Abstract

Full field x-ray propagation imaging can be severely deteriorated by wave front aberrations. Here we present an extension of ptychographic phase retrieval with simultaneous probe and object reconstruction suitable for the near-field diffractive imaging setting. Update equations used to iteratively solve the phase problem from a set of near-field images in view of reconstruction both object and probe are derived. The algorithm is tested based on numerical simulations including photon shot noise. The results indicate that the approach provides an efficient way to overcome restrictive idealizations of the illumination wave in the near-field (propagation) imaging.

© 2013 OSA

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  33. A. Burvall, U. Lundström, P. A. C. Takman, D. H. Larsson, and H. M. Hertz, “Phase retrieval in x-ray phase-contrast imaging suitable for tomography,” Opt. Express 19(11), 10359–10376 (2011).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  35. Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, “Three-dimensional visualization of a human chromosome using coherent x-ray diffraction,” Phys. Rev. Lett. 102(1), 018101 (2009).
    [CrossRef] [PubMed]
  36. H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” PNAS 107(25), 11234–11239 (2010).
    [CrossRef] [PubMed]
  37. M. Bartels, M. Priebe, R. N. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, and T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy 1(10), 1–7 (2012).
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    [CrossRef] [PubMed]
  40. T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231(1–6), 53–70 (2004).
    [CrossRef]
  41. A. V. Bronnikow, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19(3), 472–480 (2002).
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  42. Y. D. Witte, M. Boone, J. Vlassenbroeck, M. Dierick, and L. V. Hoorebeke, “Bronnikov-aided correction for x-ray computed tomography,” J. Opt. Soc. Am. A 26(4), 890–894 (2009).
    [CrossRef]
  43. M. Krenkel, M. Bartels, and T. Salditt, “Transport of intensity phase reconstruction to solve the twin image problem in holographic x-ray imaging,” Opt. Express 21(2), 2220–2235 (2013).
    [CrossRef] [PubMed]
  44. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35(2), 237–246 (1972).
  45. 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. Phys. 4, 394–398 (2008).
    [CrossRef]
  46. L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1–4), 65–75 (2001).
    [CrossRef]
  47. E. Li, Y. Liu, X. Liu, K. Zhang, Z. Wang, Y. Hong, Q. Yuan, W. Huang, A. Marcelli, P. Zhu, and Z. Wu, “Phase retrieval from a single near-field diffraction pattern with a large Fresnel number,” J. Opt. Soc. Am. A 25(11), (2008).
  48. E. Li, Y. Hwu, D. Chien, C. L. Wang, and G. Margaritondo, “Phase retrieval from integrated near-field diffraction intensity,” J. Opt. Soc. Am. A 25(11), 2651–2658 (2008).
  49. H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Phys. 2, 101–104 (2006).
    [CrossRef]
  50. D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic Fresnel coherent diffractive imaging,” Phys. Rev. A 80(6), 063823 (2009).
    [CrossRef]
  51. D. R. Luke, J. V. Burke, and R. G. Lyon, “Optical wave front reconstruction: theory and numerical methods,” SIAM Rev. 44(2), 169–224 (2002).
    [CrossRef]

2013 (5)

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87(5), 053850 (2013).
[CrossRef]

A. M. Maiden, G. R. Morrison, B. Kaulich, A. Gianoncelli, and J. M. Rodenburg, “Soft x-ray spectromicroscopy using ptychography with randomly phased illumination,” Nat. Commun. 4, 1669 (2013).
[CrossRef] [PubMed]

E. Lima, A. Diaz, M. Guizar-Sicairos, S. Gorelick, P. Pernot, T. Schleier, and A. Menzel, “Cryo-scanning x-ray diffraction microscopy of frozen-hydrated yeast,” J. Microsc. 249(1), 1–7 (2013).
[CrossRef]

Y. Takahashi, A. Suzuki, S. Furutaku, K. Yamauchi, Y. Kohmura, and T. Ishikawa, “High-resolution and high-sensitivity phase-contrast imaging by focused hard x-ray ptychography with a spatial filter,” Appl. Phys. Lett. 102(9), 094102 (2013).
[CrossRef]

