Abstract

Ptychographic X-ray computed tomography is a coherent diffractive imaging method that offers nanometer-scale resolution with quantitative contrast. It offers the possibility to study relatively thick samples by using high energy X-ray photons and exploiting the phase contrast. However, the limited depth of field forces a compromise between resolution and sample thickness. Multi-slice techniques have been used to account for propagation effects within the sample, enabling imaging beyond the depth-of-field limit. Here we introduce and experimentally demonstrate our multi-slice algorithms that allow for the reconstruction of multiple object slices and the incident illumination, as well as the retrieval of unknown object thickness. Additionally, through numerical studies, we show that smaller scanning steps surprisingly increase the depth of field, which can be further extended by the use of multi-slice methods under conditions stated by theoretical expressions. The results presented here will be instrumental for the routine implementation of the technique for X-ray nanotomography.

© 2016 Optical Society of America

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2015 (4)

K. Shimomura, A. Suzuki, M. Hirose, and Y. Takahashi, “Precession x-ray ptychography with multislice approach,” Phys. Rev. B 91, 214114 (2015).
[Crossref]

J. C. da Silva and A. Menzel, “Elementary signals in ptychography,” Opt. Express 23, 33812–33821 (2015).
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L. Tian and L. Waller, “3D intensity and phase imaging from light field measurements in an led array microscope,” Optica 2, 104–111 (2015).
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P. Li, D. J. Batey, T. B. Edo, and J. M. Rodenburg, “Separation of three-dimensional scattering effects in tilt-series fourier ptychography,” Ultramicroscopy 158, 1–7 (2015).
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2014 (6)

D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
[Crossref] [PubMed]

T. M. Godden, R. Suman, M. J. Humphry, J. M. Rodenburg, and A. M. Maiden, “Ptychographic microscope for three-dimensional imaging,” Opt. Express 22, 12513–12523 (2014).
[Crossref] [PubMed]

A. Suzuki, S. Furutaku, K. Shimomura, K. Yamauchi, Y. Kohmura, T. Ishikawa, and Y. Takahashi, “High-resolution multislice x-ray ptychography of extended thick objects,” Phys. Rev. Lett. 112, 053903 (2014).
[Crossref] [PubMed]

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Rep. 4, 3857 (2014).
[Crossref] [PubMed]

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. A. Padmore, “Chemical composition mapping with nanometre resolution by soft X-ray microscopy,” Nature Photon. 8, 765–769 (2014).
[Crossref]

X. Huang, H. Yan, R. Harder, Y. Hwu, I. K. Robinson, and Y. S. Chu, “Optimization of overlap uniformness for ptychography,” Opt. Express 22, 12634–12644 (2014).
[Crossref] [PubMed]

2013 (3)

R. Hoppe, J. Reinhardt, G. Hofmann, J. Patommel, J.-D. Grunwaldt, C. D. Damsgaard, G. Wellenreuther, G. Falkenberg, and C. Schroer, “High-resolution chemical imaging of gold nanoparticles using hard x-ray ptychography,” Appl. Phys. Lett. 102, 203104 (2013).
[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, 053850 (2013).
[Crossref]

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

2012 (4)

M. J. Humphry, B. Kraus, A. C. Hurst, A. M. Maiden, and J. M. Rodenburg, “Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging,” Nat. Commun. 3, 730 (2012).
[Crossref] [PubMed]

A. M. Maiden, M. J. Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. A 29, 1606–1614 (2012).
[Crossref]

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

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (2012).
[Crossref]

2011 (4)

R. Hovden, H. L. Xin, and D. A. Muller, “Extended depth of field for high-resolution scanning transmission electron microscopy,” Microsc. Microanal. 17, 75–80 (2011).
[Crossref]

M. Guizar-Sicairos, A. Diaz, M. Holler, M. S. Lucas, A. Menzel, R. A. Wepf, and O. Bunk, “Phase tomography from x-ray coherent diffractive imaging projections,” Opt. Express 19, 21345–21357 (2011).
[Crossref] [PubMed]

S. Gorelick, J. Vila-Comamala, V. A. Guzenko, R. Barrett, M. Salomé, and C. David, “High-efficiency Fresnel zone plates for hard x-rays by 100 kev e-beam lithography and electroplating,” J. Synchrotron Radiat. 18, 442–446 (2011).
[Crossref] [PubMed]

J. Vila-Comamala, A. Diaz, M. Guizar-Sicairos, A. Mantion, C. M. Kewish, A. Menzel, O. Bunk, and C. David, “Characterization of high-resolution diffractive x-ray optics by ptychographic coherent diffractive imaging,” Opt. Express 19, 21333–21344 (2011).
[Crossref] [PubMed]

2010 (2)

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

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[Crossref] [PubMed]

2009 (3)

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

B. Henrich, A. Bergamaschi, C. Broennimann, R. Dinapoli, E. F. Eikenberry, I. Johnson, M. Kobas, P. Kraft, A. Mozzanica, and B. Schmitt, “Pilatus: A single photon counting pixel detector for x-ray applications,” Nucl. Instrum. Methods Phys. Res. A 607, 247–249 (2009).
[Crossref]

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

2008 (3)

2007 (2)

J. Rodenburg, A. Hurst, A. Cullis, B. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref] [PubMed]

S. Mikula, I. Trotts, J. M. Stone, and E. G. Jones, “Internet-enabled high-resolution brain mapping and virtual microscopy,” Neuroimage 35, 9–15 (2007).
[Crossref] [PubMed]

2006 (3)

