Abstract

An extension of the theoretical formalism of Fresnel diffraction to the case of an inclined image plane is proposed. The resulting numerical algorithm speeds up computation times by typically three orders of magnitude, thus opening the possibility of utilizing previously inapplicable image analysis algorithms for this special type of a non shift-invariant imaging system. This is exemplified by adapting an iterative phase retrieval algorithm developed for electron microscopy to the case of hard x-ray imaging with asymmetric Bragg reflection (the so-called “Bragg Magnifier”). Numerical simulations demonstrate the convergence and feasibility of the iterative phase retrieval algorithm for the case of x-ray imaging with the Bragg Magnifier.

© 2008 Optical Society of America

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  1. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge 7th ed. 1999)
  2. C. E. Metz and K. Doi, "Transfer function analysis of radiographic imaging systems," Phys. Med. Biol. 24, 1079-1106 (1979).
    [CrossRef] [PubMed]
  3. P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron raditaion x-rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
    [CrossRef]
  4. V. Yu. Ivanov, V. P. Sivokon, and M. A. Vorontsov, "Phase retrieval from a set of intensity measurements: theory and experiment," J. Opt. Soc. Am. A 9, 1515-1524 (1992).
    [CrossRef]
  5. W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, "Phase retrieval through focus variation for ultraresolution in field-emission transmission electron microscopy," Phys. Rev. Lett. 69, 3743-3746 (1992).
    [CrossRef] [PubMed]
  6. I. A. Cunningham and R. Shaw, "Signal-to-noise optimization of medical imaging systems," J. Opt. Soc. Am. A 16621-632 (1999).
    [CrossRef]
  7. A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, Y. Mori, K. Yamauchi, K. Yamamura, and A. Saito, "Deterministic retrieval of surface waviness by means of topography with coherent X-rays," J. Synchrotron Rad. 9, 223-228 (2002).
    [CrossRef]
  8. T. Panzner, G. Dleber, T. Sant, W. Leitenberger, and U. Pietsch, "Coherence experiments at the white-beam beamline of BESSY II," Thin Sol. Films 5155563-5567 (2007).
    [CrossRef]
  9. K. D. Mielenz, "Algorithms for Fresnel diffraction at rectangular and circular apertures," J. Res. Natl. Inst. Stand. Technol. 103497-509 (1998).
    [CrossRef]
  10. P. Modregger, D. Lubbert, P. Schafer, and R. Kohler, "Magnified phase imaging using asymmetric Bragg reflection: experiment and theory," Phys. Rev. B 74 054107-1-054107-10 (2006).
    [CrossRef]
  11. N. Delen and B. Hooker, "Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach," J. Opt. Soc. Am. A 15, 857-867 (1998).
    [CrossRef]
  12. N. Delen and B. Hooker, "Verification and comparison of a fast Fourier transform-based full diffraction method for tilted and offset planes," Appl. Opt. 40, 3525-3531 (2001).
    [CrossRef]
  13. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 422015-2025 (1997).
    [CrossRef] [PubMed]
  14. P. Modregger, D. Lubbert, P. Schafer, and R. Kohler, "Two dimensional diffraction enhanced imaging algorithm," Appl. Phys. Lett. 90, 193501-1-193501-3 (2007).
    [CrossRef]
  15. A. C. Kak and M. Slaney "Principles of Computerized Tomographic Imaging," IEEE Press. (1988).
  16. Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "Polychromaticity in the combined propagationbased/ analyser-based phase-contrast imaging," J. Appl. Phys. D: Appl. Phys. 384259-4271 (2005).
    [CrossRef]

2007

T. Panzner, G. Dleber, T. Sant, W. Leitenberger, and U. Pietsch, "Coherence experiments at the white-beam beamline of BESSY II," Thin Sol. Films 5155563-5567 (2007).
[CrossRef]

