Abstract

We describe a method for characterizing focused x-ray beams using phase retrieval, with diversity achieved by transversely translating a phase-shifting or absorbing structure close to the beam focus. The required measurements can be taken with an experimental setup that is similar to that already used for fluorescent scan testing. The far-field intensity pattern is measured for each position of the translating structure, and the collected measurements are jointly used to estimate the beam profile by using a nonlinear optimization gradient search algorithm. The capability to reconstruct 1D and 2D beam foci is demonstrated through numerical simulations.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]
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  9. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett.  3, 27–29 (1978).
    [CrossRef] [PubMed]
  10. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt.  21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  11. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A  4, 118–123 (1987).
    [CrossRef]
  12. J. N. Cederquist, J. R. Fienup, J. C. Marron, and R. G. Paxman, “Phase retrieval from experimental far-field speckle data,” Opt. Lett.  13, 619–621 (1988).
    [CrossRef] [PubMed]
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    [CrossRef]
  14. S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B  68, 140101 (2003).
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    [CrossRef]
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    [CrossRef]
  27. M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
    [CrossRef] [PubMed]
  28. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
    [CrossRef] [PubMed]
  29. M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “Inspection of refractive x-ray lenses using high-resolution differential phase contrast imaging with a microfocus x-ray source,” Rev. Sci. Instrum.  78, 093707 (2007).
    [CrossRef] [PubMed]
  30. H. Mimura, H. Yumoto, S. Matsuyama, S. Handa, T. Kimura, Y. Sano, M. Yabashi, Y. Nishino, K. Tamasaku, and T. Ishikawaet al., “Direct determination of the wave field of an x-ray nanobeam,” Phys. Rev. A  77, 015812 (2008).
  31. H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett.  96, 127401 (2006).
    [CrossRef] [PubMed]
  32. A. Stein, K. Evans-Lutterodt, N. Bozovic, and A. Taylor, “Fabrication of silicon kinoform lenses for hard x-ray focusing by electron beam lithography and deep reactive ion etching,” J. Vac. Sci. Technol. B  26, 122–127 (2008).
    [CrossRef]
  33. H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Physics 2, 101–104 (2006).
    [CrossRef]
  34. Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun.  30, 304–308 (1979).
    [CrossRef]
  35. J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt.  36, 8352–8357 (1997).
    [CrossRef]
  36. M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett.  33, 156–158 (2008).
    [CrossRef] [PubMed]
  37. G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Opt. Express 17, 624–639 (2009).
    [CrossRef] [PubMed]
  38. S. T. Thurman, R. T. DeRosa, and J. R. Fienup, “Amplitude metrics for field retrieval with hard-edge and uniformly-illuminated apertures,” J. Opt. Soc. Am. A , doc. ID 101701 (posted 26 January 2009, in press).
    [CrossRef]
  39. M. Guizar-Sicairos and J. R. Fienup, “Focused x-ray beam characterization by phase retrieval with a move-able phase-shifting structure,” in Frontiers in Optics, OSA Technical Digest Series (Optical Society of America, 2008), paper FWN3.

2009 (2)

G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Opt. Express 17, 624–639 (2009).
[CrossRef] [PubMed]

S. T. Thurman, R. T. DeRosa, and J. R. Fienup, “Amplitude metrics for field retrieval with hard-edge and uniformly-illuminated apertures,” J. Opt. Soc. Am. A , doc. ID 101701 (posted 26 January 2009, in press).
[CrossRef]

2008 (11)

M. Guizar-Sicairos and J. R. Fienup, “Focused x-ray beam characterization by phase retrieval with a move-able phase-shifting structure,” in Frontiers in Optics, OSA Technical Digest Series (Optical Society of America, 2008), paper FWN3.

M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett.  33, 156–158 (2008).
[CrossRef] [PubMed]

H. Mimura, H. Yumoto, S. Matsuyama, S. Handa, T. Kimura, Y. Sano, M. Yabashi, Y. Nishino, K. Tamasaku, and T. Ishikawaet al., “Direct determination of the wave field of an x-ray nanobeam,” Phys. Rev. A  77, 015812 (2008).