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006).
[Crossref]

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” ACM Trans. Graph. 25, 924–934 (2006).
[Crossref]

K. E. Herkenhoff, S. W. Squyres, R. Anderson, B. A. Archinal, R. E. Arvidson, J. M. Barrett, K. J. Becker, J. F. Bell, C. Budney, N. A. Cabrol, M. G. Chapman, D. Cook, B. L. Ehlmann, J. Farmer, B. Franklin, L. R. Gaddis, D. M. Galuszka, P. A. Garcia, T. M. Hare, E. Howington-Kraus, J. R. Johnson, S. Johnson, K. Kinch, R. L. Kirk, E. M. Lee, C. Leff, M. Lemmon, M. B. Madsen, J. N. Maki, K. F. Mullins, B. L. Redding, L. Richter, M. R. Rosiek, M. H. Sims, L. A. Soderblom, N. Spanovich, R. Springer, R. M. Sucharski, T. Sucharski, R. Sullivan, J. M. Torson, and A. Yen, “Overview of the microscopic imager investigation during spirit’s first 450 sols in gusev crater,” J. Geophys. Res.-Planet 111, E02S04 (2006).
[Crossref]

2005 (2)

J.-A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2, 920–931 (2005).
[Crossref] [PubMed]

M. van Heel and M. Schatz, “Fourier shell correlation threshold criteria,” J. Struct. Biol. 151, 250–262 (2005).
[Crossref] [PubMed]

2004 (1)

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]

2003 (1)

1997 (1)

1996 (1)

R. H. Webb, “Confocal optical microscopy,” Rep. Prog. Phys. 59, 427 (1996).
[Crossref]

1995 (1)

1992 (1)

J. M. Rodenburg and R. H. T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Philos. Trans. R. Soc. London, Ser. A 339, 521–553 (1992).
[Crossref]

1982 (1)

1974 (1)

P. Goodman and A. Moodie, “Numerical evaluations of n-beam wave functions in electron scattering by the multi-slice method,” Acta Crystallogr. Sect. A 30, 280–290 (1974).
[Crossref]

1967 (1)

1957 (1)

J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new theoretical approach,” Acta Crystallogr. 10, 609–619 (1957).
[Crossref]

Adams, A.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” ACM Trans. Graph. 25, 924–934 (2006).
[Crossref]

Anderson, R.

K. E. Herkenhoff, S. W. Squyres, R. Anderson, B. A. Archinal, R. E. Arvidson, J. M. Barrett, K. J. Becker, J. F. Bell, C. Budney, N. A. Cabrol, M. G. Chapman, D. Cook, B. L. Ehlmann, J. Farmer, B. Franklin, L. R. Gaddis, D. M. Galuszka, P. A. Garcia, T. M. Hare, E. Howington-Kraus, J. R. Johnson, S. Johnson, K. Kinch, R. L. Kirk, E. M. Lee, C. Leff, M. Lemmon, M. B. Madsen, J. N. Maki, K. F. Mullins, B. L. Redding, L. Richter, M. R. Rosiek, M. H. Sims, L. A. Soderblom, N. Spanovich, R. Springer, R. M. Sucharski, T. Sucharski, R. Sullivan, J. M. Torson, and A. Yen, “Overview of the microscopic imager investigation during spirit’s first 450 sols in gusev crater,” J. Geophys. Res.-Planet 111, E02S04 (2006).
[Crossref]

Archinal, B. A.

K. E. Herkenhoff, S. W. Squyres, R. Anderson, B. A. Archinal, R. E. Arvidson, J. M. Barrett, K. J. Becker, J. F. Bell, C. Budney, N. A. Cabrol, M. G. Chapman, D. Cook, B. L. Ehlmann, J. Farmer, B. Franklin, L. R. Gaddis, D. M. Galuszka, P. A. Garcia, T. M. Hare, E. Howington-Kraus, J. R. Johnson, S. Johnson, K. Kinch, R. L. Kirk, E. M. Lee, C. Leff, M. Lemmon, M. B. Madsen, J. N. Maki, K. F. Mullins, B. L. Redding, L. Richter, M. R. Rosiek, M. H. Sims, L. A. Soderblom, N. Spanovich, R. Springer, R. M. Sucharski, T. Sucharski, R. Sullivan, J. M. Torson, and A. Yen, “Overview of the microscopic imager investigation during spirit’s first 450 sols in gusev crater,” J. Geophys. Res.-Planet 111, E02S04 (2006).
[Crossref]

Arvidson, R. E.

K. E. Herkenhoff, S. W. Squyres, R. Anderson, B. A. Archinal, R. E. Arvidson, J. M. Barrett, K. J. Becker, J. F. Bell, C. Budney, N. A. Cabrol, M. G. Chapman, D. Cook, B. L. Ehlmann, J. Farmer, B. Franklin, L. R. Gaddis, D. M. Galuszka, P. A. Garcia, T. M. Hare, E. Howington-Kraus, J. R. Johnson, S. Johnson, K. Kinch, R. L. Kirk, E. M. Lee, C. Leff, M. Lemmon, M. B. Madsen, J. N. Maki, K. F. Mullins, B. L. Redding, L. Richter, M. R. Rosiek, M. H. Sims, L. A. Soderblom, N. Spanovich, R. Springer, R. M. Sucharski, T. Sucharski, R. Sullivan, J. M. Torson, and A. Yen, “Overview of the microscopic imager investigation during spirit’s first 450 sols in gusev crater,” J. Geophys. Res.-Planet 111, E02S04 (2006).
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Barrett, J. M.