M. Krenkel, M. Bartels, and T. Salditt, “Transport of intensity phase reconstruction to solve the twin image problem in holographic x-ray imaging,” Opt. Express 21(2), 2220–2235 (2013).
[CrossRef] [PubMed]

2012 (5)

R. N. Wilke, M. Priebe, M. Bartels, K. Giewekemeyer, A. Diaz, P. Karvinen, and T. Salditt, “Hard x-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction,” Opt. Express 20(17), 19232–19254 (2012).
[CrossRef] [PubMed]

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, (2012).
[CrossRef]

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C.G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
[CrossRef]

M. Bartels, M. Priebe, R. N. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, and T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy 1(10), 1–7 (2012).
[CrossRef]

Y.-L. Hong, K. Zhang, Z.-L. Wang, Z.-Z. Zhu, X.-J. Zhao, W.-X. Huang, Q.-X. Yuan, P.-P. Zhu, and Z.-Y. Wu, “Reconstructing a complex field from a series of its near-field diffraction patterns,” Chinese Phys. B 21(10), 104202 (2012).
[CrossRef]

2011 (3)

C. T. Putkunz, J. N. Clark, D. J. Vine, G. J. Williams, M. A. Pfeifer, E. Balaur, I. McNulty, K. A. Nugent, and A. G. Peele, “Phase-diverse coherent diffractive imaging: high sensitivity with low dose,” Phys. Rev. Lett. 106(1), 013903 (2011).
[CrossRef] [PubMed]

M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S.-C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical Contrast in Soft X-Ray Ptychography,” Phys. Rev. Lett. 107(20), 208101 (2011).
[CrossRef] [PubMed]

A. Burvall, U. Lundström, P. A. C. Takman, D. H. Larsson, and H. M. Hertz, “Phase retrieval in x-ray phase-contrast imaging suitable for tomography,” Opt. Express 19(11), 10359–10376 (2011).
[CrossRef] [PubMed]

2010 (5)

C. M. Kewish, M. Guizar-Sicairos, C. Liu, J. Qian, B. Shi, C. Benson, A. M. Khounsary, J. Vila-Comamala, O. Bunk, J. R. Fienup, A. T. Macrander, and L. Assoufid, “Reconstruction of an astigmatic hard x-ray beam and alignment of K-B mirrors from ptychographic coherent diffraction data,” Opt. Express 18(22), 23420–23427 (2010).
[CrossRef] [PubMed]

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” PNAS 107(2), 529–534 (2010).
[CrossRef]

K. A. Nugent, “Coherent methods in the x-ray sciences,” Adv. Phys. 59(4), 1–99 (2010).
[CrossRef]

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010).
[CrossRef]

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” PNAS 107(25), 11234–11239 (2010).
[CrossRef] [PubMed]

2009 (5)

D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic Fresnel coherent diffractive imaging,” Phys. Rev. A 80(6), 063823 (2009).
[CrossRef]

Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, “Three-dimensional visualization of a human chromosome using coherent x-ray diffraction,” Phys. Rev. Lett. 102(1), 018101 (2009).
[CrossRef] [PubMed]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[CrossRef] [PubMed]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109(10), 1256–1262 (2009).
[CrossRef] [PubMed]

Y. D. Witte, M. Boone, J. Vlassenbroeck, M. Dierick, and L. V. Hoorebeke, “Bronnikov-aided correction for x-ray computed tomography,” J. Opt. Soc. Am. A 26(4), 890–894 (2009).
[CrossRef]

2008 (4)

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16(10), 7264–7278 (2008).
[CrossRef] [PubMed]

E. Li, Y. Hwu, D. Chien, C. L. Wang, and G. Margaritondo, “Phase retrieval from integrated near-field diffraction intensity,” J. Opt. Soc. Am. A 25(11), 2651–2658 (2008).

E. Li, Y. Liu, X. Liu, K. Zhang, Z. Wang, Y. Hong, Q. Yuan, W. Huang, A. Marcelli, P. Zhu, and Z. Wu, “Phase retrieval from a single near-field diffraction pattern with a large Fresnel number,” J. Opt. Soc. Am. A 25(11), (2008).