2005

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "Polychromaticity in the combined propagationbased/ analyser-based phase-contrast imaging," J. Appl. Phys. D: Appl. Phys. 384259-4271 (2005).
[CrossRef]

2002

A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, Y. Mori, K. Yamauchi, K. Yamamura, and A. Saito, "Deterministic retrieval of surface waviness by means of topography with coherent X-rays," J. Synchrotron Rad. 9, 223-228 (2002).
[CrossRef]

2001

1999

P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron raditaion x-rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

I. A. Cunningham and R. Shaw, "Signal-to-noise optimization of medical imaging systems," J. Opt. Soc. Am. A 16621-632 (1999).
[CrossRef]

1998

K. D. Mielenz, "Algorithms for Fresnel diffraction at rectangular and circular apertures," J. Res. Natl. Inst. Stand. Technol. 103497-509 (1998).
[CrossRef]

N. Delen and B. Hooker, "Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach," J. Opt. Soc. Am. A 15, 857-867 (1998).
[CrossRef]

1997

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 422015-2025 (1997).
[CrossRef] [PubMed]

1992

V. Yu. Ivanov, V. P. Sivokon, and M. A. Vorontsov, "Phase retrieval from a set of intensity measurements: theory and experiment," J. Opt. Soc. Am. A 9, 1515-1524 (1992).
[CrossRef]

W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, "Phase retrieval through focus variation for ultraresolution in field-emission transmission electron microscopy," Phys. Rev. Lett. 69, 3743-3746 (1992).
[CrossRef] [PubMed]

1979

C. E. Metz and K. Doi, "Transfer function analysis of radiographic imaging systems," Phys. Med. Biol. 24, 1079-1106 (1979).
[CrossRef] [PubMed]

Arfelli, F.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 422015-2025 (1997).
[CrossRef] [PubMed]

Baruchel, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron raditaion x-rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Chapman, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 422015-2025 (1997).
[CrossRef] [PubMed]

Cloetens, P.

P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron raditaion x-rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Coene, W.

W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, "Phase retrieval through focus variation for ultraresolution in field-emission transmission electron microscopy," Phys. Rev. Lett. 69, 3743-3746 (1992).
[CrossRef] [PubMed]

Cunningham, I. A.

Delen, N.

Dleber, G.

T. Panzner, G. Dleber, T. Sant, W. Leitenberger, and U. Pietsch, "Coherence experiments at the white-beam beamline of BESSY II," Thin Sol. Films 5155563-5567 (2007).
[CrossRef]

Doi, K.

C. E. Metz and K. Doi, "Transfer function analysis of radiographic imaging systems," Phys. Med. Biol. 24, 1079-1106 (1979).
[CrossRef] [PubMed]

Gmur, N.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 422015-2025 (1997).
[CrossRef] [PubMed]

Guigay, J. P.

P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron raditaion x-rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[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. Appl. Phys. D: Appl. Phys. 384259-4271 (2005).
[CrossRef]

Hooker, B.

Ishikawa, T.

A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, Y. Mori, K. Yamauchi, K. Yamamura, and A. Saito, "Deterministic retrieval of surface waviness by means of topography with coherent X-rays," J. Synchrotron Rad. 9, 223-228 (2002).
[CrossRef]

Ivanov, V. Yu.

Janssen, G.

W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, "Phase retrieval through focus variation for ultraresolution in field-emission transmission electron microscopy," Phys. Rev. Lett. 69, 3743-3746 (1992).
[CrossRef] [PubMed]

Johnston, R. E.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 422015-2025 (1997).
[CrossRef] [PubMed]

Kohler, R.

P. Modregger, D. Lubbert, P. Schafer, and R. Kohler, "Two dimensional diffraction enhanced imaging algorithm," Appl. Phys. Lett. 90, 193501-1-193501-3 (2007).
[CrossRef]

P. Modregger, D. Lubbert, P. Schafer, and R. Kohler, "Magnified phase imaging using asymmetric Bragg reflection: experiment and theory," Phys. Rev. B 74 054107-1-054107-10 (2006).
[CrossRef]

Leitenberger, W.

