Shapin, S. The Scientific Revolution 2nd edn (Univ. Chicago Press, 2018).
Hawkes, P. W. & Spence, J. C. H. (eds) Springer Handbook of Microscopy (Springer, 2019).
Crystallography at 100. Science 343 (Special issue), 1091–1116 (2014).
Miao, J., Charalambous, P., Kirz, J. & Sayre, D. Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens. Nature 400, 342–344 (1999). This paper reports on the experimental demonstration of unifying crystallography and microscopy by replacing the physical lens with coherent diffraction and computational algorithms.
Google Scholar
Sayre, D. in Imaging Processes and Coherence in Physics Vol. 112 (eds Schlenker, M. et al.) 229–235 (Springer, 1980).
Millane, R. P. Phase retrieval in crystallography and optics. J. Opt. Soc. Am. A 7, 394–411 (1990).
Google Scholar
Gerchberg, R. W. & Saxton, W. O. A practical algorithm for the determination of phase from image and diffraction plane pictures. Optik 35, 237–246 (1972).
Google Scholar
Fienup, J. R. Phase retrieval algorithms: a comparison. Appl. Opt. 21, 2758–2769 (1982). This was the development of error-reduction and input–output algorithms for iterative phase retrieval.
Google Scholar
Bates, R. H. T. Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory. Optik 61, 247–262 (1982).
Google Scholar
Miao, J., Sayre, D. & Chapman, H. N. Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects. J. Opt. Soc. Am. A 15, 1662–1669 (1998). The oversampling ratio was introduced; phase retrieval requires more independently measured points than unknown variables.
Google Scholar
Willmott, P. An Introduction to Synchrotron Radiation: Techniques and Applications 2nd edn (Wiley, 2019).
Boyle, W. S. & Smith, G. E. Charge coupled semiconductor devices. Bell Syst. Tech. J. 49, 587–593 (1970).
Google Scholar
Robinson, I. K. & Harder, R. Coherent X-ray diffraction imaging of strain at the nanoscale. Nat. Mater. 8, 291–298 (2009).
Google Scholar
Chapman, H. N. & Nugent, K. A. Coherent lensless X-ray imaging. Nat. Photon. 4, 833–839 (2010).
Google Scholar
Miao, J., Ishikawa, T., Robinson, I. K. & Murnane, M. M. Beyond crystallography: diffractive imaging using coherent x-ray light sources. Science 348, 530–535 (2015).
Google Scholar
Shechtman, Y. et al. Phase retrieval with application to optical imaging: a contemporary overview. IEEE Signal Process. Mag. 32, 87–109 (2015).
Google Scholar
Pfeiffer, F. X-ray ptychography. Nat. Photon. 12, 9–17 (2018).
Google Scholar
Rodenburg, J. & Maiden, A. in Springer Handbook of Microscopy (eds Hawkes, P. W. & Spence, J. C. H.) 819–904 (Springer, 2019).
Zheng, G., Shen, C., Jiang, S., Song, P. & Yang, C. Concept, implementations and applications of Fourier ptychography. Nat. Rev. Phys. 3, 207–223 (2021).
Google Scholar
Spence, J. C. H., Howells, M., Marks, L. D. & Miao, J. Lensless imaging: a workshop on “new approaches to the phase problem for non-periodic objects”. Ultramicroscopy 90, 1–6 (2001).
Google Scholar
Jiang, Y. et al. Electron ptychography of 2D materials to deep sub-ångström resolution. Nature 559, 343–349 (2018). Electron ptychography with EMPAD at 0.39 Å resolution is better than aberration-corrected electron microscopy under the same imaging conditions.
Google Scholar
Chen, Z. et al. Electron ptychography achieves atomic-resolution limits set by lattice vibrations. Science 372, 826–831 (2021).
Google Scholar
Yang, W., Sha, H., Cui, J., Mao, L. & Yu, R. Local-orbital ptychography for ultrahigh-resolution imaging. Nat. Nanotechnol. 19, 612–617 (2024).
Google Scholar
Nguyen, K. X. et al. Achieving sub-0.5-angstrom–resolution ptychography in an uncorrected electron microscope. Science 383, 865–870 (2024).
Google Scholar
Dong, Z. et al. Visualization of oxygen vacancies and self-doped ligand holes in La3Ni2O7−δ. Nature 630, 847–852 (2024). Multislice ptychography combined with electron energy-loss spectroscopy reveals oxygen vacancies in a high-Tc superconductor (La3Ni2O7−δ) and correlates nanoscale stoichiometry with electronic structure.
Google Scholar
Donnelly, C. et al. Three-dimensional magnetization structures revealed with X-ray vector nanotomography. Nature 547, 328–331 (2017). This paper reveals the development of X-ray vector ptychographic tomography for the 3D imaging of magnetic configurations around Bloch point singularities.