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. Phys. 4, 394–398 (2008).
[CrossRef]

2007 (2)

J. M. Rodenburg, A.C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-Ray Lensless Imaging of Extended Objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[CrossRef] [PubMed]

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32(12), 1617–1619 (2007).
[CrossRef] [PubMed]

2006 (3)

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

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97(2), 025506 (2006).
[CrossRef] [PubMed]

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” PNAS 103(39), 14626–14630 (2006).
[CrossRef] [PubMed]

2005 (1)

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” PNAS 102(43), 15343–15346 (2005).
[CrossRef] [PubMed]

2004 (3)

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231(1–6), 53–70 (2004).
[CrossRef]

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93(2), 023903 (2004).
[CrossRef] [PubMed]

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 4795 (2004).
[CrossRef]

2003 (2)

J. Miao, K. O. Hodgson, T. Ishikawa, C. A. Larabell, M. A. LeGros, and Y. Nishino, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67(17), 174104 (2003).
[CrossRef]

V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20(1), 40–55 (2003).
[CrossRef]

2002 (2)

A. V. Bronnikow, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19(3), 472–480 (2002).
[CrossRef]

D. R. Luke, J. V. Burke, and R. G. Lyon, “Optical wave front reconstruction: theory and numerical methods,” SIAM Rev. 44(2), 169–224 (2002).
[CrossRef]

2001 (1)

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1–4), 65–75 (2001).
[CrossRef]

1999 (2)

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]

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[CrossRef]

1996 (1)

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
[CrossRef]

1993 (1)

1983 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35(2), 237–246 (1972).

1927 (1)

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination, ” Sci. Rep. 3, 1927 (2013).

Abbey, B.

D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic Fresnel coherent diffractive imaging,” Phys. Rev. A 80(6), 063823 (2009).
[CrossRef]

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. Phys. 4, 394–398 (2008).
[CrossRef]

Allen, L. J.

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1–4), 65–75 (2001).
[CrossRef]

Assoufid, L.

Balaur, E.

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M. Bartels, M. Priebe, R. N. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, and T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy 1(10), 1–7 (2012).
[CrossRef]

M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S.-C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical Contrast in Soft X-Ray Ptychography,” Phys. Rev. Lett. 107(20), 208101 (2011).
[CrossRef] [PubMed]

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” PNAS 107(2), 529–534 (2010).
[CrossRef]

Samberg, D.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C.G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35(2), 237–246 (1972).

Sayre, D.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” PNAS 102(43), 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]

Schleier, T.

E. Lima, A. Diaz, M. Guizar-Sicairos, S. Gorelick, P. Pernot, T. Schleier, and A. Menzel, “Cryo-scanning x-ray diffraction microscopy of frozen-hydrated yeast,” J. Microsc. 249(1), 1–7 (2013).
[CrossRef]

Schlenker, M.

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” PNAS 103(39), 14626–14630 (2006).
[CrossRef] [PubMed]

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[CrossRef]

Schlichting, I.

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010).
[CrossRef]

Schroer, C.G.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C.G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
[CrossRef]

Schropp, A.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C.G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
[CrossRef]

Seiboth, F.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C.G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
[CrossRef]

Senkbeil, T.

M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S.-C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical Contrast in Soft X-Ray Ptychography,” Phys. Rev. Lett. 107(20), 208101 (2011).
[CrossRef] [PubMed]

Shapiro, D.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” PNAS 102(43), 15343–15346 (2005).
[CrossRef] [PubMed]

Shi, B.

Song, C.

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” PNAS 107(25), 11234–11239 (2010).
[CrossRef] [PubMed]

Sprung, M.

M. Bartels, M. Priebe, R. N. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, and T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy 1(10), 1–7 (2012).
[CrossRef]

Stephan, S.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C.G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
[CrossRef]

Stevenson, A. W.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
[CrossRef]

Stockmar, M.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination, ” Sci. Rep. 3, 1927 (2013).

Suzuki, A.

Y. Takahashi, A. Suzuki, S. Furutaku, K. Yamauchi, Y. Kohmura, and T. Ishikawa, “High-resolution and high-sensitivity phase-contrast imaging by focused hard x-ray ptychography with a spatial filter,” Appl. Phys. Lett. 102(9), 094102 (2013).
[CrossRef]

Takahashi, Y.