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Batey, D. J.

P. Li, D. J. Batey, T. B. Edo, and J. M. Rodenburg, “Separation of three-dimensional scattering effects in tilt-series fourier ptychography,” Ultramicroscopy 158, 1–7 (2015).
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K. E. Herkenhoff, S. W. Squyres, R. Anderson, B. A. Archinal, R. E. Arvidson, J. M. Barrett, K. J. Becker, J. F. Bell, C. Budney, N. A. Cabrol, M. G. Chapman, D. Cook, B. L. Ehlmann, J. Farmer, B. Franklin, L. R. Gaddis, D. M. Galuszka, P. A. Garcia, T. M. Hare, E. Howington-Kraus, J. R. Johnson, S. Johnson, K. Kinch, R. L. Kirk, E. M. Lee, C. Leff, M. Lemmon, M. B. Madsen, J. N. Maki, K. F. Mullins, B. L. Redding, L. Richter, M. R. Rosiek, M. H. Sims, L. A. Soderblom, N. Spanovich, R. Springer, R. M. Sucharski, T. Sucharski, R. Sullivan, J. M. Torson, and A. Yen, “Overview of the microscopic imager investigation during spirit’s first 450 sols in gusev crater,” J. Geophys. Res.-Planet 111, E02S04 (2006).
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K. E. Herkenhoff, S. W. Squyres, R. Anderson, B. A. Archinal, R. E. Arvidson, J. M. Barrett, K. J. Becker, J. F. Bell, C. Budney, N. A. Cabrol, M. G. Chapman, D. Cook, B. L. Ehlmann, J. Farmer, B. Franklin, L. R. Gaddis, D. M. Galuszka, P. A. Garcia, T. M. Hare, E. Howington-Kraus, J. R. Johnson, S. Johnson, K. Kinch, R. L. Kirk, E. M. Lee, C. Leff, M. Lemmon, M. B. Madsen, J. N. Maki, K. F. Mullins, B. L. Redding, L. Richter, M. R. Rosiek, M. H. Sims, L. A. Soderblom, N. Spanovich, R. Springer, R. M. Sucharski, T. Sucharski, R. Sullivan, J. M. Torson, and A. Yen, “Overview of the microscopic imager investigation during spirit’s first 450 sols in gusev crater,” J. Geophys. Res.-Planet 111, E02S04 (2006).
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B. Henrich, A. Bergamaschi, C. Broennimann, R. Dinapoli, E. F. Eikenberry, I. Johnson, M. Kobas, P. Kraft, A. Mozzanica, and B. Schmitt, “Pilatus: A single photon counting pixel detector for x-ray applications,” Nucl. Instrum. Methods Phys. Res. A 607, 247–249 (2009).
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K. E. Herkenhoff, S. W. Squyres, R. Anderson, B. A. Archinal, R. E. Arvidson, J. M. Barrett, K. J. Becker, J. F. Bell, C. Budney, N. A. Cabrol, M. G. Chapman, D. Cook, B. L. Ehlmann, J. Farmer, B. Franklin, L. R. Gaddis, D. M. Galuszka, P. A. Garcia, T. M. Hare, E. Howington-Kraus, J. R. Johnson, S. Johnson, K. Kinch, R. L. Kirk, E. M. Lee, C. Leff, M. Lemmon, M. B. Madsen, J. N. Maki, K. F. Mullins, B. L. Redding, L. Richter, M. R. Rosiek, M. H. Sims, L. A. Soderblom, N. Spanovich, R. Springer, R. M. Sucharski, T. Sucharski, R. Sullivan, J. M. Torson, and A. Yen, “Overview of the microscopic imager investigation during spirit’s first 450 sols in gusev crater,” J. Geophys. Res.-Planet 111, E02S04 (2006).
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J. Vila-Comamala, A. Diaz, M. Guizar-Sicairos, A. Mantion, C. M. Kewish, A. Menzel, O. Bunk, and C. David, “Characterization of high-resolution diffractive x-ray optics by ptychographic coherent diffractive imaging,” Opt. Express 19, 21333–21344 (2011).
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M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. Von Koenig, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010).
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M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 436–439 (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, 338–343 (2009).
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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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J. Rodenburg, A. Hurst, A. Cullis, B. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
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Celestre, R.

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Chapman, M. G.

K. E. Herkenhoff, S. W. Squyres, R. Anderson, B. A. Archinal, R. E. Arvidson, J. M. Barrett, K. J. Becker, J. F. Bell, C. Budney, N. A. Cabrol, M. G. Chapman, D. Cook, B. L. Ehlmann, J. Farmer, B. Franklin, L. R. Gaddis, D. M. Galuszka, P. A. Garcia, T. M. Hare, E. Howington-Kraus, J. R. Johnson, S. Johnson, K. Kinch, R. L. Kirk, E. M. Lee, C. Leff, M. Lemmon, M. B. Madsen, J. N. Maki, K. F. Mullins, B. L. Redding, L. Richter, M. R. Rosiek, M. H. Sims, L. A. Soderblom, N. Spanovich, R. Springer, R. M. Sucharski, T. Sucharski, R. Sullivan, J. M. Torson, and A. Yen, “Overview of the microscopic imager investigation during spirit’s first 450 sols in gusev crater,” J. Geophys. Res.-Planet 111, E02S04 (2006).
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Claus, D.

D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
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Cullis, A.

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Damsgaard, C. D.