T. Panzner, G. Dleber, T. Sant, W. Leitenberger, and U. Pietsch, "Coherence experiments at the white-beam beamline of BESSY II," Thin Sol. Films 5155563-5567 (2007).
[CrossRef]

Lubbert, D.

P. Modregger, D. Lubbert, P. Schafer, and R. Kohler, "Magnified phase imaging using asymmetric Bragg reflection: experiment and theory," Phys. Rev. B 74 054107-1-054107-10 (2006).
[CrossRef]

P. Modregger, D. Lubbert, P. Schafer, and R. Kohler, "Two dimensional diffraction enhanced imaging algorithm," Appl. Phys. Lett. 90, 193501-1-193501-3 (2007).
[CrossRef]

Ludwig, W.

P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron raditaion x-rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Menk, R.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 422015-2025 (1997).
[CrossRef] [PubMed]

Metz, C. E.

C. E. Metz and K. Doi, "Transfer function analysis of radiographic imaging systems," Phys. Med. Biol. 24, 1079-1106 (1979).
[CrossRef] [PubMed]

Mielenz, K. D.

K. D. Mielenz, "Algorithms for Fresnel diffraction at rectangular and circular apertures," J. Res. Natl. Inst. Stand. Technol. 103497-509 (1998).
[CrossRef]

Modregger, P.

P. Modregger, D. Lubbert, P. Schafer, and R. Kohler, "Two dimensional diffraction enhanced imaging algorithm," Appl. Phys. Lett. 90, 193501-1-193501-3 (2007).
[CrossRef]

P. Modregger, D. Lubbert, P. Schafer, and R. Kohler, "Magnified phase imaging using asymmetric Bragg reflection: experiment and theory," Phys. Rev. B 74 054107-1-054107-10 (2006).
[CrossRef]

Mori, Y.

A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, Y. Mori, K. Yamauchi, K. Yamamura, and A. Saito, "Deterministic retrieval of surface waviness by means of topography with coherent X-rays," J. Synchrotron Rad. 9, 223-228 (2002).
[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. Appl. Phys. D: Appl. Phys. 384259-4271 (2005).
[CrossRef]

Op de Beeck, M.

W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, "Phase retrieval through focus variation for ultraresolution in field-emission transmission electron microscopy," Phys. Rev. Lett. 69, 3743-3746 (1992).
[CrossRef] [PubMed]

Panzner, T.

T. Panzner, G. Dleber, T. Sant, W. Leitenberger, and U. Pietsch, "Coherence experiments at the white-beam beamline of BESSY II," Thin Sol. Films 5155563-5567 (2007).
[CrossRef]

Pietsch, U.

T. Panzner, G. Dleber, T. Sant, W. Leitenberger, and U. Pietsch, "Coherence experiments at the white-beam beamline of BESSY II," Thin Sol. Films 5155563-5567 (2007).
[CrossRef]

Pisano, E.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 422015-2025 (1997).
[CrossRef] [PubMed]

Saito, A.

A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, Y. Mori, K. Yamauchi, K. Yamamura, and A. Saito, "Deterministic retrieval of surface waviness by means of topography with coherent X-rays," J. Synchrotron Rad. 9, 223-228 (2002).
[CrossRef]

Sant, T.

T. Panzner, G. Dleber, T. Sant, W. Leitenberger, and U. Pietsch, "Coherence experiments at the white-beam beamline of BESSY II," Thin Sol. Films 5155563-5567 (2007).
[CrossRef]

Sayers, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 422015-2025 (1997).
[CrossRef] [PubMed]

Sc¨afer, P.

P. Modregger, D. Lubbert, P. Schafer, and R. Kohler, "Two dimensional diffraction enhanced imaging algorithm," Appl. Phys. Lett. 90, 193501-1-193501-3 (2007).
[CrossRef]

Schafer, P.