Y. Takahashi, A. Suzuki, S. Furutaku, K. Yamauchi, Y. Kohmura, and T. Ishikawa, “High-resolution and high-sensitivity phase-contrast imaging by focused hard x-ray ptychography with a spatial filter,” Appl. Phys. Lett. 102(9), 094102 (2013).
[CrossRef]

Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, “Three-dimensional visualization of a human chromosome using coherent x-ray diffraction,” Phys. Rev. Lett. 102(1), 018101 (2009).
[CrossRef] [PubMed]

Takman, P. A. C.

Tamanoi, F.

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” PNAS 107(25), 11234–11239 (2010).
[CrossRef] [PubMed]

Teague, M. R.

Thibault, P.

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, (2012).
[CrossRef]

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” PNAS 107(2), 529–534 (2010).
[CrossRef]

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010).
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P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[CrossRef] [PubMed]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” PNAS 102(43), 15343–15346 (2005).
[CrossRef] [PubMed]

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination, ” Sci. Rep. 3, 1927 (2013).

Tran, C. Q.

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97(2), 025506 (2006).
[CrossRef] [PubMed]

Urano, J.

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” PNAS 107(25), 11234–11239 (2010).
[CrossRef] [PubMed]

Vila-Comamala, J.

Vine, D. J.

C. T. Putkunz, J. N. Clark, D. J. Vine, G. J. Williams, M. A. Pfeifer, E. Balaur, I. McNulty, K. A. Nugent, and A. G. Peele, “Phase-diverse coherent diffractive imaging: high sensitivity with low dose,” Phys. Rev. Lett. 106(1), 013903 (2011).
[CrossRef] [PubMed]

D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic Fresnel coherent diffractive imaging,” Phys. Rev. A 80(6), 063823 (2009).
[CrossRef]

Vlassenbroeck, J.

von König, K.

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010).
[CrossRef]

Wagner, U.

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87(5), 053850 (2013).
[CrossRef]

Waigh, T. A.

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87(5), 053850 (2013).
[CrossRef]

Wang, C. L.

Wang, Z.

E. Li, Y. Liu, X. Liu, K. Zhang, Z. Wang, Y. Hong, Q. Yuan, W. Huang, A. Marcelli, P. Zhu, and Z. Wu, “Phase retrieval from a single near-field diffraction pattern with a large Fresnel number,” J. Opt. Soc. Am. A 25(11), (2008).

Wang, Z.-L.

Y.-L. Hong, K. Zhang, Z.-L. Wang, Z.-Z. Zhu, X.-J. Zhao, W.-X. Huang, Q.-X. Yuan, P.-P. Zhu, and Z.-Y. Wu, “Reconstructing a complex field from a series of its near-field diffraction patterns,” Chinese Phys. B 21(10), 104202 (2012).
[CrossRef]

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A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C.G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
[CrossRef]

Wilke, R. N.

M. Bartels, M. Priebe, R. N. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, and T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy 1(10), 1–7 (2012).
[CrossRef]

R. N. Wilke, M. Priebe, M. Bartels, K. Giewekemeyer, A. Diaz, P. Karvinen, and T. Salditt, “Hard x-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction,” Opt. Express 20(17), 19232–19254 (2012).
[CrossRef] [PubMed]

Wilkins, S. W.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231(1–6), 53–70 (2004).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
[CrossRef]

Williams, G. J.

C. T. Putkunz, J. N. Clark, D. J. Vine, G. J. Williams, M. A. Pfeifer, E. Balaur, I. McNulty, K. A. Nugent, and A. G. Peele, “Phase-diverse coherent diffractive imaging: high sensitivity with low dose,” Phys. Rev. Lett. 106(1), 013903 (2011).
[CrossRef] [PubMed]

D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic Fresnel coherent diffractive imaging,” Phys. Rev. A 80(6), 063823 (2009).
[CrossRef]

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. Phys. 4, 394–398 (2008).
[CrossRef]

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97(2), 025506 (2006).
[CrossRef] [PubMed]

Witte, Y. D.

