 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]

Honnicke, M. G.

M. G. Honnicke 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]

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]

Keyril¨ainen, J.

J. Keyrilainen, 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]

Kohler, R.

P. Modregger, D. Lubbert, P. Schafer, R. Kohler, 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. Lubbert, P. Schafer and R. Kohler, "Two dimensional diffraction enhanced imaging algorithm," Appl. Phys. Lett. 90, 193501 (2007).
[CrossRef]

P. Modregger, D. Lubbert, P. Schafer and R. Kohler, "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. Lubbert, P. Schafer and R. Kohler, "Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory," Phys. Rev. B 74, 054107 (2006).
[CrossRef]

R. Kohler and P. Schafer, "Asymmetric Bragg reflection as magnifying optics," Cryst. Res. Technol. 37, 734-746 (2002).
[CrossRef]

Korytar, D.

D. Korytar, P. Mikulık, C. Ferrari, J. Hrdy, 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]

Lubbert, D.

P. Modregger, D. Lubbert, P. Schafer, R. Kohler, 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. Lubbert, P. Schafer and R. Kohler, "Two dimensional diffraction enhanced imaging algorithm," Appl. Phys. Lett. 90, 193501 (2007).
[CrossRef]

P. Modregger, D. Lubbert, P. Schafer and R. Kohler, "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. Lubbert, P. Schafer and R. Kohler, "Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory," Phys. Rev. B 74, 054107 (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. Muller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Graber, 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]

Mikulik, P.

D. Korytar, P. Mikulık, C. Ferrari, J. Hrdy, 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. Muller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Graber, 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. Lubbert, P. Schafer, R. Kohler, 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. Lubbert, P. Schafer and R. Kohler, "Two dimensional diffraction enhanced imaging algorithm," Appl. Phys. Lett. 90, 193501 (2007).
[CrossRef]

P. Modregger, D. Lubbert, P. Schafer and R. Kohler, "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. Lubbert, P. Schafer and R. Kohler, "Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory," Phys. Rev. B 74, 054107 (2006).
[CrossRef]

Muller, B.

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Muller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Graber, 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 propagationbased/ 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. 37, 1262-1274 (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. 37, 1262-1274 (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. 37, 1262-1274 (2004).
[CrossRef]

Pfeiffer, F.

Rack, A.

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Muller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Graber, 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. Muller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Graber, 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]

Schafer, P.

P. Modregger, D. Lubbert, P. Schafer, R. Kohler, 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. Lubbert, P. Schafer and R. Kohler, "Two dimensional diffraction enhanced imaging algorithm," Appl. Phys. Lett. 90, 193501 (2007).
[CrossRef]

P. Modregger, D. Lubbert, P. Schafer and R. Kohler, "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. Lubbert, P. Schafer and R. Kohler, "Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory," Phys. Rev. B 74, 054107 (2006).
[CrossRef]

R. Kohler and P. Schafer, "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. Keyrilainen, 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. Muller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Graber, 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. Muller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Graber, 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. Lubbert, P. Schafer, R. Kohler, 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 propagationbased/ 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. 37, 1262-1274 (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]

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. Muller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Graber, 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. Forster, 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.

M. Kuriyama,W. J. Boettinger and G. G. Cohen, "Synchrotron radiation topography," Annu. Rev. Mater. Sci. 12, 23-50 (1982).
[CrossRef]

Appl. Phys. Lett.

P. Modregger, D. Lubbert, P. Schafer and R. Kohler, "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.

C. M. Sparrow, "On spectroscopic resolving power," Astrophys. J. 44, 76-86 (1916).
[CrossRef]

Cryst. Res. Technol.

R. Kohler and P. Schafer, "Asymmetric Bragg reflection as magnifying optics," Cryst. Res. Technol. 37, 734-746 (2002).
[CrossRef]

J. Phys. D: Appl. Phys.

D. Korytar, P. Mikulık, C. Ferrari, J. Hrdy, 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. 37, 1262-1274 (2004).
[CrossRef]

Ya. I. Nesterets, T. E. Gureyev and S. W. Wilkins, "Polychromaticity in the combined propagationbased/ 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.

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.

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.

E. Forster, 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

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

M. Stampanoni, G. Borchert and R. Abela, "Towards nanotomography with asymmetrically cut crystals," Nucl. Instrum. Meth. A 551, 119-124 (2005).
[CrossRef]

J. Keyrilainen, M. Fernandez and P. Suortti, "Refraction contrast in x-ray imaging," Nucl. Instrum. Meth. A 488, 419-427 (2002).
[CrossRef]

Nuovo Cimento

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.

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

Phys. Rev. B

P. Modregger, D. Lubbert, P. Schafer and R. Kohler, "Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory," Phys. Rev. B 74, 054107 (2006).
[CrossRef]

Phys. Rev. Lett.