R. Hoppe, J. Reinhardt, G. Hofmann, J. Patommel, J.-D. Grunwaldt, C. D. Damsgaard, G. Wellenreuther, G. Falkenberg, and C. Schroer, “High-resolution chemical imaging of gold nanoparticles using hard x-ray ptychography,” Appl. Phys. Lett. 102, 203104 (2013).
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J. Vila-Comamala, A. Diaz, M. Guizar-Sicairos, A. Mantion, C. M. Kewish, A. Menzel, O. Bunk, and C. David, “Characterization of high-resolution diffractive x-ray optics by ptychographic coherent diffractive imaging,” Opt. Express 19, 21333–21344 (2011).
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S. Gorelick, J. Vila-Comamala, V. A. Guzenko, R. Barrett, M. Salomé, and C. David, “High-efficiency Fresnel zone plates for hard x-rays by 100 kev e-beam lithography and electroplating,” J. Synchrotron Radiat. 18, 442–446 (2011).
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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).
[Crossref] [PubMed]

J. Rodenburg, A. Hurst, A. Cullis, B. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
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Diaz, A.

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Rep. 4, 3857 (2014).
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A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (2012).
[Crossref]

J. Vila-Comamala, A. Diaz, M. Guizar-Sicairos, A. Mantion, C. M. Kewish, A. Menzel, O. Bunk, and C. David, “Characterization of high-resolution diffractive x-ray optics by ptychographic coherent diffractive imaging,” Opt. Express 19, 21333–21344 (2011).
[Crossref] [PubMed]

M. Guizar-Sicairos, A. Diaz, M. Holler, M. S. Lucas, A. Menzel, R. A. Wepf, and O. Bunk, “Phase tomography from x-ray coherent diffractive imaging projections,” Opt. Express 19, 21345–21357 (2011).
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M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. Von Koenig, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010).
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M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[Crossref] [PubMed]

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

B. Henrich, A. Bergamaschi, C. Broennimann, R. Dinapoli, E. F. Eikenberry, I. Johnson, M. Kobas, P. Kraft, A. Mozzanica, and B. Schmitt, “Pilatus: A single photon counting pixel detector for x-ray applications,” Nucl. Instrum. Methods Phys. Res. A 607, 247–249 (2009).
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Dobson, B.

J. Rodenburg, A. Hurst, A. Cullis, B. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
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Dowski, E. R.

Edo, T. B.

P. Li, D. J. Batey, T. B. Edo, and J. M. Rodenburg, “Separation of three-dimensional scattering effects in tilt-series fourier ptychography,” Ultramicroscopy 158, 1–7 (2015).
[Crossref] [PubMed]

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, 053850 (2013).
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Ehlmann, B. L.

K. E. Herkenhoff, S. W. Squyres, R. Anderson, B. A. Archinal, R. E. Arvidson, J. M. Barrett, K. J. Becker, J. F. Bell, C. Budney, N. A. Cabrol, M. G. Chapman, D. Cook, B. L. Ehlmann, J. Farmer, B. Franklin, L. R. Gaddis, D. M. Galuszka, P. A. Garcia, T. M. Hare, E. Howington-Kraus, J. R. Johnson, S. Johnson, K. Kinch, R. L. Kirk, E. M. Lee, C. Leff, M. Lemmon, M. B. Madsen, J. N. Maki, K. F. Mullins, B. L. Redding, L. Richter, M. R. Rosiek, M. H. Sims, L. A. Soderblom, N. Spanovich, R. Springer, R. M. Sucharski, T. Sucharski, R. Sullivan, J. M. Torson, and A. Yen, “Overview of the microscopic imager investigation during spirit’s first 450 sols in gusev crater,” J. Geophys. Res.-Planet 111, E02S04 (2006).
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Eikenberry, E. F.

B. Henrich, A. Bergamaschi, C. Broennimann, R. Dinapoli, E. F. Eikenberry, I. Johnson, M. Kobas, P. Kraft, A. Mozzanica, and B. Schmitt, “Pilatus: A single photon counting pixel detector for x-ray applications,” Nucl. Instrum. Methods Phys. Res. A 607, 247–249 (2009).
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Falkenberg, G.

R. Hoppe, J. Reinhardt, G. Hofmann, J. Patommel, J.-D. Grunwaldt, C. D. Damsgaard, G. Wellenreuther, G. Falkenberg, and C. Schroer, “High-resolution chemical imaging of gold nanoparticles using hard x-ray ptychography,” Appl. Phys. Lett. 102, 203104 (2013).
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Färm, E.

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Rep. 4, 3857 (2014).
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Furutaku, S.