Google Scholar
Donnelly, C. et al. Time-resolved imaging of three-dimensional nanoscale magnetization dynamics. Nat. Nanotechnol. 15, 356–360 (2020).
Google Scholar
Rana, A. et al. Three-dimensional topological magnetic monopoles and their interactions in a ferromagnetic meta-lattice. Nat. Nanotechnol. 18, 227–232 (2023).
Google Scholar
Singer, A. et al. Nucleation of dislocations and their dynamics in layered oxide cathode materials during battery charging. Nat. Energy 3, 641–647 (2018).
Google Scholar
Csernica, P. M. et al. Persistent and partially mobile oxygen vacancies in Li-rich layered oxides. Nat. Energy 6, 642–652 (2021).
Google Scholar
Liu, T. et al. Origin of structural degradation in Li-rich layered oxide cathode. Nature 606, 305–312 (2022). In situ BCDI reveals that severe lattice strain accumulation in Li- and Mn-rich cathodes during cycling drives structural degradation and oxygen loss.
Google Scholar
Deng, H. D. et al. Correlative image learning of chemo-mechanics in phase-transforming solids. Nat. Mater. 21, 547–554 (2022).
Google Scholar
Zheng, G., Horstmeyer, R. & Yang, C. Wide-field, high-resolution Fourier ptychographic microscopy. Nat. Photon. 7, 739–745 (2013). Fourier ptychography was demonstrated by capturing a series of low-resolution intensity images to generate a wide-field, high-resolution complex-valued image.
Google Scholar
Bosch, C. et al. Non-destructive X-ray tomography of brain tissue ultrastructure. Preprint at bioRxiv https://doi.org/10.1101/2023.11.16.567403 (2023).
Dierolf, M. et al. Ptychographic X-ray computed tomography at the nanoscale. Nature 467, 436–439 (2010).
Google Scholar
Clark, J. N. et al. Three-dimensional imaging of dislocation propagation during crystal growth and dissolution. Nat. Mater. 14, 780–784 (2015).
Google Scholar
Lo, Y. H. et al. X-ray linear dichroic ptychography. Proc. Natl Acad. Sci. USA 118, e2019068118 (2021).
Google Scholar
Marrison, J., Räty, L., Marriott, P. & O’Toole, P. Ptychography – a label free, high-contrast imaging technique for live cells using quantitative phase information. Sci Rep. 3, 2369 (2013).
Google Scholar
Tian, L. et al. Computational illumination for high-speed in vitro Fourier ptychographic microscopy. Optica 2, 904–911 (2015).
Google Scholar
Sun, J., Zuo, C., Zhang, L. & Chen, Q. Resolution-enhanced Fourier ptychographic microscopy based on high-numerical-aperture illuminations. Sci Rep. 7, 1187 (2017).
Google Scholar
Zuo, C., Sun, J., Li, J., Asundi, A. & Chen, Q. Wide-field high-resolution 3D microscopy with Fourier ptychographic diffraction tomography. Opt. Lasers Eng. 128, 106003 (2020).
Google Scholar
Miao, J. et al. Imaging whole Escherichia coli bacteria by using single particle x-ray diffraction. Proc. Natl Acad. Sci. USA 100, 110–112 (2003).
Google Scholar
Shapiro, D. et al. Biological imaging by soft x-ray diffraction microscopy. Proc. Natl Acad. Sci. USA 102, 15343–15346 (2005).
Google Scholar
Giewekemeyer, K. et al. Quantitative biological imaging by ptychographic x-ray diffraction microscopy. Proc. Natl Acad. Sci. USA 107, 529–534 (2010).
Google Scholar
Jiang, H. et al. Quantitative 3D imaging of whole, unstained cells by using X-ray diffraction microscopy. Proc. Natl Acad. Sci. USA 107, 11234–11239 (2010).
Google Scholar
Deng, J. et al. Correlative 3D x-ray fluorescence and ptychographic tomography of frozen-hydrated green algae. Sci. Adv. 4, eaau4548 (2018).
Google Scholar
Song, C. et al. Quantitative imaging of single, unstained viruses with coherent X-rays. Phys. Rev. Lett. 101, 158101 (2008).
Google Scholar
Seibert, M. M. et al. Single mimivirus particles intercepted and imaged with an X-ray laser. Nature 470, 78–81 (2011).
Google Scholar
Ekeberg, T. et al. Three-dimensional reconstruction of the giant mimivirus particle with an X-ray free-electron laser. Phys. Rev. Lett. 114, 098102 (2015).
Google Scholar
Hosseinizadeh, A. et al. Conformational landscape of a virus by single-particle X-ray scattering. Nat. Methods 14, 877–881 (2017).