A. Stein, K. Evans-Lutterodt, N. Bozovic, and A. Taylor, “Fabrication of silicon kinoform lenses for hard x-ray focusing by electron beam lithography and deep reactive ion etching,” J. Vac. Sci. Technol. B  26, 122–127 (2008).
[CrossRef]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[CrossRef]

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
[CrossRef] [PubMed]

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

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, and C. Cuiet al., “Massively parallel x-ray holography,” Nat. Photonics 2, 560–563 (2008).
[CrossRef]

M. Guizar-Sicairos and J. R. Fienup, “Direct image reconstruction from a Fourier intensity pattern using HERALDO,” Opt. Lett.  33, 2668–2670 (2008).
[CrossRef] [PubMed]

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Physics 4, 394–398 (2008).
[CrossRef]

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with Fourier-weighted projections,” J. Opt. Soc. Am. A  25, 701–709 (2008).
[CrossRef]

2007 (3)

2006 (5)

G. R. Brady and J. R. Fienup, “Nonlinear optimization algorithm for retrieving the full complex pupil function,” Opt. Express 14, 474–486 (2006).
[CrossRef] [PubMed]

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Physics 2, 101–104 (2006).
[CrossRef]

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett.  96, 127401 (2006).
[CrossRef] [PubMed]

J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express 14, 498–508 (2006).
[CrossRef] [PubMed]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, and M. R. Howellset al., “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A  23, 1179–1200 (2006).
[CrossRef]

2005 (1)

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

2004 (3)

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett.  93, 023903 (2004).
[CrossRef] [PubMed]

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett.  85, 4795–4797 (2004).
[CrossRef]

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by x-ray spectro-holography,” Nature 432, 885–888 (2004).
[CrossRef] [PubMed]

2003 (1)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B  68, 140101 (2003).
[CrossRef]

2000 (1)

W. Leitenberger, T. Weitkamp, M. Drakopoulos, I. Snigireva, and A. Snigirev, “Microscopic imaging and holography with hard X-rays using Fresnel zone-plates,” Opt. Commun.  180, 233–238 (2000).
[CrossRef]

1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

1997 (1)

J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt.  36, 8352–8357 (1997).
[CrossRef]

1992 (1)

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-Resolution Imaging by Fourier Transform X-ray Holography,” Science 256, 1009–1012 (1992).
[CrossRef] [PubMed]

1988 (1)

J. N. Cederquist, J. R. Fienup, J. C. Marron, and R. G. Paxman, “Phase retrieval from experimental far-field speckle data,” Opt. Lett.  13, 619–621 (1988).
[CrossRef] [PubMed]

1987 (1)

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A  4, 118–123 (1987).
[CrossRef]

1982 (2)

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt.  21, 2758–2769 (1982).
[CrossRef] [PubMed]

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng.  21, 829–832 (1982).

1979 (2)

Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun.  30, 304–308 (1979).
[CrossRef]

R. A. Gonsalves and R. Childlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).

1978 (1)

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett.  3, 27–29 (1978).
[CrossRef] [PubMed]

Abbey, B.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Physics 4, 394–398 (2008).
[CrossRef]

Anderson, E. H.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-Resolution Imaging by Fourier Transform X-ray Holography,” Science 256, 1009–1012 (1992).
[CrossRef] [PubMed]

Bajt, S.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, and C. Cuiet al., “Massively parallel x-ray holography,” Nat. Photonics 2, 560–563 (2008).
[CrossRef]

Barty, A.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, and C. Cuiet al., “Massively parallel x-ray holography,” Nat. Photonics 2, 560–563 (2008).
[CrossRef]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, and M. R. Howellset al., “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A  23, 1179–1200 (2006).
[CrossRef]

Baumann, J.

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “Inspection of refractive x-ray lenses using high-resolution differential phase contrast imaging with a microfocus x-ray source,” Rev. Sci. Instrum.  78, 093707 (2007).
[CrossRef] [PubMed]

Beetz, T.

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

Bogan, M. J.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, and C. Cuiet al., “Massively parallel x-ray holography,” Nat. Photonics 2, 560–563 (2008).
[CrossRef]

Boutet, S.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, and C. Cuiet al., “Massively parallel x-ray holography,” Nat. Photonics 2, 560–563 (2008).
[CrossRef]

Bozovic, N.

