{{Short description|Method of microscopic imaging}} {{Use Hiberno-English|date=July 2018}} {{Use dmy dates|date=June 2023}}
alt=Collection of a ptychographic imaging data set in the simplest single-aperture configuration.|thumb|Collection of a ptychographic imaging data set in the simplest single-aperture configuration. (a) Coherent illumination incident from the left is locally confined onto an area of the specimen. A detector downstream of the specimen records an interference pattern. (b) The specimen is shifted (in this case, upwards) and a second pattern is recorded. Note that regions of illumination must overlap with one another to facilitate the ptychographic shift-invariance constraint. (c) A whole ptychographic data set uses many overlapping regions of illumination. (d) The entire data set is four-dimensional: for each 2D illumination position (''x'', ''y''), there is a 2D diffraction pattern (''k<sub>x</sub>'', ''k<sub>y</sub>'').
'''Ptychography''' (/t(a)ɪˈkɒgrəfi/ t(a)i-KO-graf-ee)<ref>{{Cite journal |last=Blaustein |first=Anna |date=2021-08-01 |title=See the Highest-Resolution Atomic Image Ever Captured |url=https://www.scientificamerican.com/article/see-the-highest-resolution-atomic-image-ever-captured/ |access-date=2025-01-28 |journal=Scientific American |volume=325 |issue=3 |pages=None |doi=10.1038/scientificamerican082021-3MPGoSUeGnLFZDu5GldVxu |pmid=39020685 |language=en}}</ref> is a computational microscopy technique that reconstructs the complex-valued image (amplitude and phase) of a specimen from a series of coherent diffraction patterns recorded as a localized probe is scanned with overlap across the sample.<ref name="Miao2025">{{Cite journal | last=Miao J | title=Computational microscopy with coherent diffractive imaging and ptychography | journal=Nature | volume=637 | pages=281–295 | year=2025 | issue=8045 | url=https://www.nature.com/articles/s41586-024-08278-z | doi=10.1038/s41586-024-08278-z| pmid=39780004 | bibcode=2025Natur.637..281M | url-access=subscription }}</ref><ref name="Rodenburg2019">{{cite book | vauthors = Rodenburg J, Maiden A |chapter=Ptychography |date=2019 |title=Springer Handbook of Microscopy |series=Springer Handbooks |pages=819–904 |veditors = Hawkes PW, Spence JC |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-030-00069-1_17 |isbn=978-3-030-00068-4 |url=https://eprints.whiterose.ac.uk/127795/1/Ptychography_Chapter-Rodenburg%2BMaiden_final.pdf}}</ref> It unifies the principles of microscopy and crystallography, combining the real-space imaging of microscopy with the reciprocal-space diffraction analysis of crystallography to produce high-resolution, quantitative images free from lens aberrations. Ptychography has been demonstrated with visible light, X-rays, electrons and extreme-ultraviolet radiation, enabling quantitative phase contrast imaging across nine orders of magnitude in length scales.<ref name="Miao2025" /><ref name="Rodenburg2019" />
Its defining characteristic is translational invariance, which means that the interference patterns are generated by one constant function (e.g. a field of illumination or an aperture stop) moving laterally by a known amount with respect to another constant function (the specimen itself or a wave field). The interference patterns occur some distance away from these two components, so that the scattered waves spread out and "fold" ({{langx|grc|πτυχή}}, "ptychē" is 'fold'<ref>{{Cite journal| vauthors = Hegerl R, Hoppe W |date=1970 |title=Dynamische Theorie der Kristallstrukturanalyse durch Elektronenbeugung im inhomogenen Primärstrahlwellenfeld |journal=Berichte der Bunsengesellschaft für physikalische Chemie |language=de |volume=74 |issue=11 |pages=1148–1154 |doi=10.1002/bbpc.19700741112}}</ref>) into one another as shown in the figure.