Google Scholar
Pei, X. et al. Cryogenic electron ptychographic single particle analysis with wide bandwidth information transfer. Nat. Commun. 14, 3027 (2023).
Google Scholar
Ayyer, K. et al. Macromolecular diffractive imaging using imperfect crystals. Nature 530, 202–206 (2016).
Google Scholar
Küçükoğlu, B. et al. Low-dose cryo-electron ptychography of proteins at sub-nanometer resolution. Nat. Commun. 15, 8062 (2024).
Jacobsen, C. X-Ray Microscopy (Cambridge Univ. Press, 2019).
Fannjiang, A. & Strohmer, T. The numerics of phase retrieval. Acta Numer. 29, 125–228 (2020).
Google Scholar
Dong, J. et al. Phase retrieval: from computational imaging to machine learning: a tutorial. IEEE Signal Process. Mag. 40, 45–57 (2023).
Google Scholar
Chen, Z. et al. Lorentz electron ptychography for imaging magnetic textures beyond the diffraction limit. Nat. Nanotechnol. 17, 1165–1170 (2022).
Google Scholar
Klose, C. et al. Coherent correlation imaging for resolving fluctuating states of matter. Nature 614, 256–261 (2023).
Google Scholar
Zayko, S. et al. Ultrafast high-harmonic nanoscopy of magnetization dynamics. Nat. Commun. 12, 6337 (2021).
Google Scholar
Gomez, L. F. et al. Shapes and vorticities of superfluid helium nanodroplets. Science 345, 906–909 (2014).
Google Scholar
Johnson, A. S. et al. Ultrafast X-ray imaging of the light-induced phase transition in VO2. Nat. Phys. 19, 215–220 (2023).
Google Scholar
Michelson, A. et al. Three-dimensional visualization of nanoparticle lattices and multimaterial frameworks. Science 376, 203–207 (2022).
Google Scholar
Xu, R. et al. Single-shot three-dimensional structure determination of nanocrystals with femtosecond X-ray free-electron laser pulses. Nat. Commun. 5, 4061 (2014).
Google Scholar
Holler, M. et al. High-resolution non-destructive three-dimensional imaging of integrated circuits. Nature 543, 402–406 (2017).
Google Scholar
Holler, M. et al. Three-dimensional imaging of integrated circuits with macro- to nanoscale zoom. Nat. Electron. 2, 464–470 (2019).
Google Scholar
Aidukas, T. et al. High-performance 4-nm-resolution X-ray tomography using burst ptychography. Nature 632, 81–88 (2024). Innovative advancements in ptychographic X-ray tomography enabled the quantitative 3D imaging of a 7-nm commercial integrated circuit, paving the way for future resolution enhancements.
Google Scholar
Horstmeyer, R., Ou, X., Zheng, G., Willems, P. & Yang, C. Digital pathology with Fourier ptychography. Comput. Med. Imaging Graph. 42, 38–43 (2015).
Google Scholar
Shahmoradian, S. H. et al. Three-dimensional imaging of biological tissue by cryo X-ray ptychography. Sci Rep. 7, 6291 (2017).
Google Scholar
Clark, J. N. et al. Ultrafast three-dimensional imaging of lattice dynamics in individual gold nanocrystals. Science 341, 56–59 (2013).
Google Scholar
Ihm, Y. et al. Direct observation of picosecond melting and disintegration of metallic nanoparticles. Nat. Commun. 10, 2411 (2019).
Google Scholar
Williams, G. J. et al. Fresnel coherent diffractive imaging. Phys. Rev. Lett. 97, 025506 (2006).
Google Scholar
Zhang, F. et al. Phase retrieval by coherent modulation imaging. Nat. Commun. 7, 13367 (2016).
Google Scholar
Latychevskaia, T., Longchamp, J.-N. & Fink, H.-W. When holography meets coherent diffraction imaging. Opt. Express 20, 28871–28892 (2012).
Google Scholar
Rivenson, Y., Zhang, Y., Günaydın, H., Teng, D. & Ozcan, A. Phase recovery and holographic image reconstruction using deep learning in neural networks. Light Sci. Appl. 7, 17141 (2018).
Google Scholar
Miao, J. et al. High resolution 3D X-ray diffraction microscopy. Phys. Rev. Lett. 89, 088303 (2002).
Google Scholar
Marchesini, S. et al. X-ray image reconstruction from a diffraction pattern alone. Phys. Rev. B 68, 140101 (2003).
Google Scholar
Chapman, H. N. et al. High-resolution ab initio three-dimensional X-ray diffraction microscopy. J. Opt. Soc. Am. A Opt. Image Sci. Vis. 23, 1179–1200 (2006).
