Abstract

We demonstrate the advantages of imaging with ptychography scans that follow a Fermat spiral trajectory. This scan pattern provides a more uniform coverage and a higher overlap ratio with the same number of scan points over the same area than the presently used mesh and concentric [13] patterns. Under realistically imperfect measurement conditions, numerical simulations show that the quality of the reconstructed image is improved significantly with a Fermat spiral compared with a concentric scan pattern. The result is confirmed by the performance enhancement with experimental data, especially under low-overlap conditions. These results suggest that the Fermat spiral pattern increases the quality of the reconstructed image and tolerance to data with imperfections.

© 2014 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. M. Guizar-Sicairos, J. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16(10), 7264–7272 (2008).
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    [CrossRef]
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    [CrossRef] [PubMed]
  15. M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. Kewish, R. Wepf, O. Bunk, F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 436–440 (2010).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  28. S. Hruszkewycz, M. Highland, M. Holt, D. Kim, C. Folkman, C. Thompson, A. Tripathi, G. Stephenson, S. Hong, P. Fuoss, “Imaging Local Polarization in Ferroelectric Thin Films by Coherent X-Ray Bragg Projection Ptychography,” Phys. Rev. Lett. 110, 177601 (2013).
    [CrossRef] [PubMed]
  29. N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equations of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
    [CrossRef]
  30. W. Press, S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University Press, 2002).

2014

2013

F. Zhang, I. Peterson, J. Vila-Comamala, A. Diaz, F. Berenguer, R. Bean, B. Chen, A. Menzel, I. Robinson, J. Rodenburg, “Translation position determination in ptychographic coherent diffraction imaging,” Opt. Express 21(11), 13592–13606 (2013).
[CrossRef] [PubMed]

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

S. Hruszkewycz, M. Highland, M. Holt, D. Kim, C. Folkman, C. Thompson, A. Tripathi, G. Stephenson, S. Hong, P. Fuoss, “Imaging Local Polarization in Ferroelectric Thin Films by Coherent X-Ray Bragg Projection Ptychography,” Phys. Rev. Lett. 110, 177601 (2013).
[CrossRef] [PubMed]

R. Wilke, M. Vassholz, T. Salditt, “Semi-transparent central stop in high-resolution X-ray ptychography using Kirkpatrick-Baez focusing,” Acta Cryst. A69, 490–497 (2013).
[CrossRef]

P. Thibault, A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[CrossRef] [PubMed]

X. Huang, H. Yan, E. Nazaretski, R. Conley, N. Bouet, J. Zhou, K. Lauer, L. Li, D. Eom, D. Legnini, R. Harder, I. Robinson, Y. Chu, “11 nm hard X-ray focus from a large-aperture multilayer Laue lens,” Sci. Rep. 3, 3562 (2013).
[PubMed]

M. Beckers, T. Senkbeil, T. Gorniak, K. Giewekemeyer, T. Salditt, A. Rosenhahn, “Drift correction in ptychographic diffractive imaging,” Ultramicroscopy 126, 44–47 (2013).
[CrossRef] [PubMed]

2012

P. Thibault, M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004 (2012).
[CrossRef]

A. Maiden, M. Humphry, M. Sarahan, B. Kraus, J. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[CrossRef] [PubMed]

M. Guizar-Sicairos, M. Holler, A. Diaz, J. Vila-Comamala, O. Bunk, A. Menzel, “Role of the illumination spatial-frequency spectrum for ptychography,” Phys. Rev. B 86, 100103R (2012).
[CrossRef]

X. Huang, M. Wojcik, N. Burdet, I. Peterson, G. Morrison, D. Vine, D. Legnini, R. Harder, Y. Chu, I. K. Robinson, “Quantitative X-ray wavefront measurements of Fresnel zone plate and K-B mirrors using phase retrieval,” Opt. Express 20(21), 24038–24048 (2012).
[CrossRef] [PubMed]

2010

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

C. Kewish, P. Thibault, M. Dierolf, O. Bunk, A. Menzel, J. Vila-Comamala, K. Jefimovs, F. Pfeiffer, “Ptychographic characterization of the wavefield in the focus of reflective hard X-ray optics,” Ultramicroscopy 110, 325–329 (2010).
[CrossRef] [PubMed]