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K. E. Herkenhoff, S. W. Squyres, R. Anderson, B. A. Archinal, R. E. Arvidson, J. M. Barrett, K. J. Becker, J. F. Bell, C. Budney, N. A. Cabrol, M. G. Chapman, D. Cook, B. L. Ehlmann, J. Farmer, B. Franklin, L. R. Gaddis, D. M. Galuszka, P. A. Garcia, T. M. Hare, E. Howington-Kraus, J. R. Johnson, S. Johnson, K. Kinch, R. L. Kirk, E. M. Lee, C. Leff, M. Lemmon, M. B. Madsen, J. N. Maki, K. F. Mullins, B. L. Redding, L. Richter, M. R. Rosiek, M. H. Sims, L. A. Soderblom, N. Spanovich, R. Springer, R. M. Sucharski, T. Sucharski, R. Sullivan, J. M. Torson, and A. Yen, “Overview of the microscopic imager investigation during spirit’s first 450 sols in gusev crater,” J. Geophys. Res.-Planet 111, E02S04 (2006).
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K. E. Herkenhoff, S. W. Squyres, R. Anderson, B. A. Archinal, R. E. Arvidson, J. M. Barrett, K. J. Becker, J. F. Bell, C. Budney, N. A. Cabrol, M. G. Chapman, D. Cook, B. L. Ehlmann, J. Farmer, B. Franklin, L. R. Gaddis, D. M. Galuszka, P. A. Garcia, T. M. Hare, E. Howington-Kraus, J. R. Johnson, S. Johnson, K. Kinch, R. L. Kirk, E. M. Lee, C. Leff, M. Lemmon, M. B. Madsen, J. N. Maki, K. F. Mullins, B. L. Redding, L. Richter, M. R. Rosiek, M. H. Sims, L. A. Soderblom, N. Spanovich, R. Springer, R. M. Sucharski, T. Sucharski, R. Sullivan, J. M. Torson, and A. Yen, “Overview of the microscopic imager investigation during spirit’s first 450 sols in gusev crater,” J. Geophys. Res.-Planet 111, E02S04 (2006).
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K. E. Herkenhoff, S. W. Squyres, R. Anderson, B. A. Archinal, R. E. Arvidson, J. M. Barrett, K. J. Becker, J. F. Bell, C. Budney, N. A. Cabrol, M. G. Chapman, D. Cook, B. L. Ehlmann, J. Farmer, B. Franklin, L. R. Gaddis, D. M. Galuszka, P. A. Garcia, T. M. Hare, E. Howington-Kraus, J. R. Johnson, S. Johnson, K. Kinch, R. L. Kirk, E. M. Lee, C. Leff, M. Lemmon, M. B. Madsen, J. N. Maki, K. F. Mullins, B. L. Redding, L. Richter, M. R. Rosiek, M. H. Sims, L. A. Soderblom, N. Spanovich, R. Springer, R. M. Sucharski, T. Sucharski, R. Sullivan, J. M. Torson, and A. Yen, “Overview of the microscopic imager investigation during spirit’s first 450 sols in gusev crater,” J. Geophys. Res.-Planet 111, E02S04 (2006).
[Crossref]

Sullivan, R.

K. E. Herkenhoff, S. W. Squyres, R. Anderson, B. A. Archinal, R. E. Arvidson, J. M. Barrett, K. J. Becker, J. F. Bell, C. Budney, N. A. Cabrol, M. G. Chapman, D. Cook, B. L. Ehlmann, J. Farmer, B. Franklin, L. R. Gaddis, D. M. Galuszka, P. A. Garcia, T. M. Hare, E. Howington-Kraus, J. R. Johnson, S. Johnson, K. Kinch, R. L. Kirk, E. M. Lee, C. Leff, M. Lemmon, M. B. Madsen, J. N. Maki, K. F. Mullins, B. L. Redding, L. Richter, M. R. Rosiek, M. H. Sims, L. A. Soderblom, N. Spanovich, R. Springer, R. M. Sucharski, T. Sucharski, R. Sullivan, J. M. Torson, and A. Yen, “Overview of the microscopic imager investigation during spirit’s first 450 sols in gusev crater,” J. Geophys. Res.-Planet 111, E02S04 (2006).
[Crossref]

Suman, R.

Suzuki, A.

K. Shimomura, A. Suzuki, M. Hirose, and Y. Takahashi, “Precession x-ray ptychography with multislice approach,” Phys. Rev. B 91, 214114 (2015).
[Crossref]

A. Suzuki, S. Furutaku, K. Shimomura, K. Yamauchi, Y. Kohmura, T. Ishikawa, and Y. Takahashi, “High-resolution multislice x-ray ptychography of extended thick objects,” Phys. Rev. Lett. 112, 053903 (2014).
[Crossref] [PubMed]

Takahashi, Y.

K. Shimomura, A. Suzuki, M. Hirose, and Y. Takahashi, “Precession x-ray ptychography with multislice approach,” Phys. Rev. B 91, 214114 (2015).
[Crossref]

A. Suzuki, S. Furutaku, K. Shimomura, K. Yamauchi, Y. Kohmura, T. Ishikawa, and Y. Takahashi, “High-resolution multislice x-ray ptychography of extended thick objects,” Phys. Rev. Lett. 112, 053903 (2014).
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Thibault, P.

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

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

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (2012).
[Crossref]

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[Crossref] [PubMed]

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. Von Koenig, O. Bunk, and 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, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
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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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Tian, L.

Torson, J. M.