Google Scholar
Chapman, H. N. et al. Femtosecond diffractive imaging with a soft-X-ray free-electron laser. Nat. Phys. 2, 839–843 (2006). Single-shot coherent X-ray diffractive imaging was demonstrated by capturing an XFEL diffraction pattern before the sample was destroyed.
Google Scholar
Sandberg, R. L. et al. Lensless diffractive imaging using tabletop coherent high-harmonic soft-X-ray beams. Phys. Rev. Lett. 99, 098103 (2007).
Google Scholar
Sandberg, R. L. et al. High numerical aperture tabletop soft x-ray diffraction microscopy with 70-nm resolution. Proc. Natl Acad. Sci. USA 105, 24–27 (2008).
Google Scholar
Ravasio, A. et al. Single-shot diffractive imaging with a table-top femtosecond soft X-ray laser-harmonics source. Phys. Rev. Lett. 103, 028104 (2009).
Google Scholar
Miao, J., Ohsuna, T., Terasaki, O., Hodgson, K. O. & O’Keefe, M. A. Atomic resolution three-dimensional electron diffraction microscopy. Phys. Rev. Lett. 89, 155502 (2002).
Google Scholar
Zuo, J. M., Vartanyants, I., Gao, M., Zhang, R. & Nagahara, L. A. Atomic resolution imaging of a carbon nanotube from diffraction intensities. Science 300, 1419–1421 (2003).
Google Scholar
Huang, W. J., Zuo, J. M., Jiang, B., Kwon, K. W. & Shim, M. Sub-ångström-resolution diffractive imaging of single nanocrystals. Nat. Phys. 5, 129–133 (2009).
Google Scholar
De Caro, L., Carlino, E., Caputo, G., Cozzoli, P. D. & Giannini, C. Electron diffractive imaging of oxygen atoms in nanocrystals at sub-ångström resolution. Nat. Nanotechnol. 5, 360–365 (2010).
Google Scholar
Neutze, R., Wouts, R., van der Spoel, D., Weckert, E. & Hajdu, J. Potential for biomolecular imaging with femtosecond X-ray pulses. Nature 406, 752–757 (2000).
Google Scholar
Robinson, I. K., Vartanyants, I. A., Williams, G. J., Pfeifer, M. A. & Pitney, J. A. Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction. Phys. Rev. Lett. 87, 195505 (2001).
Google Scholar
Pfeifer, M. A., Williams, G. J., Vartanyants, I. A., Harder, R. & Robinson, I. K. Three-dimensional mapping of a deformation field inside a nanocrystal. Nature 442, 63–66 (2006). Bragg CDI was demonstrated for quantitative 3D imaging of lattice strain at the nanoscale.
Google Scholar
Marathe, S. et al. Coherent diffraction surface imaging in reflection geometry. Opt. Express 18, 7253–7262 (2010).
Google Scholar
Roy, S. et al. Lensless X-ray imaging in reflection geometry. Nat. Photon. 5, 243–245 (2011).
Google Scholar
Sun, T., Jiang, Z., Strzalka, J., Ocola, L. & Wang, J. Three-dimensional coherent X-ray surface scattering imaging near total external reflection. Nat. Photon. 6, 586–590 (2012).
Google Scholar
Yang, Y. & Sinha, S. K. Three-dimensional imaging using coherent x rays at grazing incidence geometry. J. Opt. Soc. Am. A Opt. Image Sci. Vis. 40, 1500–1506 (2023).
Google Scholar
Seaberg, M. D. et al. Tabletop nanometer extreme ultraviolet imaging in an extended reflection mode using coherent Fresnel ptychography. Optica 1, 39–44 (2014).
Google Scholar
Zhang, B. et al. High contrast 3D imaging of surfaces near the wavelength limit using tabletop EUV ptychography. Ultramicroscopy 158, 98–104 (2015).
Google Scholar
Tanksalvala, M. et al. Nondestructive, high-resolution, chemically specific 3D nanostructure characterization using phase-sensitive EUV imaging reflectometry. Sci. Adv. 7, eabd9667 (2021).
Google Scholar
Abbey, B. et al. Keyhole coherent diffractive imaging. Nat. Phys. 4, 394–398 (2008).
Google Scholar
Kfir, O. et al. Nanoscale magnetic imaging using circularly polarized high-harmonic radiation. Sci. Adv. 3, eaao4641 (2017).
Google Scholar
Broennimann, C. et al. The PILATUS 1M detector. J. Synchrotron Radiat. 13, 120–130 (2006).
Google Scholar
Tate, M. W. et al. High dynamic range pixel array detector for scanning transmission electron microscopy. Microsc. Microanal. 22, 237–249 (2016).
Google Scholar
Grimes, M. et al. Bragg coherent diffraction imaging with the CITIUS charge-integrating detector. J. Appl. Crystallogr. 56, 1032–1037 (2023).