A. Stein, K. Evans-Lutterodt, N. Bozovic, and A. Taylor, “Fabrication of silicon kinoform lenses for hard x-ray focusing by electron beam lithography and deep reactive ion etching,” J. Vac. Sci. Technol. B  26, 122–127 (2008).
[CrossRef]

Brady, G. R.

Bruck, Y. M.

Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun.  30, 304–308 (1979).
[CrossRef]

Bunk, O.

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[CrossRef]

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

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “Inspection of refractive x-ray lenses using high-resolution differential phase contrast imaging with a microfocus x-ray source,” Rev. Sci. Instrum.  78, 093707 (2007).
[CrossRef] [PubMed]

Cai, Z.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Physics 2, 101–104 (2006).
[CrossRef]

Cederquist, J. N.

J. N. Cederquist, J. R. Fienup, J. C. Marron, and R. G. Paxman, “Phase retrieval from experimental far-field speckle data,” Opt. Lett.  13, 619–621 (1988).
[CrossRef] [PubMed]

Chapman, H. N.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, and C. Cuiet al., “Massively parallel x-ray holography,” Nat. Photonics 2, 560–563 (2008).
[CrossRef]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, and M. R. Howellset al., “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A  23, 1179–1200 (2006).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B  68, 140101 (2003).
[CrossRef]

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Childlaw, R.

R. A. Gonsalves and R. Childlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).

Clark, J. N.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Physics 4, 394–398 (2008).
[CrossRef]

Conley, R.

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett.  96, 127401 (2006).
[CrossRef] [PubMed]

Cui, C.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, and C. Cuiet al., “Massively parallel x-ray holography,” Nat. Photonics 2, 560–563 (2008).
[CrossRef]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, and M. R. Howellset al., “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A  23, 1179–1200 (2006).
[CrossRef]

David, C.

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

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “Inspection of refractive x-ray lenses using high-resolution differential phase contrast imaging with a microfocus x-ray source,” Rev. Sci. Instrum.  78, 093707 (2007).
[CrossRef] [PubMed]

de Jonge, M.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Physics 4, 394–398 (2008).
[CrossRef]

DeRosa, R. T.

S. T. Thurman, R. T. DeRosa, and J. R. Fienup, “Amplitude metrics for field retrieval with hard-edge and uniformly-illuminated apertures,” J. Opt. Soc. Am. A , doc. ID 101701 (posted 26 January 2009, in press).
[CrossRef]

Dierolf, M.

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[CrossRef]

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

Drakopoulos, M.

W. Leitenberger, T. Weitkamp, M. Drakopoulos, I. Snigireva, and A. Snigirev, “Microscopic imaging and holography with hard X-rays using Fresnel zone-plates,” Opt. Commun.  180, 233–238 (2000).
[CrossRef]

Eberhardt, W.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by x-ray spectro-holography,” Nature 432, 885–888 (2004).
[CrossRef] [PubMed]

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J. W. Goodman, Introduction to Fourier Optics, 3rd Ed. (Roberts & Company, Englewood, 2005).

Guizar-Sicairos, M.

G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Opt. Express 17, 624–639 (2009).
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M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett.  33, 156–158 (2008).
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M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with Fourier-weighted projections,” J. Opt. Soc. Am. A  25, 701–709 (2008).
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M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
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M. Guizar-Sicairos and J. R. Fienup, “Direct image reconstruction from a Fourier intensity pattern using HERALDO,” Opt. Lett.  33, 2668–2670 (2008).
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M. Guizar-Sicairos and J. R. Fienup, “Focused x-ray beam characterization by phase retrieval with a move-able phase-shifting structure,” in Frontiers in Optics, OSA Technical Digest Series (Optical Society of America, 2008), paper FWN3.

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H. Mimura, H. Yumoto, S. Matsuyama, S. Handa, T. Kimura, Y. Sano, M. Yabashi, Y. Nishino, K. Tamasaku, and T. Ishikawaet al., “Direct determination of the wave field of an x-ray nanobeam,” Phys. Rev. A  77, 015812 (2008).