Unlike conventional lens imaging, ptychography is unaffected by lens-induced aberrations or diffraction effects caused by limited numerical aperture.<ref>{{Cite journal |last=Rodenburg and Maiden |first=John and Andy |title=Ptychography |journal=Springer}}</ref> This is particularly important for atomic-scale wavelength imaging, where it is difficult and expensive to make good-quality lenses with high numerical aperture. Another advantage is its high phase sensitivity, enabling clear imaging of transparent or weakly absorbing specimens. This is because it is sensitive to the phase of the radiation that has passed through a specimen, and so it does not rely on the object absorbing radiation. In the case of visible-light biological microscopy, this means that cells do not need to be stained or labelled to create contrast.
Modern ptychography, developed in the 2000s and now the most widely used form of the technique,<ref name="Miao2025" /><ref name="Rodenburg2019" /> combines scanning microscopy with coherent diffractive imaging (CDI)<ref name="Miao1999">{{cite journal |last1=Miao |first1=J. |last2=Charalambous |first2=P. |last3=Kirz |first3=J. |last4=Sayre |first4=D. |title=Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens |journal=Nature |date=1999 |volume=400 |issue=6742 |pages=342–344 |doi=10.1038/22498|bibcode=1999Natur.400..342M |s2cid=4327928 }}</ref> through iterative phase-retrieval algorithms.<ref name="Rodenburg2007">{{cite journal | vauthors = Rodenburg JM, Hurst AC, Cullis AG, Dobson BR, Pfeiffer F, Bunk O, David C, Jefimovs K, Johnson I | display-authors = 6 | title = Hard-x-ray lensless imaging of extended objects | journal = Physical Review Letters | volume = 98 | issue = 3 | article-number = 034801 | date = January 2007 | pmid = 17358687 | doi = 10.1103/PhysRevLett.98.034801 | url = https://www.dora.lib4ri.ch/psi/islandora/object/psi%3A18137 | bibcode = 2007PhRvL..98c4801R }}</ref> In this approach, a coherent probe—such as an X-ray, electron, or optical beam—is scanned across the specimen with overlapping illumination regions, and an oversampled diffraction pattern is recorded at each position. The overlap between adjacent probe positions in real space and the oversampling of diffraction data in reciprocal space provide sufficient redundancy to enable the simultaneous reconstruction of both the probe and the sample transmission functions. This yields quantitative, aberration-free phase images that are robust to partial coherence and experimental imperfections, while providing both high spatial resolution and a large field of view. Modern ptychography has been demonstrated with X-rays, electrons, and visible light, providing sub-ångström resolution in electron microscopy and quantitative three-dimensional imaging through X-ray ptychotomography.<ref name="Rodenburg2019" />
== Phase recovery == {{main|Phase recovery}}
Although the interference patterns used in ptychography can only be measured in intensity, the mathematical constraints provided by the translational invariance of the two functions (illumination and object), together with the known shifts between them, means that the phase of the wavefield can be recovered by an inverse computation. Ptychography thus provides a general solution to the "phase problem". By recording more independent intensity measurements than unknown variables,<ref>{{cite journal |last1=Miao |first1=J. |last2=Sayre |first2=D. |last3=Chapman |first3=H. N. |title=Phase Retrieval from the Magnitude of the Fourier transform of Non-periodic Objects |journal=J. Opt. Soc. Am. A |date=1998 |volume=15 |issue=6 |pages=1662–1669 |doi=10.1364/JOSAA.15.001662|bibcode=1998JOSAA..15.1662M }}</ref> achieved through oversampling in reciprocal space and overlapping in real space, the phase information becomes encoded in the measured diffraction intensities and can be retrieved computationally using iterative algorithms.<ref>{{cite journal |last1=Faulkner |first1=H. M. L. |last2=Rodenburg |first2=J. M. |year=2004 |title=Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm |journal=Physical Review Letters |volume=93 |issue=2 |article-number=023903 |doi=10.1103/PhysRevLett.93.023903 |pmid=15323918 |bibcode=2004PhRvL..93b3903F }}</ref> This formulation has also stimulated substantial research in applied mathematics, particularly on the uniqueness, stability, and convergence properties of phase-retrieval problems.