Abstract

X-ray ptychography is becoming the standard method for sub-30 nm imaging of thick extended samples. Available algorithms and computing power have traditionally restricted sample reconstruction to 2D slices. We build on recent progress in optimization algorithms and high-performance computing to solve the ptychographic phase retrieval problem directly in 3D. Our approach addresses samples that do not fit entirely within the depth of focus of the imaging system. Such samples pose additional challenges because of internal diffraction effects within the sample. We demonstrate our approach on a computational sample modeled with 17 million complex variables.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Corrections

13 September 2018: A typographical correction was made to the funding section.


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2018 (4)

H. Öztürk, H. Yan, Y. He, M. Ge, Z. Dong, M. Lin, E. Nazaretski, I. K. Robinson, Y. S. Chu, and X. Huang, “Multi-slice ptychography with large numerical aperture multilayer Laue lenses,” Optica 5, 601–607 (2018).
[Crossref]

P. Li and A. M. Maiden, “Multi-slice ptychographic tomography,” Sci. Rep. 8, 2049 (2018).
[Crossref]

J. Lim, A. Goy, M. H. Shoreh, M. Unser, and D. Psaltis, “Learning tomography assessed using Mie theory,” Phys. Rev. Appl. 9, 034027 (2018).
[Crossref]

M. Du and C. Jacobsen, “Relative merits and limiting factors for x-ray and electron microscopy of thick, hydrated organic materials,” Ultramicroscopy 184, 293–309 (2018).
[Crossref]

2017 (11)

J. Deng, D. J. Vine, S. Chen, Q. Jin, Y. S. G. Nashed, T. Peterka, S. Vogt, and C. Jacobsen, “X-ray ptychographic and fluorescence microscopy of frozen-hydrated cells using continuous scanning,” Sci. Rep. 7, 445 (2017).
[Crossref]

C. Jacobsen, J. Deng, and Y. Nashed, “Strategies for high-throughput focused-beam ptychography,” J. Synchrotron Radiat. 24, 1078–1081 (2017).
[Crossref]

K. Li, M. Wojcik, and C. Jacobsen, “Multislice does it all—calculating the performance of nanofocusing x-ray optics,” Opt. Express 25, 1831–1846 (2017).
[Crossref]

D. J. Ching and D. Gürsoy, “Xdesign: an open-source software package for designing x-ray imaging phantoms and experiments,” J. Synchrotron Radiat. 24, 537–544 (2017).
[Crossref]

X. Huang, H. Yan, M. Ge, H. Öztürk, E. Nazaretski, I. K. Robinson, and Y. S. Chu, “Artifact mitigation of ptychography integrated with on-the-fly scanning probe microscopy,” Appl. Phys. Lett. 111, 023103 (2017).
[Crossref]

M. Holler, M. Guizar-Sicairos, E. H. R. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
[Crossref]

D. Gürsoy, “Direct coupling of tomography and ptychography,” Opt. Lett. 42, 3169–3172 (2017).
[Crossref]

Y. S. G. Nashed, T. Peterka, J. Deng, and C. Jacobsen, “Distributed automatic differentiation for ptychography,” Procedia Comput. Sci. 108, 404–414 (2017).
[Crossref]

S. Gao, P. Wang, F. Zhang, G. T. Martinez, P. D. Nellist, X. Pan, and A. I. Kirkland, “Electron ptychographic microscopy for three-dimensional imaging,” Nat. Commun. 8, 163 (2017).
[Crossref]

J. Deng, Y. P. Hong, S. Chen, Y. S. G. Nashed, T. Peterka, A. J. F. Levi, J. Damoulakis, S. Saha, T. Eiles, and C. Jacobsen, “Nanoscale x-ray imaging of circuit features without wafer etching,” Phys. Rev. B 95, 104111 (2017).
[Crossref]

D. F. Gardner, M. Tanksalvala, E. R. Shanblatt, X. Zhang, B. R. Galloway, C. L. Porter, R. Karl, C. Bevis, D. E. Adams, H. C. Kapteyn, M. M. Murnane, and G. F. Mancini, “Subwavelength coherent imaging of periodic samples using a 13.5  nm tabletop high-harmonic light source,” Nat. Photonics 11, 259–263 (2017).
[Crossref]

2016 (3)