Hau-riege, S. P.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, and C. Cuiet al., “Massively parallel x-ray holography,” Nat. Photonics 2, 560–563 (2008).
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S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by x-ray spectro-holography,” Nature 432, 885–888 (2004).
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D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, and H. Miaoet al., “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A.  102, 15343–15346 (2005).
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H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, and M. R. Howellset al., “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A  23, 1179–1200 (2006).
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S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B  68, 140101 (2003).
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I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-Resolution Imaging by Fourier Transform X-ray Holography,” Science 256, 1009–1012 (1992).
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Jacobsen, C.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, and H. Miaoet al., “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A.  102, 15343–15346 (2005).
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I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-Resolution Imaging by Fourier Transform X-ray Holography,” Science 256, 1009–1012 (1992).
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O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
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H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett.  96, 127401 (2006).
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I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-Resolution Imaging by Fourier Transform X-ray Holography,” Science 256, 1009–1012 (1992).
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H. Mimura, H. Yumoto, S. Matsuyama, S. Handa, T. Kimura, Y. Sano, M. Yabashi, Y. Nishino, K. Tamasaku, and T. Ishikawaet al., “Direct determination of the wave field of an x-ray nanobeam,” Phys. Rev. A  77, 015812 (2008).

Kirz, J.

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

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-Resolution Imaging by Fourier Transform X-ray Holography,” Science 256, 1009–1012 (1992).
[CrossRef] [PubMed]

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M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “Inspection of refractive x-ray lenses using high-resolution differential phase contrast imaging with a microfocus x-ray source,” Rev. Sci. Instrum.  78, 093707 (2007).
[CrossRef] [PubMed]

Kynde, S.

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[CrossRef]

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W. Leitenberger, T. Weitkamp, M. Drakopoulos, I. Snigireva, and A. Snigirev, “Microscopic imaging and holography with hard X-rays using Fresnel zone-plates,” Opt. Commun.  180, 233–238 (2000).
[CrossRef]

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

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H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett.  96, 127401 (2006).
[CrossRef] [PubMed]

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S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by x-ray spectro-holography,” Nature 432, 885–888 (2004).
[CrossRef] [PubMed]

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S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by x-ray spectro-holography,” Nature 432, 885–888 (2004).
[CrossRef] [PubMed]

Macrander, A. T.

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett.  96, 127401 (2006).
[CrossRef] [PubMed]

Marchesini, S.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, and C. Cuiet al., “Massively parallel x-ray holography,” Nat. Photonics 2, 560–563 (2008).
[CrossRef]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, and M. R. Howellset al., “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A  23, 1179–1200 (2006).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B  68, 140101 (2003).
[CrossRef]

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J. N. Cederquist, J. R. Fienup, J. C. Marron, and R. G. Paxman, “Phase retrieval from experimental far-field speckle data,” Opt. Lett.  13, 619–621 (1988).
[CrossRef] [PubMed]

Marti, O.

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[CrossRef]

Maser, J.

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett.  96, 127401 (2006).
[CrossRef] [PubMed]

Matsuyama, S.

H. Mimura, H. Yumoto, S. Matsuyama, S. Handa, T. Kimura, Y. Sano, M. Yabashi, Y. Nishino, K. Tamasaku, and T. Ishikawaet al., “Direct determination of the wave field of an x-ray nanobeam,” Phys. Rev. A  77, 015812 (2008).

McNulty, I.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Physics 4, 394–398 (2008).
[CrossRef]

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-Resolution Imaging by Fourier Transform X-ray Holography,” Science 256, 1009–1012 (1992).
[CrossRef] [PubMed]

Menzel, A.

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

Miao, H.

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

Miao, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Mimura, H.

H. Mimura, H. Yumoto, S. Matsuyama, S. Handa, T. Kimura, Y. Sano, M. Yabashi, Y. Nishino, K. Tamasaku, and T. Ishikawaet al., “Direct determination of the wave field of an x-ray nanobeam,” Phys. Rev. A  77, 015812 (2008).

Nishino, Y.

H. Mimura, H. Yumoto, S. Matsuyama, S. Handa, T. Kimura, Y. Sano, M. Yabashi, Y. Nishino, K. Tamasaku, and T. Ishikawaet al., “Direct determination of the wave field of an x-ray nanobeam,” Phys. Rev. A  77, 015812 (2008).