P. Modregger, D. Lubbert, P. Schafer, and R. Kohler, "Magnified phase imaging using asymmetric Bragg reflection: experiment and theory," Phys. Rev. B 74 054107-1-054107-10 (2006).
[CrossRef]

Schlenker, M.

P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron raditaion x-rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Shaw, R.

Sivokon, V. P.

Souvorov, A.

A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, Y. Mori, K. Yamauchi, K. Yamamura, and A. Saito, "Deterministic retrieval of surface waviness by means of topography with coherent X-rays," J. Synchrotron Rad. 9, 223-228 (2002).
[CrossRef]

Tamasaku, K.

A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, Y. Mori, K. Yamauchi, K. Yamamura, and A. Saito, "Deterministic retrieval of surface waviness by means of topography with coherent X-rays," J. Synchrotron Rad. 9, 223-228 (2002).
[CrossRef]

Thomlinson, W.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 422015-2025 (1997).
[CrossRef] [PubMed]

van Dyck, D.

P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron raditaion x-rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, "Phase retrieval through focus variation for ultraresolution in field-emission transmission electron microscopy," Phys. Rev. Lett. 69, 3743-3746 (1992).
[CrossRef] [PubMed]

van Landuyt, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron raditaion x-rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Vorontsov, M. A.

Washburn, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 422015-2025 (1997).
[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. Appl. Phys. D: Appl. Phys. 384259-4271 (2005).
[CrossRef]

Yabashi, M.

A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, Y. Mori, K. Yamauchi, K. Yamamura, and A. Saito, "Deterministic retrieval of surface waviness by means of topography with coherent X-rays," J. Synchrotron Rad. 9, 223-228 (2002).
[CrossRef]

Yamamura, K.

A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, Y. Mori, K. Yamauchi, K. Yamamura, and A. Saito, "Deterministic retrieval of surface waviness by means of topography with coherent X-rays," J. Synchrotron Rad. 9, 223-228 (2002).
[CrossRef]

Yamauchi, K.

A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, Y. Mori, K. Yamauchi, K. Yamamura, and A. Saito, "Deterministic retrieval of surface waviness by means of topography with coherent X-rays," J. Synchrotron Rad. 9, 223-228 (2002).
[CrossRef]

Zhong, Z.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 422015-2025 (1997).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron raditaion x-rays," Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

J. Appl. Phys. D: Appl. Phys.

Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "Polychromaticity in the combined propagationbased/ analyser-based phase-contrast imaging," J. Appl. Phys. D: Appl. Phys. 384259-4271 (2005).
[CrossRef]

J. Opt. Soc. Am. A

J. Res. Natl. Inst. Stand. Technol.

K. D. Mielenz, "Algorithms for Fresnel diffraction at rectangular and circular apertures," J. Res. Natl. Inst. Stand. Technol. 103497-509 (1998).
[CrossRef]

J. Synchrotron Rad.

A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, Y. Mori, K. Yamauchi, K. Yamamura, and A. Saito, "Deterministic retrieval of surface waviness by means of topography with coherent X-rays," J. Synchrotron Rad. 9, 223-228 (2002).
[CrossRef]

Phys. Med. Biol.

C. E. Metz and K. Doi, "Transfer function analysis of radiographic imaging systems," Phys. Med. Biol. 24, 1079-1106 (1979).
[CrossRef] [PubMed]

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 422015-2025 (1997).
[CrossRef] [PubMed]

Phys. Rev. Lett.