E. H. R. Tsai, I. Usov, A. Diaz, A. Menzel, and M. Guizar-Sicairos, “X-ray ptychography with extended depth of field,” Opt. Express 24, 29089–29108 (2016).
[Crossref]

U. S. Kamilov, I. N. Papadopoulos, M. H. Shoreh, A. Goy, C. Vonesch, M. Unser, and D. Psaltis, “Optical tomographic image reconstruction based on beam propagation and sparse regularization,” IEEE Trans. Comput. Imaging 2, 59–70 (2016).
[Crossref]

T. Pock and S. Sabach, “Inertial proximal alternating linearized minimization (iPALM) for nonconvex and nonsmooth problems,” SIAM J. Imag. Sci. 9, 1756–1787 (2016).
[Crossref]

2015 (5)

R. Hesse, D. R. Luke, S. Sabach, and M. K. Tam, “Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging,” SIAM J. Imaging Sci. 8, 426–457 (2015).
[Crossref]

U. S. Kamilov, I. N. Papadopoulos, M. H. Shoreh, A. Goy, C. Vonesch, M. Unser, and D. Psaltis, “Learning approach to optical tomography,” Optica 2, 517–522 (2015).
[Crossref]

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

J. Deng, Y. S. G. Nashed, S. Chen, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, C. Jacobsen, and D. J. Vine, “Continuous motion scan ptychography: characterization for increased speed in coherent x-ray imaging,” Opt. Express 23, 5438–5451 (2015).
[Crossref]

X. Huang, K. Lauer, J. N. Clark, W. Xu, E. Nazaretski, R. Harder, I. K. Robinson, and Y. S. Chu, “Fly-scan ptychography,” Sci. Rep. 5, 9074 (2015).
[Crossref]

2014 (9)

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. L. 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,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

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

J. N. Clark, X. Huang, R. J. Harder, and I. K. Robinson, “A continuous scanning mode for ptychography,” Opt. Lett. 39, 6066–6069 (2014).
[Crossref]

P. M. Pelz, M. Guizar-Sicairos, P. Thibault, I. Johnson, M. Holler, and A. Menzel, “On-the-fly scans for x-ray ptychography,” Appl. Phys. Lett. 105, 251101 (2014).
[Crossref]

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]

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]

D. Gürsoy, F. De Carlo, X. Xiao, and C. Jacobsen, “TomoPy: a framework for the analysis of synchrotron tomographic data,” J. Synchrotron Radiat. 21, 1188–1193 (2014).
[Crossref]

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]

Y. S. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 22, 32082–32097 (2014).
[Crossref]

2013 (3)

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

J. Bolte, S. Sabach, and M. Teboulle, “Proximal alternating linearized minimization for nonconvex and nonsmooth problems,” Math. Program. Ser. B 146, 459–494 (2013).
[Crossref]

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

2012 (3)

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).
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K. Katayama, Y. Takeda, K. Kuwabara, and S. Kuwahara, “A novel photocatalytic microreactor bundle that does not require an electric power source,” Chem. Commun. 48, 7368–7370 (2012).
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A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, and J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
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2011 (3)

T. Schoonjans, A. Brunetti, B. Golosio, M. S. del Rio, V. A. Solé, C. Ferrero, and L. Vincze, “The xraylib library for x-ray-matter interactions. recent developments,” Spectrochim. Acta B 66, 776–784 (2011).
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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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G. E. Ice, J. D. Budai, and J. W. Pang, “The race to x-ray microbeam and nanobeam science,” Science 334, 1234–1239 (2011).
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2010 (4)

A. E. Sakdinawat and D. T. Attwood, “Nanoscale x-ray imaging,” Nat. Photonics 4, 840–848 (2010).
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H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10  nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (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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A. Schropp and C. G. Schroer, “Dose requirements for resolving a given feature in an object by coherent x-ray diffraction imaging,” New J. Phys. 12, 035016 (2010).
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2009 (4)

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
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X. Huang, H. Miao, J. Steinbrener, J. Nelson, D. Shapiro, A. Stewart, J. Turner, and C. Jacobsen, “Signal-to-noise and radiation exposure considerations in conventional and diffraction x-ray microscopy,” Opt. Express 17, 13541–13553 (2009).
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A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18, 2419–2434 (2009).
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Y. Takahashi, Y. Nishino, R. Tsutsumi, H. Kubo, H. Furukawa, H. Mimura, S. Matsuyama, N. Zettsu, E. Matsubara, T. Ishikawa, and K. Yamauchi, “High-resolution diffraction microscopy using the plane-wave field of a nearly diffraction limited focused x-ray beam,” Phys. Rev. B 80, 054103 (2009).
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2008 (2)