Wu, Z.

E. Li, Y. Liu, X. Liu, K. Zhang, Z. Wang, Y. Hong, Q. Yuan, W. Huang, A. Marcelli, P. Zhu, and Z. Wu, “Phase retrieval from a single near-field diffraction pattern with a large Fresnel number,” J. Opt. Soc. Am. A 25(11), (2008).

Wu, Z.-Y.

Y.-L. Hong, K. Zhang, Z.-L. Wang, Z.-Z. Zhu, X.-J. Zhao, W.-X. Huang, Q.-X. Yuan, P.-P. Zhu, and Z.-Y. Wu, “Reconstructing a complex field from a series of its near-field diffraction patterns,” Chinese Phys. B 21(10), 104202 (2012).
[CrossRef]

Xu, R.

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” PNAS 107(25), 11234–11239 (2010).
[CrossRef] [PubMed]

Yamauchi, K.

Y. Takahashi, A. Suzuki, S. Furutaku, K. Yamauchi, Y. Kohmura, and T. Ishikawa, “High-resolution and high-sensitivity phase-contrast imaging by focused hard x-ray ptychography with a spatial filter,” Appl. Phys. Lett. 102(9), 094102 (2013).
[CrossRef]

Yuan, Q.

E. Li, Y. Liu, X. Liu, K. Zhang, Z. Wang, Y. Hong, Q. Yuan, W. Huang, A. Marcelli, P. Zhu, and Z. Wu, “Phase retrieval from a single near-field diffraction pattern with a large Fresnel number,” J. Opt. Soc. Am. A 25(11), (2008).

Yuan, Q.-X.

Y.-L. Hong, K. Zhang, Z.-L. Wang, Z.-Z. Zhu, X.-J. Zhao, W.-X. Huang, Q.-X. Yuan, P.-P. Zhu, and Z.-Y. Wu, “Reconstructing a complex field from a series of its near-field diffraction patterns,” Chinese Phys. B 21(10), 104202 (2012).
[CrossRef]

Zanette, I.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination, ” Sci. Rep. 3, 1927 (2013).

Zhang, K.

Y.-L. Hong, K. Zhang, Z.-L. Wang, Z.-Z. Zhu, X.-J. Zhao, W.-X. Huang, Q.-X. Yuan, P.-P. Zhu, and Z.-Y. Wu, “Reconstructing a complex field from a series of its near-field diffraction patterns,” Chinese Phys. B 21(10), 104202 (2012).
[CrossRef]

E. Li, Y. Liu, X. Liu, K. Zhang, Z. Wang, Y. Hong, Q. Yuan, W. Huang, A. Marcelli, P. Zhu, and Z. Wu, “Phase retrieval from a single near-field diffraction pattern with a large Fresnel number,” J. Opt. Soc. Am. A 25(11), (2008).

Zhao, X.-J.

Y.-L. Hong, K. Zhang, Z.-L. Wang, Z.-Z. Zhu, X.-J. Zhao, W.-X. Huang, Q.-X. Yuan, P.-P. Zhu, and Z.-Y. Wu, “Reconstructing a complex field from a series of its near-field diffraction patterns,” Chinese Phys. B 21(10), 104202 (2012).
[CrossRef]

Zhu, P.

E. Li, Y. Liu, X. Liu, K. Zhang, Z. Wang, Y. Hong, Q. Yuan, W. Huang, A. Marcelli, P. Zhu, and Z. Wu, “Phase retrieval from a single near-field diffraction pattern with a large Fresnel number,” J. Opt. Soc. Am. A 25(11), (2008).

Zhu, P.-P.

Y.-L. Hong, K. Zhang, Z.-L. Wang, Z.-Z. Zhu, X.-J. Zhao, W.-X. Huang, Q.-X. Yuan, P.-P. Zhu, and Z.-Y. Wu, “Reconstructing a complex field from a series of its near-field diffraction patterns,” Chinese Phys. B 21(10), 104202 (2012).
[CrossRef]

Zhu, Z.-Z.