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. Kewish, R. Wepf, O. Bunk, F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 436–440 (2010).
[CrossRef] [PubMed]

2009

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

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

X. Huang, H. Miao, J. Steinbrener, J. Nelson, A. Stewart, D. Shapiro, C. Jacobsen, “Signal to noise considerations in diffraction and conventional microscopy,” Opt. Express 17(16), 13541–13553 (2009).
[CrossRef] [PubMed]

2008

M. Guizar-Sicairos, J. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16(10), 7264–7272 (2008).
[CrossRef] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, 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, F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[CrossRef]

2007

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

2005

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. Risbud, T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95, 085503 (2005).
[CrossRef] [PubMed]

1979

H. Vogel, “A better way to construct the sunflower head,” Math. Biosci. 44, 179–189 (1979).
[CrossRef]

1970

R. Hegerl, W. Hoppe, “Dynamic theory of crystalline structure analysis by electron diffraction in inhomogeneous primary wave field,” Ber. Bunsenges. Phys. Chem. 74, 1148 (1970).
[CrossRef]

1953

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equations of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Bean, R.

Beckers, M.

M. Beckers, T. Senkbeil, T. Gorniak, K. Giewekemeyer, T. Salditt, A. Rosenhahn, “Drift correction in ptychographic diffractive imaging,” Ultramicroscopy 126, 44–47 (2013).
[CrossRef] [PubMed]

Berenguer, F.

Bouet, N.

X. Huang, H. Yan, E. Nazaretski, R. Conley, N. Bouet, J. Zhou, K. Lauer, L. Li, D. Eom, D. Legnini, R. Harder, I. Robinson, Y. Chu, “11 nm hard X-ray focus from a large-aperture multilayer Laue lens,” Sci. Rep. 3, 3562 (2013).
[PubMed]

Bunk, O.

M. Guizar-Sicairos, M. Holler, A. Diaz, J. Vila-Comamala, O. Bunk, A. Menzel, “Role of the illumination spatial-frequency spectrum for ptychography,” Phys. Rev. B 86, 100103R (2012).
[CrossRef]

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. Kewish, R. Wepf, O. Bunk, F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 436–440 (2010).
[CrossRef] [PubMed]

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

C. Kewish, P. Thibault, M. Dierolf, O. Bunk, A. Menzel, J. Vila-Comamala, K. Jefimovs, F. Pfeiffer, “Ptychographic characterization of the wavefield in the focus of reflective hard X-ray optics,” Ultramicroscopy 110, 325–329 (2010).
[CrossRef] [PubMed]

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

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, 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, F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[CrossRef]

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

Burdet, N.

Chen, B.

Chu, Y.

X. Huang, H. Yan, E. Nazaretski, R. Conley, N. Bouet, J. Zhou, K. Lauer, L. Li, D. Eom, D. Legnini, R. Harder, I. Robinson, Y. Chu, “11 nm hard X-ray focus from a large-aperture multilayer Laue lens,” Sci. Rep. 3, 3562 (2013).
[PubMed]

X. Huang, M. Wojcik, N. Burdet, I. Peterson, G. Morrison, D. Vine, D. Legnini, R. Harder, Y. Chu, I. K. Robinson, “Quantitative X-ray wavefront measurements of Fresnel zone plate and K-B mirrors using phase retrieval,” Opt. Express 20(21), 24038–24048 (2012).
[CrossRef] [PubMed]

Clark, J.

J. Clark, X. Huang, R. Harder, I. Robinson, “Dynamic imaging using ptychography,” Phys. Rev. Lett. 112, 113901 (2014).
[CrossRef] [PubMed]

Conley, R.

X. Huang, H. Yan, E. Nazaretski, R. Conley, N. Bouet, J. Zhou, K. Lauer, L. Li, D. Eom, D. Legnini, R. Harder, I. Robinson, Y. Chu, “11 nm hard X-ray focus from a large-aperture multilayer Laue lens,” Sci. Rep. 3, 3562 (2013).
[PubMed]

Cullis, A.

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

David, C.

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

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

Diaz, A.