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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M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
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2007 (3)

H. Yan, J. Maser, A. Macrander, Q. Shen, S. Vogt, G. Stephenson, and H. Kang, “Takagi-Taupin description of x-ray dynamical diffraction from diffractive optics with large numerical aperture,” Phys. Rev. B 76, 115438 (2007).
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G. J. Williams, M. Pfeifer, I. Vartanyants, and I. K. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Crystallogr. Sect. A 63, 36–42 (2007).
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F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
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2004 (2)

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
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C. Larabell and M. Le Gros, “X-ray tomography generates 3-D reconstructions of the yeast, Saccharomyces cerevisiae, at 60-nm resolution,” Mol. Biol. Cell 15, 957–962 (2004).
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2002 (1)

J. C. H. Spence, U. Weierstall, and M. Howells, “Phase recovery and lensless imaging by iterative methods in optical, x-ray and electron diffraction,” Philos. Trans. R. Soc. A 360, 875–895 (2002).
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2000 (1)

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1997 (1)

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1993 (1)

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1992 (1)

J. Maser and G. Schmahl, “Coupled wave description of the diffraction by zone plates with high aspect ratios,” Opt. Commun. 89, 355–362 (1992).
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1988 (1)

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1970 (1)

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1969 (1)

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1957 (2)

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1953 (1)

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M. Holler, M. Guizar-Sicairos, E. H. R. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
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A. E. Sakdinawat and D. T. Attwood, “Nanoscale x-ray imaging,” Nat. Photonics 4, 840–848 (2010).
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Budai, J. D.

G. E. Ice, J. D. Budai, and J. W. Pang, “The race to x-ray microbeam and nanobeam science,” Science 334, 1234–1239 (2011).
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M. Holler, M. Guizar-Sicairos, E. H. R. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
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J. Deng, Y. P. Hong, S. Chen, Y. S. G. Nashed, T. Peterka, A. J. F. Levi, J. Damoulakis, S. Saha, T. Eiles, and C. Jacobsen, “Nanoscale x-ray imaging of circuit features without wafer etching,” Phys. Rev. B 95, 104111 (2017).
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J. Deng, D. J. Vine, S. Chen, Q. Jin, Y. S. G. Nashed, T. Peterka, S. Vogt, and C. Jacobsen, “X-ray ptychographic and fluorescence microscopy of frozen-hydrated cells using continuous scanning,” Sci. Rep. 7, 445 (2017).
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J. Deng, Y. S. G. Nashed, S. Chen, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, C. Jacobsen, and D. J. Vine, “Continuous motion scan ptychography: characterization for increased speed in coherent x-ray imaging,” Opt. Express 23, 5438–5451 (2015).
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D. J. Ching and D. Gürsoy, “Xdesign: an open-source software package for designing x-ray imaging phantoms and experiments,” J. Synchrotron Radiat. 24, 537–544 (2017).
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H. Öztürk, H. Yan, Y. He, M. Ge, Z. Dong, M. Lin, E. Nazaretski, I. K. Robinson, Y. S. Chu, and X. Huang, “Multi-slice ptychography with large numerical aperture multilayer Laue lenses,” Optica 5, 601–607 (2018).
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X. Huang, H. Yan, M. Ge, H. Öztürk, E. Nazaretski, I. K. Robinson, and Y. S. Chu, “Artifact mitigation of ptychography integrated with on-the-fly scanning probe microscopy,” Appl. Phys. Lett. 111, 023103 (2017).
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X. Huang, K. Lauer, J. N. Clark, W. Xu, E. Nazaretski, R. Harder, I. K. Robinson, and Y. S. Chu, “Fly-scan ptychography,” Sci. Rep. 5, 9074 (2015).
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Clark, J. N.