Y.-L. Hong, K. Zhang, Z.-L. Wang, Z.-Z. Zhu, X.-J. Zhao, W.-X. Huang, Q.-X. Yuan, P.-P. Zhu, and Z.-Y. Wu, “Reconstructing a complex field from a series of its near-field diffraction patterns,” Chinese Phys. B 21(10), 104202 (2012).
[CrossRef]

Adv. Phys. (1)

K. A. Nugent, “Coherent methods in the x-ray sciences,” Adv. Phys. 59(4), 1–99 (2010).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (4)

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 4795 (2004).
[CrossRef]

Y. Takahashi, A. Suzuki, S. Furutaku, K. Yamauchi, Y. Kohmura, and T. Ishikawa, “High-resolution and high-sensitivity phase-contrast imaging by focused hard x-ray ptychography with a spatial filter,” Appl. Phys. Lett. 102(9), 094102 (2013).
[CrossRef]

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C.G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[CrossRef]

Chinese Phys. B (1)

Y.-L. Hong, K. Zhang, Z.-L. Wang, Z.-Z. Zhu, X.-J. Zhao, W.-X. Huang, Q.-X. Yuan, P.-P. Zhu, and Z.-Y. Wu, “Reconstructing a complex field from a series of its near-field diffraction patterns,” Chinese Phys. B 21(10), 104202 (2012).
[CrossRef]

J. Microsc. (1)

E. Lima, A. Diaz, M. Guizar-Sicairos, S. Gorelick, P. Pernot, T. Schleier, and A. Menzel, “Cryo-scanning x-ray diffraction microscopy of frozen-hydrated yeast,” J. Microsc. 249(1), 1–7 (2013).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Nat. Commun. (1)

A. M. Maiden, G. R. Morrison, B. Kaulich, A. Gianoncelli, and J. M. Rodenburg, “Soft x-ray spectromicroscopy using ptychography with randomly phased illumination,” Nat. Commun. 4, 1669 (2013).
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Nat. Phys. (2)

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

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. Phys. 4, 394–398 (2008).
[CrossRef]

Nature (2)

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
[CrossRef]

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]

New J. Phys. (2)

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010).
[CrossRef]

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, (2012).
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Opt. Commun. (2)

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1–4), 65–75 (2001).
[CrossRef]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231(1–6), 53–70 (2004).
[CrossRef]

Opt. Express (5)

Opt. Lett. (1)

Opt. Nanoscopy (1)

M. Bartels, M. Priebe, R. N. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, and T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy 1(10), 1–7 (2012).
[CrossRef]

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35(2), 237–246 (1972).

Phys. Rev. A (2)

D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic Fresnel coherent diffractive imaging,” Phys. Rev. A 80(6), 063823 (2009).
[CrossRef]

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87(5), 053850 (2013).
[CrossRef]

Phys. Rev. B (1)

J. Miao, K. O. Hodgson, T. Ishikawa, C. A. Larabell, M. A. LeGros, and Y. Nishino, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67(17), 174104 (2003).
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Phys. Rev. Lett. (6)

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93(2), 023903 (2004).
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J. M. Rodenburg, A.C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-Ray Lensless Imaging of Extended Objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[CrossRef] [PubMed]

M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S.-C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical Contrast in Soft X-Ray Ptychography,” Phys. Rev. Lett. 107(20), 208101 (2011).
[CrossRef] [PubMed]

C. T. Putkunz, J. N. Clark, D. J. Vine, G. J. Williams, M. A. Pfeifer, E. Balaur, I. McNulty, K. A. Nugent, and A. G. Peele, “Phase-diverse coherent diffractive imaging: high sensitivity with low dose,” Phys. Rev. Lett. 106(1), 013903 (2011).
[CrossRef] [PubMed]

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97(2), 025506 (2006).
[CrossRef] [PubMed]

Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, “Three-dimensional visualization of a human chromosome using coherent x-ray diffraction,” Phys. Rev. Lett. 102(1), 018101 (2009).
[CrossRef] [PubMed]

PNAS (4)

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” PNAS 107(25), 11234–11239 (2010).
[CrossRef] [PubMed]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” PNAS 102(43), 15343–15346 (2005).
[CrossRef] [PubMed]

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” PNAS 103(39), 14626–14630 (2006).
[CrossRef] [PubMed]

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” PNAS 107(2), 529–534 (2010).
[CrossRef]

Sci. Rep. (1)

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination, ” Sci. Rep. 3, 1927 (2013).