F. Zhang, I. Peterson, J. Vila-Comamala, A. Diaz, F. Berenguer, R. Bean, B. Chen, A. Menzel, I. Robinson, J. Rodenburg, “Translation position determination in ptychographic coherent diffraction imaging,” Opt. Express 21(11), 13592–13606 (2013).
[CrossRef] [PubMed]

M. Guizar-Sicairos, M. Holler, A. Diaz, J. Vila-Comamala, O. Bunk, A. Menzel, “Role of the illumination spatial-frequency spectrum for ptychography,” Phys. Rev. B 86, 100103R (2012).
[CrossRef]

Dierolf, M.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. Kewish, R. Wepf, O. Bunk, F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 436–440 (2010).
[CrossRef] [PubMed]

C. Kewish, P. Thibault, M. Dierolf, O. Bunk, A. Menzel, J. Vila-Comamala, K. Jefimovs, F. Pfeiffer, “Ptychographic characterization of the wavefield in the focus of reflective hard X-ray optics,” Ultramicroscopy 110, 325–329 (2010).
[CrossRef] [PubMed]

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

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

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, 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, F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[CrossRef]

Dobson, B.

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

Eom, D.

X. Huang, H. Yan, E. Nazaretski, R. Conley, N. Bouet, J. Zhou, K. Lauer, L. Li, D. Eom, D. Legnini, R. Harder, I. Robinson, Y. Chu, “11 nm hard X-ray focus from a large-aperture multilayer Laue lens,” Sci. Rep. 3, 3562 (2013).
[PubMed]

Fienup, J.

Flannery, B.

W. Press, S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University Press, 2002).

Folkman, C.

S. Hruszkewycz, M. Highland, M. Holt, D. Kim, C. Folkman, C. Thompson, A. Tripathi, G. Stephenson, S. Hong, P. Fuoss, “Imaging Local Polarization in Ferroelectric Thin Films by Coherent X-Ray Bragg Projection Ptychography,” Phys. Rev. Lett. 110, 177601 (2013).
[CrossRef] [PubMed]

Fuoss, P.

S. Hruszkewycz, M. Highland, M. Holt, D. Kim, C. Folkman, C. Thompson, A. Tripathi, G. Stephenson, S. Hong, P. Fuoss, “Imaging Local Polarization in Ferroelectric Thin Films by Coherent X-Ray Bragg Projection Ptychography,” Phys. Rev. Lett. 110, 177601 (2013).
[CrossRef] [PubMed]

Gianoncelli, A.

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

Giewekemeyer, K.

M. Beckers, T. Senkbeil, T. Gorniak, K. Giewekemeyer, T. Salditt, A. Rosenhahn, “Drift correction in ptychographic diffractive imaging,” Ultramicroscopy 126, 44–47 (2013).
[CrossRef] [PubMed]

Gorniak, T.

M. Beckers, T. Senkbeil, T. Gorniak, K. Giewekemeyer, T. Salditt, A. Rosenhahn, “Drift correction in ptychographic diffractive imaging,” Ultramicroscopy 126, 44–47 (2013).
[CrossRef] [PubMed]

Guizar-Sicairos, M.

P. Thibault, M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004 (2012).
[CrossRef]

M. Guizar-Sicairos, M. Holler, A. Diaz, J. Vila-Comamala, O. Bunk, A. Menzel, “Role of the illumination spatial-frequency spectrum for ptychography,” Phys. Rev. B 86, 100103R (2012).
[CrossRef]

M. Guizar-Sicairos, J. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16(10), 7264–7272 (2008).
[CrossRef] [PubMed]

Harder, R.

J. Clark, X. Huang, R. Harder, I. Robinson, “Dynamic imaging using ptychography,” Phys. Rev. Lett. 112, 113901 (2014).
[CrossRef] [PubMed]

X. Huang, H. Yan, E. Nazaretski, R. Conley, N. Bouet, J. Zhou, K. Lauer, L. Li, D. Eom, D. Legnini, R. Harder, I. Robinson, Y. Chu, “11 nm hard X-ray focus from a large-aperture multilayer Laue lens,” Sci. Rep. 3, 3562 (2013).
[PubMed]

X. Huang, M. Wojcik, N. Burdet, I. Peterson, G. Morrison, D. Vine, D. Legnini, R. Harder, Y. Chu, I. K. Robinson, “Quantitative X-ray wavefront measurements of Fresnel zone plate and K-B mirrors using phase retrieval,” Opt. Express 20(21), 24038–24048 (2012).
[CrossRef] [PubMed]

Hegerl, R.