Google Scholar
Miao, J., Förster, F. & Levi, O. Equally sloped tomography with oversampling reconstruction. Phys. Rev. B 72, 052103 (2005).
Google Scholar
Scott, M. C. et al. Electron tomography at 2.4-ångström resolution. Nature 483, 444–447 (2012). The experimental demonstration of atomic electron tomography at 2.4 Å resolution was enabled by an advanced computational algorithm.
Google Scholar
Miao, J., Ercius, P. & Billinge, S. J. L. Atomic electron tomography: 3D structures without crystals. Science 353, aaf2157 (2016).
Google Scholar
Pham, M., Yuan, Y., Rana, A., Osher, S. & Miao, J. Accurate real space iterative reconstruction (RESIRE) algorithm for tomography. Sci Rep. 13, 5624 (2023).
Google Scholar
Yang, Y. et al. Determining the three-dimensional atomic structure of an amorphous solid. Nature 592, 60–64 (2021). The 3D atomic structure of a metallic glass was determined by atomic electron tomography, revealing both short- and medium-range order in the atomic arrangement.
Google Scholar
Yuan, Y. et al. Three-dimensional atomic packing in amorphous solids with liquid-like structure. Nat. Mater. 21, 95–102 (2022).
Google Scholar
Chang, D. J. et al. Ptychographic atomic electron tomography: towards three-dimensional imaging of individual light atoms in materials. Phys. Rev. B 102, 174101 (2020).
Google Scholar
Pelz, P. M. et al. Solving complex nanostructures with ptychographic atomic electron tomography. Nat. Commun. 14, 7906 (2023).
Google Scholar
Hoppe, W. Diffraction in inhomogeneous primary wave fields: 1. Principle of phase determination from electron diffraction interference [in German]. Acta Crystallogr. A 25, 495–501 (1969).
Google Scholar
Hoppe, W. Diffraction in inhomogeneous primary wave fields: 3. Amplitude and phase determination for nonperiodic objects [in German]. Acta Crystallogr. A 25, 508–515 (1969).
Google Scholar
Rodenburg, J. M. & Bates, R. H. T. The theory of super-resolution electron microscopy via Wigner-distribution deconvolution. Philos. Trans. A Math. Phys. Eng. Sci. 339, 521–553 (1992).
Google Scholar
Nellist, P. D., McCallum, B. C. & Rodenburg, J. M. Resolution beyond the ‘information limit’ in transmission electron microscopy. Nature 374, 630–632 (1995).
Google Scholar
Chapman, H. N. Phase-retrieval X-ray microscopy by Wigner-distribution deconvolution. Ultramicroscopy 66, 153–172 (1996).
Google Scholar
Rodenburg, J. M. et al. Hard-X-ray lensless imaging of extended objects. Phys. Rev. Lett. 98, 034801 (2007). Modern ptychography was demonstrated by scanning an X-ray probe across an extended sample and using an iterative phase-retrieval algorithm to reconstruct its complex transmission function.
Google Scholar
Faulkner, H. M. L. & Rodenburg, J. M. Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm. Phys. Rev. Lett. 93, 023903 (2004).
Google Scholar
Thibault, P. et al. High-resolution scanning X-ray diffraction microscopy. Science 321, 379–382 (2008). This is the experimental demonstration of reconstructing the complex-valued probe and sample functions using iterative phase retrieval in modern ptychography.
Google Scholar
Thibault, P. & Menzel, A. Reconstructing state mixtures from diffraction measurements. Nature 494, 68–71 (2013).
Google Scholar
Guizar-Sicairos, M. & Fienup, J. R. Phase retrieval with transverse translation diversity: a nonlinear optimization approach. Opt. Express 16, 7264–7278 (2008).
Google Scholar
Maiden, A. M., Humphry, M. J., Sarahan, M. C., Kraus, B. & Rodenburg, J. M. An annealing algorithm to correct positioning errors in ptychography. Ultramicroscopy 120, 64–72 (2012).
Google Scholar
Raines, K. S. et al. Three-dimensional structure determination from a single view. Nature 463, 214–217 (2010).
Google Scholar
Wei, H. Fundamental limits of ‘ankylography’ due to dimensional deficiency. Nature 480, E1 (2011).
Google Scholar
Wang, G., Yu, H., Cong, W. & Katsevich, A. Non-uniqueness and instability of ‘ankylography’. Nature 480, E2–E3 (2011).
Google Scholar
Maiden, A. M., Humphry, M. J. & Rodenburg, J. M. Ptychographic transmission microscopy in three dimensions using a multi-slice approach. J. Opt. Soc. Am. A Opt. Image Sci. Vis. 29, 1606–1614 (2012).