K. E. Herkenhoff, S. W. Squyres, R. Anderson, B. A. Archinal, R. E. Arvidson, J. M. Barrett, K. J. Becker, J. F. Bell, C. Budney, N. A. Cabrol, M. G. Chapman, D. Cook, B. L. Ehlmann, J. Farmer, B. Franklin, L. R. Gaddis, D. M. Galuszka, P. A. Garcia, T. M. Hare, E. Howington-Kraus, J. R. Johnson, S. Johnson, K. Kinch, R. L. Kirk, E. M. Lee, C. Leff, M. Lemmon, M. B. Madsen, J. N. Maki, K. F. Mullins, B. L. Redding, L. Richter, M. R. Rosiek, M. H. Sims, L. A. Soderblom, N. Spanovich, R. Springer, R. M. Sucharski, T. Sucharski, R. Sullivan, J. M. Torson, and A. Yen, “Overview of the microscopic imager investigation during spirit’s first 450 sols in gusev crater,” J. Geophys. Res.-Planet 111, E02S04 (2006).
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S. Mikula, I. Trotts, J. M. Stone, and E. G. Jones, “Internet-enabled high-resolution brain mapping and virtual microscopy,” Neuroimage 35, 9–15 (2007).
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A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (2012).
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J. Vila-Comamala, A. Diaz, M. Guizar-Sicairos, A. Mantion, C. M. Kewish, A. Menzel, O. Bunk, and C. David, “Characterization of high-resolution diffractive x-ray optics by ptychographic coherent diffractive imaging,” Opt. Express 19, 21333–21344 (2011).
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M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. Von Koenig, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010).
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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, 053850 (2013).
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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, 053850 (2013).
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D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. A. Padmore, “Chemical composition mapping with nanometre resolution by soft X-ray microscopy,” Nature Photon. 8, 765–769 (2014).
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M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
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K. E. Herkenhoff, S. W. Squyres, R. Anderson, B. A. Archinal, R. E. Arvidson, J. M. Barrett, K. J. Becker, J. F. Bell, C. Budney, N. A. Cabrol, M. G. Chapman, D. Cook, B. L. Ehlmann, J. Farmer, B. Franklin, L. R. Gaddis, D. M. Galuszka, P. A. Garcia, T. M. Hare, E. Howington-Kraus, J. R. Johnson, S. Johnson, K. Kinch, R. L. Kirk, E. M. Lee, C. Leff, M. Lemmon, M. B. Madsen, J. N. Maki, K. F. Mullins, B. L. Redding, L. Richter, M. R. Rosiek, M. H. Sims, L. A. Soderblom, N. Spanovich, R. Springer, R. M. Sucharski, T. Sucharski, R. Sullivan, J. M. Torson, and A. Yen, “Overview of the microscopic imager investigation during spirit’s first 450 sols in gusev crater,” J. Geophys. Res.-Planet 111, E02S04 (2006).
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M. van Heel and M. Schatz, “Fourier shell correlation threshold criteria,” J. Struct. Biol. 151, 250–262 (2005).
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S. Gorelick, J. Vila-Comamala, V. A. Guzenko, R. Barrett, M. Salomé, and C. David, “High-efficiency Fresnel zone plates for hard x-rays by 100 kev e-beam lithography and electroplating,” J. Synchrotron Radiat. 18, 442–446 (2011).
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Microsc. Microanal. (1)

R. Hovden, H. L. Xin, and D. A. Muller, “Extended depth of field for high-resolution scanning transmission electron microscopy,” Microsc. Microanal. 17, 75–80 (2011).
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Nat. Commun. (1)

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Nature (2)

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
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P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
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Nature Photon. (1)

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. A. Padmore, “Chemical composition mapping with nanometre resolution by soft X-ray microscopy,” Nature Photon. 8, 765–769 (2014).
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Neuroimage (1)

S. Mikula, I. Trotts, J. M. Stone, and E. G. Jones, “Internet-enabled high-resolution brain mapping and virtual microscopy,” Neuroimage 35, 9–15 (2007).
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M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. Von Koenig, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010).
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K. Shimomura, A. Suzuki, M. Hirose, and Y. Takahashi, “Precession x-ray ptychography with multislice approach,” Phys. Rev. B 91, 214114 (2015).
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A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (2012).
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Figures (5)

Fig. 1
Fig. 1

Illustration of multi-slice ptychography, which reconstructs multiple axial slices of the object along the propagation direction to account for propagation effects including multiple scattering, diffraction, and propagation within the sample.

Fig. 2
Fig. 2

Experimental reconstructions of microspheres on membranes at 6.2 keV photon energy. Comparison of (a) conventional difference map algorithm, (b) 3DM, and (c) 3ML. In (b) and (c), individual slices are combined according to Eq. (19). Reconstructions of individual slices for 3DM are shown in (d) and (f), and for 3ML in (e) and (g). Yellow arrowheads point to impurities in the sample, clearly reconstructed on slice 2 only. Insets of DM and 3ML are shown in (h) and (i), respectively. Pink arrowheads point to particular artifacts of the central area of particles in the DM reconstruction.

Fig. 3
Fig. 3

Reconstruction using 3ML with fixed wrong slice separation, shown by (a), (c), and (e), and with separation determination, shown by (b), (d), and (f). (a)–(b) Reconstructed phase for slice 1. (c)–(d) Reconstructed phase for slice 2. (e)–(f) The slice separation, Δz, over the iterations. The separation was fixed at 100 μm in (e) while it converged to 220 μm in (f).

Fig. 4
Fig. 4

Ptychography simulation at 6.2 keV, imaging objects with three layers and thicknesses of 40 μm, 200 μm, and 2 mm. With increasing layer separation, the single-slice method gives worse reconstructions, shown by the first row, while the multi-slice method gives visibly a constant reconstruction quality, shown by the projections in the second row. The light blue lines indicate the FRC curves for the case of 40 μm-thickness, the blue for 200 μm, dark blue for 2 mm, and the red lines give the 1-bit threshold. Rows three to five show the individual slices. The sectioning effectiveness improves with increasing separation and, with small separation, low spatial frequency features cannot be effectively sectioned, as shown by the reconstructed images and pointed by the yellow arrowheads at the FRC curves.