R. Hegerl, W. Hoppe, “Dynamic theory of crystalline structure analysis by electron diffraction in inhomogeneous primary wave field,” Ber. Bunsenges. Phys. Chem. 74, 1148 (1970).
[CrossRef]

Highland, M.

S. Hruszkewycz, M. Highland, M. Holt, D. Kim, C. Folkman, C. Thompson, A. Tripathi, G. Stephenson, S. Hong, P. Fuoss, “Imaging Local Polarization in Ferroelectric Thin Films by Coherent X-Ray Bragg Projection Ptychography,” Phys. Rev. Lett. 110, 177601 (2013).
[CrossRef] [PubMed]

Holler, M.

M. Guizar-Sicairos, M. Holler, A. Diaz, J. Vila-Comamala, O. Bunk, A. Menzel, “Role of the illumination spatial-frequency spectrum for ptychography,” Phys. Rev. B 86, 100103R (2012).
[CrossRef]

Holt, M.

S. Hruszkewycz, M. Highland, M. Holt, D. Kim, C. Folkman, C. Thompson, A. Tripathi, G. Stephenson, S. Hong, P. Fuoss, “Imaging Local Polarization in Ferroelectric Thin Films by Coherent X-Ray Bragg Projection Ptychography,” Phys. Rev. Lett. 110, 177601 (2013).
[CrossRef] [PubMed]

Hong, S.

S. Hruszkewycz, M. Highland, M. Holt, D. Kim, C. Folkman, C. Thompson, A. Tripathi, G. Stephenson, S. Hong, P. Fuoss, “Imaging Local Polarization in Ferroelectric Thin Films by Coherent X-Ray Bragg Projection Ptychography,” Phys. Rev. Lett. 110, 177601 (2013).
[CrossRef] [PubMed]

Hoppe, W.

R. Hegerl, W. Hoppe, “Dynamic theory of crystalline structure analysis by electron diffraction in inhomogeneous primary wave field,” Ber. Bunsenges. Phys. Chem. 74, 1148 (1970).
[CrossRef]

Hruszkewycz, S.

S. Hruszkewycz, M. Highland, M. Holt, D. Kim, C. Folkman, C. Thompson, A. Tripathi, G. Stephenson, S. Hong, P. Fuoss, “Imaging Local Polarization in Ferroelectric Thin Films by Coherent X-Ray Bragg Projection Ptychography,” Phys. Rev. Lett. 110, 177601 (2013).
[CrossRef] [PubMed]

Huang, X.

Humphry, M.

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J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. Risbud, T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95, 085503 (2005).
[CrossRef] [PubMed]

S. Hruszkewycz, M. Highland, M. Holt, D. Kim, C. Folkman, C. Thompson, A. Tripathi, G. Stephenson, S. Hong, P. Fuoss, “Imaging Local Polarization in Ferroelectric Thin Films by Coherent X-Ray Bragg Projection Ptychography,” Phys. Rev. Lett. 110, 177601 (2013).
[CrossRef] [PubMed]

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

J. Clark, X. Huang, R. Harder, I. Robinson, “Dynamic imaging using ptychography,” Phys. Rev. Lett. 112, 113901 (2014).
[CrossRef] [PubMed]

Sci. Rep.

X. Huang, H. Yan, E. Nazaretski, R. Conley, N. Bouet, J. Zhou, K. Lauer, L. Li, D. Eom, D. Legnini, R. Harder, I. Robinson, Y. Chu, “11 nm hard X-ray focus from a large-aperture multilayer Laue lens,” Sci. Rep. 3, 3562 (2013).
[PubMed]

Science

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

Ultramicroscopy

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

A. Maiden, M. Humphry, M. Sarahan, B. Kraus, J. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[CrossRef] [PubMed]

M. Beckers, T. Senkbeil, T. Gorniak, K. Giewekemeyer, T. Salditt, A. Rosenhahn, “Drift correction in ptychographic diffractive imaging,” Ultramicroscopy 126, 44–47 (2013).
[CrossRef] [PubMed]