<ref>{{cite journal |last1=Candès |first1=E. J. |last2=Strohmer |first2=T. |last3=Voroninski |first3=V. |year=2013 |title=PhaseLift: Exact and stable signal recovery from magnitude measurements via convex programming |journal=Communications on Pure and Applied Mathematics |volume=66 |issue=8 |pages=1241–1274 |doi=10.1002/cpa.21432 }}</ref><ref>{{cite journal |last1=Fannjiang |first1=A. |last2=Strohmer |first2=T. |year=2020 |title=The numerics of phase retrieval |journal=Acta Numerica |volume=29 |pages=125–228 |doi=10.1017/S0962492920000013 |doi-broken-date=18 November 2025 }}</ref><ref>{{cite journal |last1=Dong |first1=J. |last2=Valzania |first2=L. |last3=Maillard |first3=A. |last4=Pham |first4=T.-A. |last5=Gigan |first5=S. |last6=Unser |first6=M. |year=2023 |title=Phase retrieval: From computational imaging to machine learning: A tutorial |journal=IEEE Signal Processing Magazine |volume=40 |issue=1 |pages=45–57 |doi=10.1109/MSP.2022.10004797 |doi-broken-date=18 November 2025 }}</ref>
Once this is achieved, all the information relating to the scattered wave (modulus and phase) has been recovered, and so virtually perfect images of the object can be obtained. There are various strategies for performing this inverse phase-retrieval calculation, including direct Wigner distribution deconvolution (WDD)<ref name=":2">{{cite journal | vauthors = Rodenburg J, Bates RH |date=15 June 1992 |title=The theory of super-resolution electron microscopy via Wigner-distribution deconvolution |journal=Phil. Trans. R. Soc. Lond. A |volume=339 |issue=1655 |pages=521–553 |doi=10.1098/rsta.1992.0050 |bibcode=1992RSPTA.339..521R |s2cid=123384269}}</ref> and iterative methods.<ref name=":3">{{Cite journal |vauthors = Rodenburg JM, Faulkner HM |date=15 November 2004 |title=A phase retrieval algorithm for shifting illumination |journal=Applied Physics Letters |volume=85 |issue=20 |pages=4795–4797 |doi=10.1063/1.1823034 |bibcode=2004ApPhL..85.4795R}}</ref><ref>{{cite journal | vauthors = Guizar-Sicairos M, Fienup JR | title = Phase retrieval with transverse translation diversity: a nonlinear optimization approach | journal = Optics Express | volume = 16 | issue = 10 | pages = 7264–78 | date = May 2008 | pmid = 18545432 | doi = 10.1364/OE.16.007264 | name-list-style = vanc | doi-access = free | bibcode = 2008OExpr..16.7264G }}</ref><ref>{{cite journal | vauthors = Thibault P, Dierolf M, Menzel A, Bunk O, David C, Pfeiffer F | title = High-resolution scanning x-ray diffraction microscopy | journal = Science | volume = 321 | issue = 5887 | pages = 379–82 | date = July 2008 | pmid = 18635796 | doi = 10.1126/science.1158573 | s2cid = 30125688 | bibcode = 2008Sci...321..379T | url = http://infoscience.epfl.ch/record/125719 }}</ref><ref name=":0">{{cite journal | vauthors = Thibault P, Dierolf M, Bunk O, Menzel A, Pfeiffer F | title = Probe retrieval in ptychographic coherent diffractive imaging | journal = Ultramicroscopy | volume = 109 | issue = 4 | pages = 338–43 | date = March 2009 | pmid = 19201540 | doi = 10.1016/j.ultramic.2008.12.011 | url = http://infoscience.epfl.ch/record/159879 }}</ref><ref name=":1">{{cite journal | vauthors = Maiden AM, Rodenburg JM | title = An improved ptychographical phase retrieval algorithm for diffractive imaging | journal = Ultramicroscopy | volume = 109 | issue = 10 | pages = 1256–62 | date = September 2009 | pmid = 19541420 | doi = 10.1016/j.ultramic.2009.05.012 }}</ref> The difference map algorithm developed by Thibault and co-workers<ref name=":0" /> is available in a downloadable package called [https://ptycho.github.io/ptypy/index.html PtyPy].<ref name=":4">{{cite journal | vauthors = Enders B, Thibault P | title = A computational framework for ptychographic reconstructions | journal = Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences| volume = 472 | issue = 2196 | article-number = 20160640 | date = December 2016 | pmid = 28119552 | pmc = 5247528 | doi = 10.1098/rspa.2016.0640 | bibcode = 2016RSPSA.47260640E }}</ref>
== Optical configurations == There are many optical configurations for ptychography: mathematically, it requires two invariant functions that move across one another while an interference pattern generated by the product of the two functions is measured. The interference pattern can be a diffraction pattern, a Fresnel diffraction pattern or, in the case of Fourier ptychography, an image. The "ptycho" convolution in a Fourier ptychographic image derived from the impulse response function of the lens.