X. Huang, K. Lauer, J. N. Clark, W. Xu, E. Nazaretski, R. Harder, I. K. Robinson, and Y. S. Chu, “Fly-scan ptychography,” Sci. Rep. 5, 9074 (2015).
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J. N. Clark, X. Huang, R. J. Harder, and I. K. Robinson, “Dynamic imaging using ptychography,” Phys. Rev. Lett. 112, 113901 (2014).
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J. N. Clark, X. Huang, R. J. Harder, and I. K. Robinson, “A continuous scanning mode for ptychography,” Opt. Lett. 39, 6066–6069 (2014).
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Cowley, J. M.

J. M. Cowley and A. F. Moodie, “Fourier images, I: the point source,” Proc. Phys. Soc. London Sect. B 70, 486–496 (1957).
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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).
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Crowther, R. A.

R. A. Crowther, D. J. DeRosier, and A. Klug, “The reconstruction of a three-dimensional structure from projections and its application to electron microscopy,” Proc. R. Soc. London Ser. A 317, 319–340 (1970).
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Damoulakis, J.

J. Deng, Y. P. Hong, S. Chen, Y. S. G. Nashed, T. Peterka, A. J. F. Levi, J. Damoulakis, S. Saha, T. Eiles, and C. Jacobsen, “Nanoscale x-ray imaging of circuit features without wafer etching,” Phys. Rev. B 95, 104111 (2017).
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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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Davis, J. C.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50–30, 000  eV, Z = 1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
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C. Jacobsen, J. Deng, and Y. Nashed, “Strategies for high-throughput focused-beam ptychography,” J. Synchrotron Radiat. 24, 1078–1081 (2017).
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J. Deng, D. J. Vine, S. Chen, Q. Jin, Y. S. G. Nashed, T. Peterka, S. Vogt, and C. Jacobsen, “X-ray ptychographic and fluorescence microscopy of frozen-hydrated cells using continuous scanning,” Sci. Rep. 7, 445 (2017).
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J. Deng, Y. S. G. Nashed, S. Chen, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, C. Jacobsen, and D. J. Vine, “Continuous motion scan ptychography: characterization for increased speed in coherent x-ray imaging,” Opt. Express 23, 5438–5451 (2015).
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Diaz, A.

Dierolf, M.

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, 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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M. Holler, M. Guizar-Sicairos, E. H. R. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
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Eiles, T.

J. Deng, Y. P. Hong, S. Chen, Y. S. G. Nashed, T. Peterka, A. J. F. Levi, J. Damoulakis, S. Saha, T. Eiles, and C. Jacobsen, “Nanoscale x-ray imaging of circuit features without wafer etching,” Phys. Rev. B 95, 104111 (2017).
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J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
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H. Öztürk, H. Yan, Y. He, M. Ge, Z. Dong, M. Lin, E. Nazaretski, I. K. Robinson, Y. S. Chu, and X. Huang, “Multi-slice ptychography with large numerical aperture multilayer Laue lenses,” Optica 5, 601–607 (2018).
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X. Huang, H. Yan, M. Ge, H. Öztürk, E. Nazaretski, I. K. Robinson, and Y. S. Chu, “Artifact mitigation of ptychography integrated with on-the-fly scanning probe microscopy,” Appl. Phys. Lett. 111, 023103 (2017).
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Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
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M. Holler, M. Guizar-Sicairos, E. H. R. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
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Figures (6)