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

P. Modregger, D. Lubbert, P. Schafer and R. Kohler, "Spatial resolution in Bragg-magnified X-ray images as determined by Fourier analysis," Phys. Status Solidi(a) 204, 2746-2752 (2007).
[CrossRef]

Proc. SPIE

A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Muller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Graber, 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.

B. Batterman and H. Cole, "Dynamical diffraction of x rays by perfect crystals," Rev. Mod. Phys. 36, 681-716 (1964).
[CrossRef]

Rev. Sci. Instrum.

M. G. Honnicke 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

A. Authier: Dynamical Theory of X-Ray Diffraction, Vol. 11 of IUCr Monographs on Crystallography, 2nd ed. (Oxford University Press, Oxford 2001).

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

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

J. Als-Niehlsen and D. McMorrow, Elements of Modern X-Ray Physics, (Wiley & Sons, 2001).

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Figures (12)

Fig. 1.
Fig. 1.

Scheme of the Bragg Magnifier. The arrows labeled s and p indicate the corresponding polarization directions. The mean propagation distances in the used experimental setup are d 1=35 mm sample to first analyzer, d 2=45 mm first to second analyzer and d 3=150 mm second analyzer to CCD camera.

Fig. 2.
Fig. 2.

The figure illustrates that for imaging the incident divergence σinc as seen from the sample is of importance - not the intrinsic angular size of the source. More generally, the spectral and angular distribution of the beam at the location of the sample (Ib α,λ); see also Sec. 4.3) defines the possible effects of partially coherent illumination. The dotted rectangles named DCM and BM indicate the position of the double crystal monochromator and the Bragg Magnifier. The kink of beam fan at the position of the Bragg Magnifier illustrates the fact that the divergence of the beam is reduced after asymmetric Bragg reflection according to Eq. (1).

Fig. 3.
Fig. 3.

Ewald sphere construction for two wave vectors with fixed incident direction but different wavelengths for an asymmetric Bragg reflection. α and α′ are the glancing incidence and exit angles of the reference beam (thick arrows) respectively. The exit angle of the wavelength corresponding to kk is defined by the intersection of its Ewald sphere and the crystal truncation rod (CTR). Its obvious that the exit angle depends on the wavelength. Thus, even a perfectly collimated but polychromatic beam will be divergent after asymmetric reflection, an effect that is well known from dynamical theory and may be called dispersion induced divergence.

Fig. 4.
Fig. 4.

Definition of quantities for the theoretical formalism: the angle of incidence α of the beam on the analyzer crystal, the coordinate system for the output wave field (x; z), the position on the analyzer surface s, the mean propagation distance between sample and analyzer crystal z 0 and the additional propagation distance z(x) that is dependent on the position of the analyzer surface. Note: Since the influence of free-space propagation after reflection is practically negligible, the observable wave field at the detector (not shown) equals the output wave field after reflection on the surface of the analyzer crystal.

Fig. 5.
Fig. 5.

Numerically calculated polychromatic response function (RF poly ) and the monochromatic response function (i.e. ζ=0) for the first reflection and a free-space propagation distance of 5 mm.

Fig. 6.
Fig. 6.

(a) Numerically calculated contrast of a Gaussian-shaped sample in dependence of the relative spectral width ζ. While the contrast clearly decreases with increasing ζ the general shape of the intensity distribution is preserved. (b) Contrast ratio between monochromatic and polychromatic case for the 224 reflection (first analyzer) and the 004 reflection (second analyzer) in dependence on ζ. While the full lines indicate the spectral widths corresponding to 50% contrast loss, the dashed lines reflect the typical experimental situation with a spectral width of ζ=1.3×10-4.

Fig. 7.
Fig. 7.

Spatial resolution of the Bragg Magnifier as determined by the Sparrow criterion for the monochromatic and dispersive cases. The numerical calculations were performed for the first reflection (Si-224, σ) at 8.048 keV, corresponding to a magnification factor of 40. The working point was chosen on the left slope. As expected the resolution decreases in the dispersive case but is still well in the sub-micrometer regime.

Fig. 8.
Fig. 8.

Intensity maps with the two working points ω 1 and ω 2 on the analyzer crystals of the Bragg Magnifier as parameters: (a) experiment and (b) theory. In order to improve the comparability a background value was added in (b) corresponding to a background present in the experiment.

Fig. 9.
Fig. 9.

Observable halfwidths of the experimental intensity map in Fig. 8(a). The FWHMs of each horizontal line (corresponding to the half width in ω 1) and each vertical line (corresponding to the half width in ω 2) in the intensity map was determined. The resulting FWHMs are in each case shown in dependence of the other working point. It is clearly visible that the half width of one rocking curve depends on the working point position of the other rocking curve. This can be understood as a dispersive effect.

Fig. 10.
Fig. 10.