Google Scholar
Suzuki, A. et al. High-resolution multislice X-ray ptychography of extended thick objects. Phys. Rev. Lett. 112, 053903 (2014).
Google Scholar
O’Leary, C. M. et al. Three-dimensional structure of buried heterointerfaces revealed by multislice ptychography. Phys. Rev. Appl. 22, 014016 (2024).
Google Scholar
Ophus, C. Four-dimensional scanning transmission electron microscopy (4D-STEM): from scanning nanodiffraction to ptychography and beyond. Microsc. Microanal. 25, 563–582 (2019).
Google Scholar
Lozano, J. G., Martinez, G. T., Jin, L., Nellist, P. D. & Bruce, P. G. Low-dose aberration-free imaging of Li-rich cathode materials at various states of charge using electron ptychography. Nano Lett. 18, 6850–6855 (2018).
Google Scholar
Song, J. et al. Atomic resolution defocused electron ptychography at low dose with a fast, direct electron detector. Sci Rep. 9, 3919 (2019).
Google Scholar
O’Leary, C. M. et al. Phase reconstruction using fast binary 4D STEM data. Appl. Phys. Lett. 116, 124101 (2020).
Google Scholar
Chen, Z. et al. Mixed-state electron ptychography enables sub-angstrom resolution imaging with picometer precision at low dose. Nat. Commun. 11, 2994 (2020).
Google Scholar
Peng, X. et al. Observation of formation and local structures of metal-organic layers via complementary electron microscopy techniques. Nat. Commun. 13, 5197 (2022).
Google Scholar
Sha, H. et al. Ptychographic measurements of varying size and shape along zeolite channels. Sci. Adv. 9, eadf1151 (2023).
Google Scholar
Zhang, H. et al. Three-dimensional inhomogeneity of zeolite structure and composition revealed by electron ptychography. Science 380, 633–638 (2023).
Google Scholar
Kirkland, A. I., Saxton, W. O., Chau, K.-L., Tsuno, K. & Kawasaki, M. Super-resolution by aperture synthesis: tilt series reconstruction in CTEM. Ultramicroscopy 57, 355–374 (1995).
Google Scholar
Haigh, S. J., Sawada, H. & Kirkland, A. I. Atomic structure imaging beyond conventional resolution limits in the transmission electron microscope. Phys. Rev. Lett. 103, 126101 (2009).
Google Scholar
Horstmeyer, R. & Yang, C. A phase space model of Fourier ptychographic microscopy. Opt. Express 22, 338–358 (2014).
Google Scholar
Tian, L., Li, X., Ramchandran, K. & Waller, L. Multiplexed coded illumination for Fourier ptychography with an LED array microscope. Biomed. Opt. Express 5, 2376–2389 (2014).
Google Scholar
Horstmeyer, R., Chung, J., Ou, X., Zheng, G. & Yang, C. Diffraction tomography with Fourier ptychography. Optica 3, 827–835 (2016).
Google Scholar
Tian, L. & Waller, L. 3D intensity and phase imaging from light field measurements in an LED array microscope. Optica 2, 104–111 (2015).
Google Scholar
Chowdhury, S. et al. High-resolution 3D refractive index microscopy of multiple-scattering samples from intensity images. Optica 6, 1211–1219 (2019).
Google Scholar
Sha, H., Cui, J. & Yu, R. Deep sub-angstrom resolution imaging by electron ptychography with misorientation correction. Sci. Adv. 8, eabn2275 (2022).
Moniri, S. et al. Three-dimensional atomic structure and local chemical order of medium- and high-entropy nanoalloys. Nature 624, 564–569 (2023).
Google Scholar
Yang, Y. et al. Atomic-scale identification of active sites of oxygen reduction nanocatalysts. Nat. Catal. 7, 796–806 (2024).
Google Scholar
Basov, D. N., Averitt, R. D. & Hsieh, D. Towards properties on demand in quantum materials. Nat. Mater. 16, 1077–1088 (2017).
Google Scholar
Nagaosa, N. & Tokura, Y. Topological properties and dynamics of magnetic skyrmions. Nat. Nanotechnol. 8, 899–911 (2013).
Google Scholar
Yang, Y. et al. Deciphering chemical order/disorder and material properties at the single-atom level. Nature 542, 75–79 (2017).
Google Scholar
Rupp, D. et al. Coherent diffractive imaging of single helium nanodroplets with a high harmonic generation source. Nat. Commun. 8, 493 (2017).
Google Scholar
Langbehn, B. et al. Three-dimensional shapes of spinning helium nanodroplets. Phys. Rev. Lett. 121, 255301 (2018).
Google Scholar
Feinberg, A. J. et al. X-ray diffractive imaging of highly ionized helium nanodroplets. Phys. Rev. Res. 4, L022063 (2022).