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[CrossRef]

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

C. Kewish, P. Thibault, M. Dierolf, O. Bunk, A. Menzel, J. Vila-Comamala, K. Jefimovs, F. Pfeiffer, “Ptychographic characterization of the wavefield in the focus of reflective hard X-ray optics,” Ultramicroscopy 110, 325–329 (2010).
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Figures (10)

Fig. 1
Fig. 1

Scan positions for ptychograph: (a) 11 × 11 mesh pattern with 121 points, (b) concentric pattern with 5 points in the first ring, 50 pixel radial increment steps and 119 points in total, (c) Fermat spiral pattern with 31 pixel increments and 119 points in total, which contains two sets of equiangular spirals: 13 clockwise (red curves) and 21 counterclockwise (blue curves). It is significant that 13 and 21 are two consecutive Fibonacci numbers.

Fig. 2
Fig. 2

Top row: Amplitudes and phases of the simulated objects. Bottom row: Amplitudes and phases of the simulated illuminations. (a) for simulation set 1, (b) for simulation set 2.

Fig. 3
Fig. 3

(a) The original object amplitude. The reconstructed amplitude using mesh pattern (b), concentric pattern (c) and Fermat spiral pattern (d). (e)(f)(g) are the difference images between the original amplitude and the reconstructed amplitudes.

Fig. 4
Fig. 4

(a) The original object phase. The reconstructed phase using mesh pattern (b), concentric pattern (c), and Fermat spiral pattern (d). (e)(f)(g) are the difference images between the original phase and the reconstructed phases.

Fig. 5
Fig. 5

(a)(b)(c): Fourier transforms of the mesh, concentric and Fermat spiral scan patterns. (d) Azimuthally averaged power spectrum density curves. Δ is denoted for the detector’s pixel size. The variation and normalized power are calculated in high spatial frequency region as indicated by the purple box. The summed power is normalized by the number of points.

Fig. 6
Fig. 6

Histograms of the summed distance for mesh (a), concentric (b) and Fermat spiral (c), respectively. A smaller distance distribution range and a less histogram variation indicate a more uniform overall point separation.

Fig. 7
Fig. 7

Reconstruction results with noisy data and positioning error: (a) and (b) are a typical reconstructed objects’ amplitude and phase with concentric scan pattern, (c) and (d) are typical reconstructions with a Fermat scan pattern.

Fig. 8
Fig. 8

Reconstructed probe amplitude (a) and object phase (b) from experimental data with Fermat spiral scan pattern at a high overlap condition. Reconstructed probe amplitude (c) and object phase (d) from experimental data with a concentric scan pattern at a high overlap condition. Reconstructed probe amplitude (e) and object phase (f) from experimental data with a Fermat spiral scan pattern at a low overlap condition. Reconstructed probe amplitude (g) and object phase (h) from experimental data with a concentric scan pattern at a low overlap condition.

Fig. 9
Fig. 9

(a)(b): Hexagonal scan pattern and its Fourier transform. (c)(d): Mesh scan pattern disturbed by a random positioning noise and its Fourier transform. (e) Power spectrum density curves of (c) and (d), where the variation and normalized power are calculated in the area indicated by the purple box. Deviations in local overlap are present in the noisy mesh scan pattern as shown in (c), where the overlap ratio is high in the area indicated by the red box and low in the area enclosed by the blue box.

Fig. 10
Fig. 10

Top row: The default scan trajectories for mesh, concentric and Fermat spiral patterns. Bottom row: The scan trajectories sorted by minimizing the total movement distances using the Metropolis algorithm.

Equations (4)

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r = c n , θ = n ϕ 0 ,
R = I 0 I recon 2 I 0 + I recon 2 ,
SNR = r c c 1 r c c , r c c = ( I recon 1 I recon 1 ) ( I recon 2 I recon 2 ) ( I recon 1 I recon 1 ) 2 ( I recon 2 I recon 2 ) 2 ,
σ = j = 1 N 1 | P ( r r j ) | | P ( r r j + 1 ) | j = 1 N 1 | P ( r r j ) | 2 ,

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