SIAM Rev. (1)

D. R. Luke, J. V. Burke, and R. G. Lyon, “Optical wave front reconstruction: theory and numerical methods,” SIAM Rev. 44(2), 169–224 (2002).
[CrossRef]

Ultramicroscopy (2)

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[CrossRef] [PubMed]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109(10), 1256–1262 (2009).
[CrossRef] [PubMed]

Other (3)

D. M. Paganin, Coherent X-Ray Optics (Oxford series on synchrotron radiation, Oxford University, 2006).
[CrossRef]

M. Guizar-Sicairos and J. R. Fienup, “Focused x-ray beam characterization by phase retrieval with a moveable phase-shifting structure,” Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing, OSA Technical Digest (CD), (2008).

K. Giewekemeyer, A Study on New Approaches in Coherent X-Ray Microscopy of Biological Specimens(Göttingen series in x-ray physics, Volume 5, Universitätsverlag Göttingen, 2011).

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Figures (4)

Fig. 1
Fig. 1

Schematic of the phase retrieval algorithm. In the outer loop, a reference plane for the probe update is chosen. In the shown simulations we define the first plane to be this reference plane. In general any plane can be chosen. Three different error metrics are calculated (see Eqs. (11), (12) and (13)). The update plane j for the object is selected randomly. A Gerchberg-Saxton step using two empty beam measurements at two detector positions a and b is needed to predefine the probe. The complex valued field of the exit wave behind the object is calculated as the product of the object and the current, propagated probe. Next comes a modulus constraint according to the respective image recorded with the object in plane j, i.e. propagated intensities are set to the measured data, followed by back propagation. Object o and probe p are separated using to Eqs. (7) and (8).

Fig. 2
Fig. 2

Simulation setup: (a) Complex valued model chosen to represent the object. Left: amplitude, right: phase. (b) Complex valued field chosen for the probe in plane one. Left: amplitude, right: phase. (c) Simulation set-up: Six defocus measurements (only three intensity patterns and their corresponding Fresnel numbers F are shown for illustration) are recorded with the detector fixed at distance z from the first plane. (d) A first empty beam intensity pattern is recorded at the same detector position z as before, while for the second empty beam image, the detector is placed at position z/2 from the first plane. A parallel beam geometry and coordinate system is used.

Fig. 3
Fig. 3

(a) Phase retrieval for known probe. The factor β in Eq. (9) is set to zero. No Gerchberg-Saxton iteration is performed. (b) Reconstruction of the probe without the Gerchberg-Saxton step using full information of the object (the factor α in Eq. (10) is set to zero). (c) Simultaneous reconstruction of object and probe without a Gerchberg-Saxton step.

Fig. 4
Fig. 4

(a) Reconstruction of object and probe using two additional empty beam measurements treated by a Gerchberg-Saxton step. The algorithm was initialized with ones for o and p (zeros for the respective phases) and ran for 50 iterations, setting α = 0.3 and β = 0.2 in Eqs. (9) and (10). (b) Poisson noise was added to the simulated intensity recordings. After 50 iterations, object and probe are reconstructed. (c), (d) Convergence for the simulations of (a), (b) by the error metrics described in Eqs. (11), (12) and (13).

Tables (2)

Tables Icon

Table 1 Fresnel numbers of the simulated holograms and empty beam patterns computed for 10 pixels each.

Tables Icon

Table 2 Simulation parameters.