Google Scholar
Johnson, A. S. et al. Quantitative hyperspectral coherent diffractive imaging spectroscopy of a solid-state phase transition in vanadium dioxide. Sci. Adv. 7, eabf1386 (2021).
Google Scholar
Shapiro, D. A. et al. Chemical composition mapping with nanometre resolution by soft X-ray microscopy. Nat. Photon. 8, 765–769 (2014).
Google Scholar
Kim, D. et al. Active site localization of methane oxidation on Pt nanocrystals. Nat. Commun. 9, 3422 (2018).
Google Scholar
Yau, A., Cha, W., Kanan, M. W., Stephenson, G. B. & Ulvestad, A. Bragg coherent diffractive imaging of single-grain defect dynamics in polycrystalline films. Science 356, 739–742 (2017).
Google Scholar
Loh, N. D. et al. Fractal morphology, imaging and mass spectrometry of single aerosol particles in flight. Nature 486, 513–517 (2012).
Google Scholar
Barty, A. et al. Ultrafast single-shot diffraction imaging of nanoscale dynamics. Nat. Photon. 2, 415–419 (2008).
Google Scholar
Abbey, B. et al. Quantitative coherent diffractive imaging of an integrated circuit at a spatial resolution of 20 nm. Appl. Phys. Lett. 93, 214101 (2008).
Google Scholar
Nogales, E. & Mahamid, J. Bridging structural and cell biology with cryo-electron microscopy. Nature 628, 47–56 (2024).
Google Scholar
Huang, X. et al. Soft X-ray diffraction microscopy of a frozen hydrated yeast cell. Phys. Rev. Lett. 103, 198101 (2009).
Google Scholar
Rodriguez, J. A. et al. Three-dimensional coherent X-ray diffractive imaging of whole, frozen-hydrated cells. IUCrJ. 2, 575–583 (2015).
Google Scholar
Zhou, L. et al. Low-dose phase retrieval of biological specimens using cryo-electron ptychography. Nat. Commun. 11, 2773 (2020).
Google Scholar
Chapman, H. N. et al. Femtosecond X-ray protein nanocrystallography. Nature 470, 73–77 (2011). Serial femtosecond crystallography was demonstrated to determine protein nanocrystal structures by capturing snapshot diffraction patterns with XFEL pulses.
Google Scholar
Barends, T. R. M., Stauch, B., Cherezov, V. & Schlichting, I. Serial femtosecond crystallography. Nat. Rev. Methods Primers 2, 59 (2022).
Google Scholar
Rodriguez, J. A. et al. Structure of the toxic core of α-synuclein from invisible crystals. Nature 525, 486–490 (2015).
Google Scholar
Kimura, T. et al. Imaging live cell in micro-liquid enclosure by X-ray laser diffraction. Nat. Commun. 5, 3052 (2014).
Google Scholar
van der Schot, G. et al. Imaging single cells in a beam of live cyanobacteria with an X-ray laser. Nat. Commun. 6, 5704 (2015).
Google Scholar
Hantke, M. F. et al. High-throughput imaging of heterogeneous cell organelles with an X-ray laser. Nat. Photon. 8, 943–949 (2014).
Google Scholar
Sheldrick, G. M. A short history of SHELX. Acta Crystallogr. A 64, 112–122 (2008).
Google Scholar
Wakonig, K. et al. PtychoShelves, a versatile high-level framework for high-performance analysis of ptychographic data. J. Appl. Crystallogr. 53, 574–586 (2020).
Google Scholar
Savitzky, B. H. et al. py4DSTEM: a software package for four-dimensional scanning transmission electron microscopy data analysis. Microsc. Microanal. 27, 712–743 (2021).
Google Scholar
Sinha, A., Lee, J., Li, S. & Barbastathis, G. Lensless computational imaging through deep learning. Optica 4, 1117–1125 (2017).
Google Scholar
Kappeler, A., Ghosh, S., Holloway, J., Cossairt, O. & Katsaggelos, A. Ptychnet: CNN based fourier ptychography. In Proc. IEEE Int. Conf. Image Processing 1712–1716 (IEEE, 2017).
Wang, K. et al. On the use of deep learning for phase recovery. Light Sci. Appl. 13, 4 (2024).
Google Scholar
Cherukara, M. J., Nashed, Y. S. G. & Harder, R. J. Real-time coherent diffraction inversion using deep generative networks. Sci Rep. 8, 16520 (2018).
Google Scholar
Wu, L. et al. Three-dimensional coherent X-ray diffraction imaging via deep convolutional neural networks. NPJ Comput. Mater. 7, 175 (2021).
Google Scholar
Chang, D. J. et al. Deep-learning electron diffractive imaging. Phys. Rev. Lett. 130, 016101 (2023).