Noy, A.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, and M. R. Howellset al., “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A  23, 1179–1200 (2006).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B  68, 140101 (2003).
[CrossRef]

Nugent, K. A.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Physics 4, 394–398 (2008).
[CrossRef]

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Physics 2, 101–104 (2006).
[CrossRef]

Paganin, D. M.

Paterson, D.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Physics 2, 101–104 (2006).
[CrossRef]

Pavlov, K. M.

Paxman, R. G.

J. N. Cederquist, J. R. Fienup, J. C. Marron, and R. G. Paxman, “Phase retrieval from experimental far-field speckle data,” Opt. Lett.  13, 619–621 (1988).
[CrossRef] [PubMed]

R. G. Paxman, “Diversity imaging,” in Signal Recovery and Synthesis, 2001 OSA Technical Digest Series (Optical Society of America, 2001), paper SWA1.

Peele, A. G.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Physics 4, 394–398 (2008).
[CrossRef]

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Physics 2, 101–104 (2006).
[CrossRef]

Pfeifer, M. A.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Physics 4, 394–398 (2008).
[CrossRef]

Pfeiffer, F.

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

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[CrossRef]

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “Inspection of refractive x-ray lenses using high-resolution differential phase contrast imaging with a microfocus x-ray source,” Rev. Sci. Instrum.  78, 093707 (2007).
[CrossRef] [PubMed]

Podorov, S. G.

Quiney, H. M.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Physics 2, 101–104 (2006).
[CrossRef]

Rodenburg, J. M.

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett.  93, 023903 (2004).
[CrossRef] [PubMed]

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett.  85, 4795–4797 (2004).
[CrossRef]

Sakdinawat, A. E.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, and C. Cuiet al., “Massively parallel x-ray holography,” Nat. Photonics 2, 560–563 (2008).
[CrossRef]

Sano, Y.

H. Mimura, H. Yumoto, S. Matsuyama, S. Handa, T. Kimura, Y. Sano, M. Yabashi, Y. Nishino, K. Tamasaku, and T. Ishikawaet al., “Direct determination of the wave field of an x-ray nanobeam,” Phys. Rev. A  77, 015812 (2008).

Sayre, D.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Schlotter, W. F.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by x-ray spectro-holography,” Nature 432, 885–888 (2004).
[CrossRef] [PubMed]

Schuster, M.

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “Inspection of refractive x-ray lenses using high-resolution differential phase contrast imaging with a microfocus x-ray source,” Rev. Sci. Instrum.  78, 093707 (2007).
[CrossRef] [PubMed]

Shapiro, D.

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

Snigirev, A.

W. Leitenberger, T. Weitkamp, M. Drakopoulos, I. Snigireva, and A. Snigirev, “Microscopic imaging and holography with hard X-rays using Fresnel zone-plates,” Opt. Commun.  180, 233–238 (2000).
[CrossRef]

Snigireva, I.

W. Leitenberger, T. Weitkamp, M. Drakopoulos, I. Snigireva, and A. Snigirev, “Microscopic imaging and holography with hard X-rays using Fresnel zone-plates,” Opt. Commun.  180, 233–238 (2000).
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Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun.  30, 304–308 (1979).
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Stein, A.

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Stöhr, J.

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Taylor, A.

A. Stein, K. Evans-Lutterodt, N. Bozovic, and A. Taylor, “Fabrication of silicon kinoform lenses for hard x-ray focusing by electron beam lithography and deep reactive ion etching,” J. Vac. Sci. Technol. B  26, 122–127 (2008).
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P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
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D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, and H. Miaoet al., “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A.  102, 15343–15346 (2005).
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H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett.  96, 127401 (2006).
[CrossRef] [PubMed]

Weierstall, U.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B  68, 140101 (2003).
[CrossRef]

Weitkamp, T.

W. Leitenberger, T. Weitkamp, M. Drakopoulos, I. Snigireva, and A. Snigirev, “Microscopic imaging and holography with hard X-rays using Fresnel zone-plates,” Opt. Commun.  180, 233–238 (2000).
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B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Physics 4, 394–398 (2008).
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Yumoto, H.