=== The single aperture === alt=Diagram showing the optical configuration for ptychographu using a single aperture.|thumb|Optical configuration for ptychography using a single aperture This is conceptually the simplest ptychographical arrangement.<ref name=":12" /> The detector can either be a long way from the object (i.e. in the Fraunhofer diffraction plane), or closer by, in the Fresnel regime. An advantage of the Fresnel regime is that there is no longer a very high-intensity beam at the centre of the diffraction pattern, which can otherwise saturate the detector pixels there.
=== Focused-probe ptychography === alt=Diagram showing the optical configuration for ptychography using a focussed probe.|thumb|Optical configuration for ptychography using a focused probe A lens is used to form a tight crossover of the illuminating beam at the plane of the specimen. The configuration is used in the scanning transmission electron microscope (STEM),<ref name=":6" /><ref name=":17">{{cite journal | vauthors = Yang H, Rutte RN, Jones L, Simson M, Sagawa R, Ryll H, Huth M, Pennycook TJ, Green ML, Soltau H, Kondo Y, Davis BG, Nellist PD | display-authors = 6 | title = Simultaneous atomic-resolution electron ptychography and Z-contrast imaging of light and heavy elements in complex nanostructures | language = En | journal = Nature Communications | volume = 7 | article-number = 12532 | date = August 2016 | pmid = 27561914 | pmc = 5007440 | doi = 10.1038/ncomms12532 | bibcode = 2016NatCo...712532Y }}</ref> and often in high-resolution X-ray ptychography. The specimen is sometimes shifted up or downstream of the probe crossover so as to allow the size of the patch of illumination to be increased, thus requiring fewer diffraction patterns to scan a wide field of view.