Fig. 5
Fig. 5

Numerical simulation for resolution versus object thickness. With s = 1.5 μm, the blue curve a1 gives the result for conventional method, a2 for 3DM reconstructed with two slices, and a3 for 3DM reconstructed with three slices. Similarly for b1, b2, and b3, with s = 2.0 μm. The intersections of DM and 3DM with two slices indicate the point where multi-slice ptychography becomes beneficial and are indicated by circles: blue circles for s = 1.5 μm, red ones for s = 2.0 μm, and green for intermediate scanning steps. Gray circles are the intersecting points between DM and 3DM with three slices. For simplicity, full curves are not shown for these intermediate scanning step data sets. Equations (3) and (4) are shown as gray and black lines, respectively.

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

ψ j , r = 𝒫 r r j O r ,
T a ( δ r ) / λ ,
T 2 ( δ r ) 2 / λ
T 5.2 ( δ r ) 2 / λ .
ψ j , r ( N ) = 𝒫 Δ z N 1 , N { 𝒫 Δ z N 2 , N 1 { 𝒫 Δ z 2 , 3 { 𝒫 Δ z 1 , 2 { P r r j O r ( 1 ) } O r ( 2 ) } O r ( 3 ) } O r ( N 1 ) } O r ( N ) ,
I j , q = | ψ ˜ j , q ( N ) | 2 = | { ψ j , r ( N ) } | 2 ,
Π F { ψ j , r ( N ) } : ψ j , r ( N ) ψ j , r ( N ) F = 1 ( ψ ˜ j , q ( N ) I j , q | ψ ˜ j , q ( N ) | ) .
Π O { ψ j , r ( n ) } : ψ j , r ( n ) ψ j , r ( n ) O = P ^ j , r ( n ) O ^ r ( n ) , n ,
= n j m r | ψ j , r n P ^ j , r ( n ) O ^ r ( n ) | 2 ,
P ^ r = j O r + r j ( 1 ) * ψ j , r ( 1 ) j | O r + r j ( 1 ) * | 2 , for n = 1 ,
P ^ j , r ( n ) = O r ( n ) * ψ j , r ( n ) | O r ( n ) | 2 , for 1 n N ,
O ^ r ( n ) = j P j , r ( n ) * ψ j , r ( n ) j | P j , r ( n ) | 2 , n ,
𝒫 Δ z { ψ r } = 1 { { ψ r } exp ( i k Δ z 1 | λ q | 2 ) } ,
ψ j , r ( N ) [ k + 1 ] = ψ j , r ( N ) [ k ] + Π F { 2 Π O { ψ j , r ( N ) [ k ] } ψ j , r ( N ) [ k ] } Π O { ψ j , r ( N ) [ k ] } .
= j m q ( | { P j , r ( N ) O r ( N ) } | I j , q ) 2 ,
O r ( n ) = { j P j , r ( n ) ( 𝒫 Δ z n , n + 1 { P j , r ( n + 1 ) * } ) * ; 1 n N j P j , r ( N ) χ j , r ( N ) * ; n = N
P j , r ( n ) = { O r ( n ) ( 𝒫 Δ z n , n + 1 { P j , r ( n + 1 ) * } ) * ; 1 n N O r ( N ) χ j , r ( N ) * ; n = N
χ j , r ( N ) = 1 { ( 1 I j , q | ψ ˜ j , q ( N ) | ) P ˜ j , q ( N ) } .
P r = j P j , r + r j ( 1 ) .
Δ z n , n + 1 = 2 j m q ( 1 I j , q | ψ ˜ j , q ( N ) | ) { ψ ˜ j , q ( N ) * × { 𝒫 Δ z N 1 , N { 𝒫 Δ z n + 1 , n + 2 { 1 { { P j , r ( n ) O r ( n ) } × i k 1 | λ q | 2 exp ( i k Δ z n , n + 1 1 | λ q | 2 ) } × O r ( n + 1 ) } O r ( n + 2 ) } O r ( N ) } } ,
O r = n = 1 N O r ( n ) .
ψ r ( N ) = P r ( N ) O r ( N ) = 𝒫 Δ z N 1 , N { ψ r N 1 } O r ( N ) = 𝒫 Δ z N 1 , N { P r ( N 1 ) O r ( N 1 ) } O r ( N ) ,
ψ r ( N ) P r ( N 1 ) O r ( N 1 ) O r ( N ) .
H q = exp ( i k Δ z ) exp ( i π λ Δ z | q | 2 ) ,
Δ z = 1 λ | q | 2 ,
O r ( n ) = 2 j m q ( | ψ ˜ j , q ( N ) | I j , q ) | ψ ˜ j , q ( N ) | O r ( n ) ,
| ψ ˜ j , q ( N ) | O r ( n ) = ψ ˜ j , q ( N ) ψ ˜ j , q ( N ) * O r ( n ) = 1 2 | ψ ˜ j , q ( N ) | ( ψ ˜ j , q ( N ) O r ( n ) ψ ˜ j , q ( N ) * + ψ ˜ j , q ( N ) * O r ( n ) ψ ˜ j , q ( N ) ) = 1 | ψ ˜ j , q ( N ) | { ψ ˜ j , q ( N ) O r ( n ) ψ ˜ j , q ( N ) * } = 1 | ψ ˜ j , q ( N ) | { [ { P j , r ( N ) O r ( N ) } ] O r ( n ) ψ ˜ j , q ( N ) * } .