Equations (28)

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ψ j : = o p j = o D Δ j 1 , j [ p j 1 ] .
D Δ a , b [ ψ a ] = 1 [ [ ψ a ] exp { i Δ a , b 2 k ( u 2 + v 2 ) } ] ,
{ M j } j = 1 , , N = { | D Δ j , det [ ψ j ] | 2 } j = 1 , , N .
P M [ ψ j ] = D Δ j , det [ Δ j , det D Δ j , det [ ψ j ] ]
Δ j , det : = ( 1 | D Δ j , det [ ψ j ] | 2 + 2 ε ( | D Δ j , det [ ψ j ] | 2 + ε ) 3 / 2 ( | D Δ j , det [ ψ j ] | 2 | D Δ j , det [ ψ j ] | 2 + ε M j ) )
S : = j x j , y j S j ( x j , y j ) : = j x j , y j | ψ j ψ ^ j | 2 .
o ^ = ψ j p ^ j * | p ^ j | 2 ,
p ^ 1 = D Δ 1 , j [ ψ j o ^ * | o ^ | 2 ] .
p 1 = p 1 + β D Δ 1 , j [ ( ψ j ' ψ j ) o * | o | 2 ] ,
o = o + α ( ψ j ' ψ j ) p j * | p j | 2 .
χ M 2 = 1 N j = 1 N 1 K L k , l = 1 K , L ( | D Δ j , det [ ψ j ] ( k Δ d , l Δ d ) | M j ( k Δ d , l Δ d ) M j ( k Δ d , l Δ d ) ) 2 .
χ o 2 = 1 K L k , l = 1 K , L ( | o rec ( k Δ d , l Δ d ) | | o orig ( k Δ d , l Δ d ) | | o orig ( k Δ d , l Δ d ) | ) 2 .
χ p 2 = 1 K L k , l = 1 K , L ( | p rec ( k Δ d , l Δ d ) | | p orig ( k Δ d , l Δ d ) | | p orig ( k Δ d , l Δ d ) | ) 2 .
F : = ( 10 Δ d ) 2 λ z .
S : = j x j , y j S j : = j x j , y j | ψ j ψ ^ j | 2
[ S j o ^ ] x ˜ j , y ˜ j = 0 ,
[ S j p ^ i ] x ˜ j , y ˜ j = 0 ,
[ S j o ^ * ] x ˜ j , y ˜ j = 0 ,
[ S j p ^ i * ] x ˜ j , y ˜ j = 0 .
S j = [ ψ j ( x j , y j ) ψ ^ j ( x j , y j ) ] [ ψ j ( x j , y j ) ψ ^ j ( x j , y j ) ] *
= ψ j ψ j * ψ ^ j * ψ j ψ ^ j ψ j * + ψ ^ j ψ ^ j *
= ψ j ψ j * o ^ * p ^ j * ψ j o ^ p ^ j ψ j * + o ^ p ^ j o ^ * p ^ j * .
[ S j o ^ ] x ˜ j , y ˜ j = p ^ j ( x ˜ j , y ˜ j ) ψ j * ( x ˜ j , y ˜ j ) + o ^ * ( x ˜ j , y ˜ j ) | p ^ j ( x ˜ j , y ˜ j ) | 2 ,
[ S j o ^ * ] x ˜ j , y ˜ j = p ^ j * ( x ˜ j , y ˜ j ) ψ j ( x ˜ j , y ˜ j ) + o ^ ( x ˜ j , y ˜ j ) | p ^ j ( x ˜ j , y ˜ j ) | 2 ,
[ S j p ^ j ] x ˜ j , y ˜ j = o ^ ( x ˜ j , y ˜ j ) ψ j * ( x ˜ j , y ˜ j ) + p ^ j * ( x ˜ j , y ˜ j ) | o ^ ( x ˜ j , y ˜ j ) | 2 ,
[ S j p ^ j * ] x ˜ j , y ˜ j = o ^ * ( x ˜ j , y ˜ j ) ψ j ( x ˜ j , y ˜ j ) + p ^ j ( x ˜ j , y ˜ j ) | o ^ ( x ˜ j , y ˜ j ) | 2 .
o ^ ( x ˜ j , y ˜ j ) = ψ j ( x ˜ j , y ˜ j ) p ^ j * ( x ˜ j , y ˜ j ) | p ^ j ( x ˜ j , y ˜ j ) | 2 ,
p ^ j ( x ˜ j , y ˜ j ) = ψ j ( x ˜ j , y ˜ j ) o ^ * ( x ˜ j , y ˜ j ) | o ^ ( x ˜ j , y ˜ j ) | 2 .

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