H. Mimura, H. Yumoto, S. Matsuyama, S. Handa, T. Kimura, Y. Sano, M. Yabashi, Y. Nishino, K. Tamasaku, and T. Ishikawaet al., “Direct determination of the wave field of an x-ray nanobeam,” Phys. Rev. A  77, 015812 (2008).

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M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with Fourier-weighted projections,” J. Opt. Soc. Am. A  25, 701–709 (2008).
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S. T. Thurman, R. T. DeRosa, and J. R. Fienup, “Amplitude metrics for field retrieval with hard-edge and uniformly-illuminated apertures,” J. Opt. Soc. Am. A , doc. ID 101701 (posted 26 January 2009, in press).
[CrossRef]

J. Vac. Sci. Technol (1)

A. Stein, K. Evans-Lutterodt, N. Bozovic, and A. Taylor, “Fabrication of silicon kinoform lenses for hard x-ray focusing by electron beam lithography and deep reactive ion etching,” J. Vac. Sci. Technol. B  26, 122–127 (2008).
[CrossRef]

Nat. Photonics (1)

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-riege, A. Szöke, and C. Cuiet al., “Massively parallel x-ray holography,” Nat. Photonics 2, 560–563 (2008).
[CrossRef]

Nat. Physics (2)

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Physics 4, 394–398 (2008).
[CrossRef]

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Physics 2, 101–104 (2006).
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[CrossRef] [PubMed]

Opt. Commun (2)

W. Leitenberger, T. Weitkamp, M. Drakopoulos, I. Snigireva, and A. Snigirev, “Microscopic imaging and holography with hard X-rays using Fresnel zone-plates,” Opt. Commun.  180, 233–238 (2000).
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M. Guizar-Sicairos and J. R. Fienup, “Direct image reconstruction from a Fourier intensity pattern using HERALDO,” Opt. Lett.  33, 2668–2670 (2008).
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[CrossRef] [PubMed]

Phys. Rev (2)

H. Mimura, H. Yumoto, S. Matsuyama, S. Handa, T. Kimura, Y. Sano, M. Yabashi, Y. Nishino, K. Tamasaku, and T. Ishikawaet al., “Direct determination of the wave field of an x-ray nanobeam,” Phys. Rev. A  77, 015812 (2008).

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B  68, 140101 (2003).
[CrossRef]

Phys. Rev. Lett (2)

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett.  96, 127401 (2006).
[CrossRef] [PubMed]

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D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, and H. Miaoet al., “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A.  102, 15343–15346 (2005).
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Proc. SPIE (1)

R. A. Gonsalves and R. Childlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).

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M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “Inspection of refractive x-ray lenses using high-resolution differential phase contrast imaging with a microfocus x-ray source,” Rev. Sci. Instrum.  78, 093707 (2007).
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M. Guizar-Sicairos and J. R. Fienup, “Focused x-ray beam characterization by phase retrieval with a move-able phase-shifting structure,” in Frontiers in Optics, OSA Technical Digest Series (Optical Society of America, 2008), paper FWN3.

Supplementary Material (3)

» Media 1: MOV (2699 KB)     
» Media 2: MOV (2556 KB)     
» Media 3: MOV (3148 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

Experimental setups for x-ray beam diagnostics for (a) 2D and (b) 1D beam foci. In both cases the far-field intensity patterns are measured for each position of the moveable translating structure.

Fig. 2.
Fig. 2.

(a) Amplitude transmissivity of the cylindrical lenses L 1 and L 2. Phase error (deviation from cylinder) for (b) L 1 and (c) L 2. Colorbar for phase in (b) and (c) is given in radians.

Fig. 3.
Fig. 3.

(a) Amplitude and (b) phase (deviation from sphere) of the field after L 2. (c) Square-root of the beam amplitude at the nominal focus. Line plots show cuts through amplitude of the focused beam through the origin. Colorbar in (b) is in radians.

Fig. 4.
Fig. 4.