Fig. 1.
Fig. 1. Depth of focus as a function of x-ray photon energy for a variety of transverse resolution δ r values. Also shown is the exp [ 1 ] penetration depth μ 1 for amorphous ice (for frozen hydrated biological specimens) and silicon (for microelectronics specimens) as proxies for the thickness range of x-ray imaging as a function of photon energy. As the transverse resolution δ r in x-ray microscopy is improved to finer values, the DOF decreases with the square of the resolution improvement [Eq. (1)], leading to a decrease in the size of a specimen that can be imaged within the projection approximation required by standard tomography.
Fig. 2.
Fig. 2. Experimental geometry used for our simulated experiment. A lens was assumed to produce a Gaussian coherent illumination probe of size 14 nm FWHM through which the object is scanned at each object rotation angle. A far-field diffraction pattern is then captured in each scan.
Fig. 3.
Fig. 3. Left: an isosurface rendering of the true object with 200 nm length along the cone’s axis. The simulated object was a conical glass capillary tube with embedded Ti nanospheres. Right: the as-designed Gaussian probe function with 14 nm FWHM size, as represented at the midpoint of the object region; brightness indicates the amplitude of the wave, and hue indicates the phase (see color wheel inset).
Fig. 4.
Fig. 4. Comparisons of two cuts (zonal, top row; and meridonal, bottom row) of the reconstructed phase-shifting part δ of the x-ray refractive index n = 1 δ i β for different methods and data collections. These cuts show the reconstructed value of δ in the voxels at the selected planes. The true object is shown at left, followed by the MOOR reconstruction using 360° and then 180° rotation axes. Finally, a reconstruction from the standard PPA to ptychographic tomography is shown. As noted in the text, the pure projection approximation does not properly reproduce the object, and it also suffers from regular artifacts due to insufficient probe overlap for the reconstruction method used [59]. For this reason, we also show a column of “PPA+filtering” images with post-processing. These figures show that 360° MOOR gives a reconstructed image that represents the true object with a high degree of fidelity.
Fig. 5.
Fig. 5. Comparisons of 3D isosurface renderings of the true object and MOOR reconstructions using 360° and 180° object rotation. Rotation over a full 360° range gives improved results. The 3D object reconstructed using the PPA is not shown, as its errors (Fig. 4) do not make it possible to obtain a clean isosurface rendering.
Fig. 6.
Fig. 6. Histograms of the distribution of absolute error in the per-voxel values of the phase-shifting part δ of the x-ray refractive index n = 1 δ i β for both our MOOR reconstruction approach and for the filtered pure projection approximation (PPA, and PPA + filtering) reconstructions. At left are shown the histogram for object-containing voxels, while at right are shown the histograms for object-absent voxels. Also indicated are the values of δ for Si ( 1.98 × 10 5 at 5 keV) and TiO 2 ( 3.01 × 10 5 at 5 keV). As can be seen for the object-containing voxels (left), the MOOR reconstruction approach gives results with absolute errors that are small compared to the actual values of δ , while for PPA the absolute errors are almost as large as the expected values of δ . For the object-absent voxels (right), again the errors in the MOOR reconstruction are very small compared to the expected values of δ in the object-present regions, while the PPA reconstructions have errors in the object-absent voxels that approach the true values expected in the object-present voxels.

Tables (1)

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Algorithm 1 A proximal alternating linearized minimization algorithm for solving Eq. (11)

Equations (18)

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DOF = 2 0.61 2 δ r 2 λ 5.4 δ r δ r λ ,
δ r = 0.61 λ NA ,
I ( z ) [ sin ( u ( 1 b f 2 ) ) u ] 2 ,
ψ j ( x , y ) ψ j ( x , y ) exp [ 2 π Δ z λ ( i δ ( x , y , z j ) β ( x , y , z j ) ) ] = M j [ ψ j ( x , y ) ] .
h ( x , y ) = exp [ i π λ Δ z ( x 2 + y 2 ) ] ,
ψ j + 1 ( x , y ) = ψ j ( x , y ) h ( x , y ) = F 1 [ F ( ψ j ( x , y ) ) H ( x , y ) ] ,
= P z [ ψ j ( x , y ) ] ,
ψ d ( x , y ) = F [ ψ s ( x , y ) ] .
ψ d ( x , y ) = F { P z [ M s [ P z [ M 0 [ ψ 0 ( x , y ) ] ] repeated nesting until    s 1 ] ] } .
β = log ( | y | ) λ 2 π Δ z , δ = arg ( y ) λ 2 π Δ z .
| F { ( q = 1 s C diag ( S q , j y ) ) x } | = d j ,
min ( x , y , z ) X × Y × M F ( x , y , z ) + κ TV 3 ( y ) ,
X = { x C l 2 : | x | ν x } , Y = { y C n 3 : | y 1 | ν y } , and M = { z C l 2 × m : | F { z j } | = d j , j = 1 , , m } .
F ( x , y , z ) = j = 1 m ( q = 1 s C diag ( S q , j y ) ) x z j 2 2
[ proj X ( x ) ] j = { x j if    | x j | ν x x j | x j | ν x otherwise
[ proj M ( z ) ] J j = F 1 { z ^ } ,
z ^ w = { d j , w [ F { z J j } ] w | [ F { z J j } ] w | if    [ F { z J j } ] w 0 d j , w otherwise .
prox α y τ Y + κ TV 3 ( y ) = arg min y ^ Y TV 3 ( y ^ ) + α y 2 y ^ y 2