Schematic visualization of the integral in Eq. 32 used to explain the dependence of FWHMω 2 on ω 1 (bottom line in Fig. 9). In this case the integral describes a convolution of the second reflection curve |R̂2|2 with the function |W×R̂1|2. The figures show the intensity distribution of the quantities as indicated in the legend with increasing ω 1. It is obvious that the width of the function |W×R̂1|2 depends on the chosen working point ω 1.

Fig. 11.
Fig. 11.

Dispersion at the monochromator with asymmetric Bragg reflection (here: b<1). A fixed exit direction expressed with respect to the angular deviation ϕ from the reference beam is traced through the crystal via the vacuum dispersion surfaces. Δθd λ/λ tan θm denotes the angular offset for a symmetric reflection (i.e. b=1). The inset in the middle bottom part of the figure describes the relative orientations of the incident wave vector k0, the exit wave vector k⃗h and the lattice vector h⃗, respectively. Bearing in mind that an angular deviation on the exit side Δθout transforms to an angular deviation on the incident side Δθin via Δθin =bΔθout , it is possible to retrieve Eq. (34) and Eq. (35) from this figure.

Fig. 12.
Fig. 12.

Ray-tracing at one monochromator crystal.

Tables (2)

Tables Icon

Table 1. Experimental parameters of a typical monochromator and the Bragg Magnifier. The index of reflection and the corresponding polarization is given by hkl. θB is the Bragg angle, b the magnification factor, ωD the theoretical Darwin width and ζ is the corresponding relative spectral width (or spectral acceptance) according to Eq. (6). All values apply for a x-ray photon energy of 8:048 keV.

Tables Icon

Table 2. Spatial coherence lengths and incident divergences of two beamlines: the TOPO-TOMO beamline at ANKA [19] and the ID19 beamline at the ESRF [20]. The values apply for a wavelength of 1.5 Å and are valid under the assumption that no additional optical elements (e.g. windows) deteriorate the beam coherence.

Equations (40)

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σout=σinb
Δxver=σverd1+σverb(d2+d3){0.18μm(TOPOTOMO)0.008μm(ID19)
Δxhor=σhor(d1+d2)+σhorbd3{1.4μm(TOPOTOMO)0.07μm(ID19)
b=sinα′sinα=sin(θB+ρ)sin(θBρ)
Δθd=ΔλλtanθB
ζ=ωDcotθB.
Δθad=(11b)ΔλλtanθB
Δx=dΔθad=d (11b)ζtanθB.
Dout(x)= dq D̂in (q)R̂1(qK+ω1)exp(iqxiz0+z(x)2Kq2)
x=ssinαandz=s cos α
q(f)=KtanαK2tan2α2Kcosαf.
Dout(s)= d f P̂ (q(f))D̂in(q(f))eifs
P̂(f)=exp(iz02Kq(f)2)sinαcosαKq(f)R̂(q(f)K+ω1).
I(x)= d λ d Δ α Ib (Δα,λ) DΔα,λ(x)2
Δα=1Kϕx
Din (x)exp (iΔαKx)Din(x)
D̂in(q)D̂in(qq0).
q(f)=KtanαK2tan2α2Kcosαfq0
P̂Δα(f)=exp(iz02K(q(f)+q0)2)sinαcosαK(q(f)+q0)R̂(q(f)K+ω1Δα)
Δθλi=tan θi λλrefλref (i=1,2)
P̂Δα,λ(f)=exp(iz02K(q(f)+q0)2)sinαcosαK(q(f)+q0)R̂(q(f)K+ω1Δα+Δθλi).
Ib(Δα,λ)δD(Δα)W(λ)
P̂λ(f)=exp(iz02Kq(f)2)sinαcosαKq(f)R̂(q(f)K+ω1+Δθλ1).
Dλ(s)= d f P̂λ (f)D̂in(q(f))eifs
Idisp(s)= d λ W (λ)Dλ(s)2 .
RFpoly(s)= d λ W (λ)dfP̂λ(q(f))eifs2
h(x)=h1(1exp(x22σ2))
Din(x)=exp (ih(x)(n1)K)
Din(x)=1D̂in(q)=δD(q).
Iλ(x;λ)=R̂1(Δθλ1+ω1)2.
Iλ(λ)=R̂1(Δθλ1+ω1)2R̂2(Δθλ2+ω2)2,
I(ω1,ω2)= d λ W (λ)R̂1(ω1+Δθλ1)2R̂2(ω2+Δθλ2)2
FWHMout=FWHMWR12+FWHMR22
Δθm=bm(ϕ+Δλλtanθm)
ϕ′=bmϕ+(bm1)Δλλtanθm.
x′=1b1(x+z1ϕ)+z2ϕ′
x″=1b2(x′+z3ϕ′)+z4ϕ″
ϕ″=b2b1ϕ+(b2b11)ΔλλtanθB
Ib(ϕ,λ,x)=R̂c1(Δθ1)2R̂c2(Δθ2)2Is(ϕ″,λ,x″)
Δλλ=xzgcotθB

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