Google Scholar
Schloz, M. et al. Deep reinforcement learning for data-driven adaptive scanning in ptychography. Sci Rep. 13, 8732 (2023).
Google Scholar
Babu, A. V. et al. Deep learning at the edge enables real-time streaming ptychographic imaging. Nat. Commun. 14, 7059 (2023).
Google Scholar
Lee, J., Lee, M., Park, Y., Ophus, C. & Yang, Y. Multislice electron tomography using four-dimensional scanning transmission electron microscopy. Phys. Rev. Appl. 19, 054062 (2023).
Google Scholar
Popmintchev, T. et al. Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers. Science 336, 1287–1291 (2012).
Google Scholar
Li, P. et al. 4th generation synchrotron source boosts crystalline imaging at the nanoscale. Light Sci. Appl. 11, 73 (2022).
Google Scholar
Lo, Y. H. et al. In situ coherent diffractive imaging. Nat. Commun. 9, 1826 (2018).
Google Scholar
Lu, X. et al. Computational microscopy beyond perfect lenses. Phys. Rev. E 110, 054407 (2024).
Young, L. et al. Roadmap of ultrafast x-ray atomic and molecular physics. J. Phys. B At. Mol. Opt. Phys. 51, 032003 (2018).
Google Scholar
Wakonig, K. et al. X-ray Fourier ptychography. Sci. Adv. 5, eaav0282 (2019).
Google Scholar
Cao, R., Shen, C. & Yang, C. High-resolution, large field-of-view label-free imaging via aberration-corrected, closed-form complex field reconstruction. Nat. Commun. 15, 4713 (2024).
Google Scholar
Schnars, U., Falldorf, C., Watson, J. & Jüptner, W. in Digital Holography and Wavefront Sensing: Principles, Techniques and Applications 2nd edn (eds Schnars, U. et al.) 39–68 (Springer, 2015).
Taylor, G. The phase problem. Acta Crystallogr. D Biol. Crystallogr. 59, 1881–1890 (2003).
Google Scholar
Sayre, D. Some implications of a theorem due to Shannon. Acta Crystallogr. 5, 843 (1952).
Google Scholar
Fienup, J. R. Reconstruction of an object from the modulus of its Fourier transform. Opt. Lett. 3, 27–29 (1978).
Google Scholar
Bruck, Y. M. & Sodin, L. G. On the ambiguity of the image reconstruction problem. Opt. Commun. 30, 304–308 (1979).
Google Scholar
Hayes, M. H. The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform. IEEE Trans. Acoust. Speech Signal Process. 30, 140–154 (1982).
Google Scholar
Candès, E. J., Li, X. & Soltanolkotabi, M. Phase retrieval via Wirtinger flow: theory and algorithms. IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
Google Scholar
Grohs, P., Koppensteiner, S. & Rathmair, M. Phase retrieval: uniqueness and stability. SIAM Rev. 62, 301–350 (2020).
Google Scholar
Elser, V. Phase retrieval by iterated projections. J. Opt. Soc. Am. A 20, 40–55 (2003).
Google Scholar
Luke, D. R. Relaxed averaged alternating reflections for diffraction imaging. Inverse Probl. 21, 37–50 (2005).
Google Scholar
Rodriguez, J. A., Xu, R., Chen, C.-C., Zou, Y. & Miao, J. Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities. J. Appl. Cryst. 46, 312–318 (2013).
Google Scholar
Maiden, A. M. & Rodenburg, J. M. An improved ptychographical phase retrieval algorithm for diffractive imaging. Ultramicroscopy 109, 1256–1262 (2009). The extended ptychographic iterative engine was developed to simultaneously reconstruct the complex-valued probe and sample functions.
Google Scholar
Thibault, P. & Guizar-Sicairos, M. Maximum-likelihood refinement for coherent diffractive imaging. New J. Phys. 14, 063004 (2012).
Google Scholar
Odstrčil, M., Menzel, A. & Guizar-Sicairos, M. Iterative least-squares solver for generalized maximum-likelihood ptychography. Opt. Express 26, 3108–3123 (2018).
Google Scholar
Szameit, A. et al. Sparsity-based single-shot subwavelength coherent diffractive imaging. Nat. Mater. 11, 455–459 (2012).
Google Scholar
Candès, E. J., Strohmer, T. & Voroninski, V. PhaseLift: Exact and stable signal recovery from magnitude measurements via convex programming. Commun. Pure Appl. Math. 66, 1241–1274 (2013). Phase retrieval was formulated as a convex optimization problem, known as PhaseLift, sparking intense research into numerical algorithms in applied mathematics.
Google Scholar
Waldspurger, I., d’Aspremont, A. & Mallat, S. Phase recovery, MaxCut and complex semidefinite programming. Math. Program. 149, 47–81 (2015).