| ψ ˜ j , q ( N ) | O r ( n ) = 1 | ψ ˜ j , q ( N ) | { ψ ˜ j , q ( N ) * O r ( n ) [ m r 1 M 𝒫 j , r ( N ) O r ( N ) exp ( i 2 π q r ) ] } ,
| ψ ˜ j , q ( N ) | O r ( N ) = 1 | ψ ˜ j , q ( N ) | { ψ ˜ j , q ( N ) * [ m r 1 M 𝒫 j , r ( N ) δ ( r r ) exp ( i 2 π q r ) ] } , = 1 | ψ ˜ j , q ( N ) | { 1 M ψ ˜ j , q ( N ) * P j , r ( N ) exp ( i 2 π q r ) }
O r ( N ) = 2 j m q ( | ψ ˜ j , q ( N ) | I j , q ) 1 | ψ ˜ j , q ( N ) | { 1 M ψ ˜ j , q ( N ) * P j , r ( N ) exp ( i 2 π q r ) } = 2 j { χ j , r ( N ) * P j , r ( N ) }
| ψ ˜ j , q ( N ) | O r ( N ) = 1 | ψ ˜ j , q ( N ) | { i 1 M ψ ˜ j , q ( N ) * P j , r ( N ) exp ( i 2 π q r ) }
O r ( N ) = 2 j { χ j , r ( N ) * P j , r ( N ) } ,
O r ( N ) = j χ j , r ( N ) * P j , r ( N ) .
P j , r ( N ) = χ j , r ( N ) * O j , r ( N ) .
O r ( n ) = j m r ( P j , r ( n + 1 ) P j , r ( n + 1 ) O r ( n ) + 𝒫 j , r ( n + 1 ) P j , r ( n + 1 ) O r ( n ) ) = 2 j m r ( { P r ( n + 1 ) } { P j , r ( n + 1 ) O r ( n ) } { P j , r ( n + 1 ) } { P j , r ( n + 1 ) O r ( n ) } ) = 2 j m r { P r ( n + 1 ) P j , r ( n + 1 ) O r ( n ) } = 2 j m r { P r ( n + 1 ) O r ( n ) 𝒫 Δ z n , n + 1 { P j , r ( n ) O r ( n ) } } .
O r ( n ) = 2 M j m r { P r ( n + 1 ) O r ( n ) m q m s P j , s ( n ) O s ( n ) exp ( i 2 π q s ) × exp ( i k Δ z n , n + 1 1 | λ q | 2 ) exp ( i 2 π q r ) } 2 M j m r { P r ( n + 1 ) m q m s P j , s ( n ) δ ( r s ) exp ( i 2 π q s ) × exp ( i k Δ z n , n + 1 1 | λ q | 2 ) exp ( i 2 π q r ) } = 2 M j m r { P r ( n + 1 ) m q P j , r ( n ) exp ( i 2 π q r ) × exp ( i k Δ z n , n + 1 1 | λ q | 2 ) exp ( i 2 π q r ) } = 2 M j { P j , r ( n ) m q [ m r P j , r ( n + 1 ) exp ( i 2 π q r ) ] × exp ( i 2 π q r ) exp ( i k Δ z n , n + 1 1 | λ q | 2 ) } = 2 j { P j , r ( n ) ( 𝒫 Δ z n , n + 1 { P j , r ( n + 1 ) * } ) * }
O r ( n ) = 2 j { P j , r ( n ) ( 𝒫 Δ z n , n + 1 { P j , r ( n + 1 ) * } ) * } ,
O r ( n ) = j P j , r ( n ) ( 𝒫 Δ z n , n + 1 { P j , r ( n + 1 ) * } ) * .
P j , r ( n ) = O r ( n ) ( 𝒫 Δ z n , n + 1 { P j , r ( n + 1 ) * } ) * .
Δ z n , n + 1 = 2 j m q ( 1 I j , q | ψ ˜ j , q ( N ) | ) { ψ ˜ j , q ( N ) * ψ ˜ j , q ( N ) Δ z n , n + 1 } = 2 j m q ( 1 I j , q | ψ ˜ j , q ( N ) | ) { ψ ˜ j , q ( N ) * Δ z n , n + 1 [ 1 M m r P j , r ( N ) O r ( N ) exp ( i 2 π q r ) ] } = 2 j m q ( 1 I j , q | ψ ˜ j , q ( N ) | ) { ψ ˜ j , q ( N ) * × Δ z n , n + 1 [ 1 M m r 𝒫 Δ z N 1 , N { 𝒫 Δ z n + 1 , n + 2 { 𝒫 Δ z n , n + 1 { 𝒫 j , r ( n ) O r ( n ) } × O r ( n + 1 ) } O r ( n + 2 ) } O r ( N ) exp ( i 2 π q r ) ] } = 2 j m q ( 1 I j , q | ψ ˜ j , q ( N ) | ) { ψ ˜ j , q ( N ) * × 1 M m r 𝒫 Δ z N 1 , N { 𝒫 Δ z n + 1 , n + 2 { 1 M m q { P j , r ( n ) O r ( n ) } × i k 1 | λ q | 2 exp ( i k Δ z n , n + 1 1 | λ q | 2 ) exp ( i 2 π q r ) O r ( n + 1 ) } × O r ( n + 2 ) } O r ( N ) exp ( i 2 π q r ) } = 2 j m q ( 1 I j , q | ψ ˜ j , q ( N ) | ) { ψ ˜ j , q ( N ) * × { 𝒫 Δ z N 1 , N { 𝒫 Δ z n + 1 , n + 2 { 1 { { P j , r ( n ) O r ( n ) } × i k 1 | λ q | 2 exp ( i k Δ z n , n + 1 1 | λ q | 2 ) } O r ( n + 1 ) } O r ( n + 1 ) } O r ( n + 1 ) } } .

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