(a) Square-root of x-ray beam amplitude at 1 mm downstream from focus. Line plots show vertical and horizontal cuts through the beam amplitude through the origin. (b) The 392 nm radius translating structure was moved to 16 positions along the transverse extent of the beam. (c) A second set of measurements with a 981 nm radius translating structure was used for an independent reconstruction.

Fig. 5.
Fig. 5.

X-ray beam intensity at the detector plane, 7.37 m from the translating structure. Square-root of the beam amplitude is shown for visualization purposes. (a) and (b) show far-field intensity patterns for the two positions of the 392 nm radius translating structure indicated by red circles in Fig. 4(b). (c) and (d) show the intensity patterns for the 981 nm radius structure for positions indicated by red circles in Fig. 4(c). (Media 1) and (Media 2), show the 16 intensity patterns for the 392 and 981 nm structures, respectively. Amplitude to the 0.75 power is shown in the multimedia files.

Fig. 6.
Fig. 6.

Square-root of the amplitude of the field reconstructed from detector measurements using (a) the 392 nm radius structure and (b) the 981 nm radius structure. (c) and (d) show cuts through the amplitude of the initial estimate (dashed curves), the true x-ray beam (solid curves) and the reconstructions with the 392 nm (circles) and the 981 nm radius structure (points).

Fig. 7.
Fig. 7.

Square-root of amplitude of the numerically refocused reconstructions obtained from measurements using the (a) 392 nm and (b) 981 nm structures. (c) Horizontal and (d) vertical cuts through the amplitude of the true x-ray beam at best focus (solid curves) and the numerically refocused reconstructions using the small (circles) and the large (points) structure.

Fig. 8.
Fig. 8.

Square-root of beam amplitude (a) at the nominal focus and (b) 1 mm downstream (plane of the translating structure). Line plots show horizontal and vertical cuts through the x-ray beam through the origin. (c) Phase of 1D binary phase-shifting structure, having unity transmissivity and phase values of 0 and π radians.

Fig. 9.
Fig. 9.

(a) Beam intensity at measurement plane. (b) After including the effect of integration due to finite pixel size (notice the very pronounced sampling effect along the y direction). (c) Result of integrating the intensity measurement in (b) along the undersampled direction. Multimedia file (Media 3) shows the 25 intensity patterns at the detector plane and the integrated measurements used for 1D phase retrieval. Amplitude to the 0.5 power is shown in (a) and (b) and to the 0.75 power in (Media 3).

Fig. 10.
Fig. 10.

(a) Initial estimate for reconstruction (dashed line). Projection of the true field at the position of the translating structure (solid curve) and 1D reconstruction (points). (b) Amplitude of the true beam projection at 2 μm downstream the nominal focus (solid curve) and the numerically refocused reconstruction (points).

Equations (10)

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u 1 ( x 1 , y 1 ) = t 1 ( x 1 , y 1 ) exp ( ik x 1 2 2 f ) ,
E 2 = min α , x , y x , y αû ( x x , y y ) u ( x , y ) 2 x , y u ( x , y ) 2 .
u out ( x , y ) d y = 𝓟 x , y { u in ; z } d y = 𝓟 x { u in ( x , y ) d y ; z } ,
u out ( x , y ) = 𝓟 x , y { u in ; z } = 1 iλz exp [ ik 2 z ( x 2 + y 2 ) ]
× u in ( x , y ) exp [ ik 2 z ( x 2 + y 2 ) ] exp [ i 2 π λz ( xx + yy ) ] d x d y
u out ( x , y ) = ( f x , f y ) ( x , y ) 1 { exp [ iπλz ( f x 2 + f y 2 ) ] ( x , y ) ( f x , f y ) { u in ( x , y ) } } ,
( x , y ) ( f x , f y ) { u ( x , y ) } = u ( x , y ) exp [ i 2 π ( x f x + y f y ) ] d x d y .
u in ( x , y ) = t ( x , y ) exp ( ik x 2 2 f ) ,
𝓟 x , y { u in ; z } = 1 iλz exp ( ik 2 z x 2 ) t ˜ ( x , y ) exp [ ik x 2 ( f z fz ) ] exp ( i 2 π λz xx ) d x ,
t ˜ ( x , y ) = f y y 1 { exp ( iπλz f y 2 ) y f y { t ( x , y ) } } ,

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