W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, "Phase retrieval through focus variation for ultraresolution in field-emission transmission electron microscopy," Phys. Rev. Lett. 69, 3743-3746 (1992).
[CrossRef] [PubMed]

Thin Sol. Films

T. Panzner, G. Dleber, T. Sant, W. Leitenberger, and U. Pietsch, "Coherence experiments at the white-beam beamline of BESSY II," Thin Sol. Films 5155563-5567 (2007).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge 7th ed. 1999)

P. Modregger, D. Lubbert, P. Schafer, and R. Kohler, "Two dimensional diffraction enhanced imaging algorithm," Appl. Phys. Lett. 90, 193501-1-193501-3 (2007).
[CrossRef]

A. C. Kak and M. Slaney "Principles of Computerized Tomographic Imaging," IEEE Press. (1988).

P. Modregger, D. Lubbert, P. Schafer, and R. Kohler, "Magnified phase imaging using asymmetric Bragg reflection: experiment and theory," Phys. Rev. B 74 054107-1-054107-10 (2006).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

(Color online) (a) Experimental image of a copper mesh obtained with the Bragg Magnifier. (b) Intensity of the first and last period along the line indicated in (a). The different intensity distribution (see arrows) of identical objects features due to varying propagation distances within the image is clearly visible. The mean propagation distances between sample and the surface of the analyzer crystal are about 5mm for the first and 39mm for the last period, respectively.

Fig. 2.
Fig. 2.

Coordinate system for the theory.

Fig. 3.
Fig. 3.

(Color online) Simulated Fresnel diffraction in case of an inclined image plane (α=0.5°). The figure compares the intensity distribution of a double rectangular input wave field (5µm width; 10µm apart) as computed by many Fourier transforms for planes of constant propagation distance (here 512) plus interpolation and by the proposed algorithm according to equations (4)–(6). The effect of varying propagation distance is clearly visible. While the intensity distributions show negligible difference, the computation time of the proposed algorithm is reduced by a factor of about 200.

Fig. 4.
Fig. 4.

(Color online) (a) The Shepp-Logan phantom, which is used as a realistic model sample. It is illuminated in the direction indicated by the arrows. (b) Numerically simulated intensities after free-space propagation (mean distance 30mm) and asymmetric reflection according to eq. (9). The two intensity distributions corresponding to opposite slope position (top left; bottom right) of the reflection curve, are used as input for the iteration algorithm (see Fig. 5).

Fig. 5.
Fig. 5.

(Color online) (a) The comparison of the original and reconstructed phase distribution of the Shepp-Logan phantom (see Fig. 4a) shows a very good agreement. (b) The original intensity distribution on the right slope (same as bottom line in Fig. 4(b)) and the intensity distribution corresponding to the input wave field after reconstruction calculated according to eq. (9). The comparison shows the consistency of the reconstructed data with the input intensity.

Equations (11)

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D out ( x ) dq D ̂ in ( q ) exp ( i z + z 0 2 K q 2 ) exp ( iqx )
D out ( s ) dq D ̂ in ( q ) exp ( i z 0 2 K q 2 ) exp ( is ( q sin α cos α 2 K q 2 ) )
f = q sin α cos α 2 K q 2 .
q = q ( f ) = K tan α ± K 2 tan 2 α 2 K cos α f
D out ( s ) d f P ̂ ( f ) D ̂ in ( q ( f ) ) e ifs
P ̂ ( f ) = 1 sin α cos α K q ( f ) exp ( i z 0 2 K q ( f ) 2 ) .
D out ( s , y ) d f d p D ̂ in ( q ( f ) , p ) P ̂ ( f ) exp ( i f s + i p y i z 0 2 K p 2 ) .
P ̂ B M ( f ) = 1 sin α cos α K q ( f ) R ̂ ( q ( f ) K + Δ θ ) exp ( i z 0 2 K q ( f ) 2 )
I ( s ) = d f P ̂ B M ( q ( f ) ) D ̂ ( q ( f ) ) e i f s 2 .
S = m ds ( I m exp ( s ) I m ( s ) ) 2 .
D ̂ j + 1 ( f ) = D ̂ j ( f ) + γ M m = 1 M P ̂ m ( f ) F { ( I m exp ( s ) I m j ( s ) ) F 1 { P ̂ m ( f ) D ̂ j ( f ) } }

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