=== Multislice ptychography === thumb|A focused probe scans the sample, collecting diffraction patterns at each position. The algorithm reconstructs multiple phase slices at different depths, decomposing the 3D structure from a single-view scan.<ref name="OLeary2024">{{cite journal | last1 = O'Leary | first1 = Colum M. | last2 = Sha | first2 = Haozhi | last3 = Zhang | first3 = Jianhua | last4 = Su | first4 = Cong | last5 = Kahn | first5 = Salman | last6 = Jiang | first6 = Huaidong | last7 = Zettl | first7 = Alex | last8 = Ciston | first8 = Jim | last9 = Miao | first9 = Jianwei | title = Three-dimensional structure of buried heterointerfaces revealed by multislice ptychography | journal = Physical Review Applied | volume = 22 | issue = 1 | article-number = 014016 | date = July 8, 2024 | doi = 10.1103/PhysRevApplied.22.014016 | arxiv = 2308.15471 | bibcode = 2024PhRvP..22a4016O }}</ref>
Multislice ptychography extends iterative ptychography to account for multiple scattering and three-dimensional (3D) structure by modeling the specimen as a sequence of transmission slices along the beam propagation direction. This approach addresses the limitations of the single-slice (projection) approximation, which breaks down for thicker samples where electron or X-ray wavefronts undergo significant longitudinal evolution. The conceptual origin for using a single view to recover 3D structural information was demonstrated in ankylography by Miao and collaborator in 2010,<ref name="Miao2010"> {{cite journal | last1=Raines | first1=Kevin S. | last2=Salha | first2=Sara | last3=Sandberg | first3=Richard L. | last4=Jiang | first4=Huaidong | last5=Rodríguez | first5=José A. | last6=Fahimian | first6=Benjamin P. | last7=Kapteyn | first7=Henry C. | last8=Du | first8=Jincheng | last9=Miao | first9=Jianwei | title=Three-dimensional structure determination from a single view | journal=Nature | date=14 January 2010 | volume=463 | issue=7280 | pages=214–217 | doi=10.1038/nature08705 | arxiv=0905.0269 | bibcode=2010Natur.463..214R }}</ref> which showed that coherent diffraction patterns oversampled on a curved Ewald sphere can encode depth information without the requirement of tilting or depth scanning. The general computational formulation of multislice ptychography was introduced in 2012 by Maiden, Humphry, and Rodenburg,<ref name="Maiden2012">{{cite journal | vauthors = Maiden AM, Humphry MJ, Rodenburg JM | title = Ptychographic transmission microscopy in three dimensions using a multi-slice approach | journal = Journal of the Optical Society of America A | volume = 29 | issue = 8 | pages = 1606–1614 | date = August 2012 | pmid = 23201876 | doi = 10.1364/JOSAA.29.001606 | bibcode = 2012JOSAA..29.1606M }}</ref> who developed the multislice iterative engine (MIE), incorporating transverse scanning and multislice propagation to reconstruct multiple axial slices from overlapping, oversampled diffraction patterns. Since then, multislice ptychography has been implemented with X-ray,<ref>{{cite journal | last1 = Suzuki | first1 = A. | title = High-resolution multislice X-ray ptychography of extended thick objects | journal = Physical Review Letters | volume = 112 | issue = 5 | article-number = 053903 | date = February 5, 2014 | doi = 10.1103/PhysRevLett.112.053903 | pmid = 24580593 | bibcode = 2014PhRvL.112e3903S }}</ref><ref>{{cite journal | last1 = Tsai | first1 = Esther H. | last2 = Usov | first2 = Ivan | last3 = Diaz | first3 = Ana | last4 = Menzel | first4 = Andreas | last5 = Guizar-Sicairos | first5 = Manuel | title = X-ray ptychography with extended depth of field | journal = Optics Express | volume = 24 | issue = 26 | pages = 29089–29108 | date = December 26, 2016 | doi = 10.1364/OE.24.029089 | pmid = 27958573 | bibcode = 2016OExpr..2429089T }}</ref> electron,<ref>{{cite journal | last1 = Gao | first1 = Si | last2 = Wang | first2 = Peng | last3 = Zhang | first3 = Fucai | last4 = Martinez | first4 = Gerardo T. | last5 = Nellist | first5 = Peter D. | last6 = Pan | first6 = Xiaoqing | last7 = Kirkland | first7 = Angus I. | title = Electron ptychographic microscopy for three-dimensional imaging | journal = Nature Communications | volume = 8 | article-number = 163 | date = July 21, 2017 | doi = 10.1038/s41467-017-00150-1 | pmid = 28740133 | pmc = 5524651 | bibcode = 2017NatCo...8..163G | url = https://escholarship.org/uc/item/7w56p653 }}</ref><ref name="Chen2021">{{Cite journal|last1=Chen|first1=Zhen|last2=Jiang|first2=Yi|last3=Shao|first3=Yu-Tsun|last4=Holtz|first4=Megan E.|last5=Odstrčil|first5=Michal|last6=Guizar-Sicairos|first6=Manuel|last7=Hanke|first7=Isabelle|last8=Ganschow|first8=Steffen|last9=Schlom|first9=Darrell G.|last10=Muller|first10=David A.|date=21 May 2021|title=Electron ptychography achieves atomic-resolution limits set by lattice vibrations|url=https://www.science.org/doi/10.1126/science.abg2533|journal=Science|language=en|volume=372|issue=6544|pages=826–831|doi=10.1126/science.abg2533|arxiv=2101.00465|issn=0036-8075|pmid=34016774|bibcode=2021Sci...372..826C |s2cid=230435950}}</ref><ref name="Sha2023">{{cite journal | last1 = Sha | first1 = Haozhi | last2 = Cui | first2 = Jizhe | last3 = Li | first3 = Jialu | last4 = Zhang | first4 = Yuxuan | last5 = Yang | first5 = Wenfeng | last6 = Li | first6 = Yadong | last7 = Yu | first7 = Rong | title = Ptychographic measurements of varying size and shape along zeolite channels | journal = Science Advances | volume = 9 | issue = 11 | article-number = eadf1151 | date = March 17, 2023 | doi = 10.1126/sciadv.adf1151 | pmid = 36921047 | pmc = 10017048 | bibcode = 2023SciA....9F1151S }}</ref><ref name="Zhang2023">{{cite journal | last1 = Zhang | first1 = Hui | last2 = Li | first2 = Guanxing | last3 = Zhang | first3 = Jiaxing | last4 = Zhang | first4 = Daliang | last5 = Chen | first5 = Zhen | last6 = Liu | first6 = Xiaona | last7 = Guo | first7 = Peng | last8 = Zhu | first8 = Yihan | last9 = Chen | first9 = Cailing | last10 = Liu | first10 = Lingmei | last11 = Guo | first11 = Xinwen | last12 = Han | first12 = Yu | title = Three-dimensional inhomogeneity of zeolite structure and composition revealed by electron ptychography | journal = Science | volume = 380 | issue = 6645 | pages = 633–638 | date = May 12, 2023 | doi = 10.1126/science.adg3183 | pmid = 37167385 | bibcode = 2023Sci...380..633Z }}</ref><ref name="Dong2024">{{cite journal | last1 = Dong | first1 = Zehao | last2 = Huo | first2 = Mengwu | last3 = Li | first3 = Jie | last4 = Li | first4 = Jingyuan | last5 = Li | first5 = Pengcheng | last6 = Sun | first6 = Hualei | last7 = Gu | first7 = Lin | last8 = Lu | first8 = Yi | last9 = Wang | first9 = Meng | last10 = Wang | first10 = Yayu | last11 = Chen | first11 = Zhen | title = Visualization of oxygen vacancies and self-doped ligand holes in La<sub>3</sub>Ni<sub>2</sub>O<sub>7−δ</sub> | journal = Nature | volume = 630 | issue = 8018 | pages = 847–852 | date = June 27, 2024 | doi = 10.1038/s41586-024-07482-1 | pmid = 38839959 | arxiv = 2312.15727 | bibcode = 2024Natur.630..847D }}</ref><ref name="Huang2023">{{cite journal | last1 = Zhang | first1 = Yichao | last2 = Ahammed | first2 = Ballal | last3 = Bae | first3 = Sang Hyun | last4 = Lee | first4 = Chia-Hao | last5 = Huang | first5 = Jeffrey | last6 = Hossain | first6 = Mohammad Abir | last7 = Rakib | first7 = Tawfiqur | last8 = van der Zande | first8 = Arend M. | last9 = Ertekin | first9 = Elif | last10 = Huang | first10 = Pinshane Y. | title = Atom-by-atom imaging of moiré phasons with electron ptychography | journal = Science | volume = 389 | issue = 6758 | pages = 423–428 | date = 24 July 2025 | doi = 10.1126/science.adw7751 | pmid = 40705895 | arxiv = 2505.03060 | bibcode = 2025Sci...389..423Z }}</ref> and optical<ref>{{cite journal | last1 = Tian | first1 = Lei | last2 = Waller | first2 = Laura | title = 3D intensity and phase imaging from light field measurements in an LED array microscope | journal = Optica | volume = 2 | issue = 2 | pages = 104–111 | date = February 2015 | doi = 10.1364/OPTICA.2.000104 | bibcode = 2015Optic...2..104T }}</ref><ref>{{cite journal | last1 = Chowdhury | first1 = Shwetadwip | last2 = Chen | first2 = Michael | last3 = Eckert | first3 = Regina | last4 = Ren | first4 = David | last5 = Wu | first5 = Fan | last6 = Repina | first6 = Nicole | last7 = Waller | first7 = Laura | title = High-resolution 3D refractive index microscopy of multiple-scattering samples from intensity images | journal = Optica | volume = 6 | issue = 9 | pages = 1211–1219 | date = September 16, 2019 | doi = 10.1364/OPTICA.6.001211 | pmid = 38515960 | pmc = 10956703 | arxiv = 1909.00023 | bibcode = 2019Optic...6.1211C }}</ref> instruments, enabling slice resolved (partial 3D) imaging in thick specimens. An important multislice electron ptychography experiment was reported by Chen, Muller, and collaborators in 2021, producing a phase image of a PrScO<sub>3</sub> crystal with 0.23 Å resolution.<ref name="Chen2021" />
The achievable depth resolution ''d''<sub>z</sub> is bounded by<ref name="Miao2010" /><ref name="OLeary2024" /> <math display="block"> d_z = \frac{\lambda}{2\sin^2(\theta / 2)}, </math> where ''λ'' is the wavelength of the illumination, and ''θ'' is the maximum scattering angle captured by the detector. The depth resolution is typically much poorer than the lateral resolution, which is the primary current limitation.
Recent applications of multislice ptychography include atomic-scale imaging of oxygen vacancies in zeolites<ref name="Sha2023" /><ref name="Zhang2023" /> and high temperature superconductors,<ref name="Dong2024" /> oxygen anion displacements in 3D in ferroelectrics,<ref>{{Cite journal |last1=Kp |first1=Harikrishnan |last2=Xu |first2=Ruijuan |last3=Patel |first3=Kinnary |last4=Crust |first4=Kevin J. |last5=Khandelwal |first5=Aarushi |last6=Zhang |first6=Chenyu |last7=Prosandeev |first7=Sergey |last8=Zhou |first8=Hua |last9=Shao |first9=Yu-Tsun |last10=Bellaiche |first10=Laurent |last11=Hwang |first11=Harold Y. |last12=Muller |first12=David A. |date=September 2025 |title=Electron ptychography reveals a ferroelectricity dominated by anion displacements |url=https://www.nature.com/articles/s41563-025-02205-x |journal=Nature Materials |language=en |volume=24 |issue=9 |pages=1433–1440 |doi=10.1038/s41563-025-02205-x |pmid=40269146 |arxiv=2408.14795 |bibcode=2025NatMa..24.1433K |issn=1476-1122}}</ref> interface mapping in van der Waals heterostructures<ref name="OLeary2024" /> and moiré phasons in 2D materials.<ref name="Huang2023" /> Multislice ptychography continues to evolve rapidly.
=== Near-field ptychography === alt=A diagram showing the optical configuration for near-field ptychography.|thumb|Optical configuration for near-field ptychograhy This uses a wide field of illumination. To provide magnification, a diverging beam is incident on the specimen. An out-of-focus image, which appears as a Fresnel interference pattern, is projected onto the detector. The illumination must have phase distortions in it, often provided by a diffuser that scrambles the phase of the incident wave before it reaches the specimen, otherwise the image remains constant as the specimen is moved, so there is no new ptychographical information from one position to the next.<ref name=":13" /> In the electron microscope, a lens can be used to map the magnified Fresnel image onto the detector.
=== Fourier ptychography === {{See also|Fourier ptychography}} alt=Diagram showing the optical configuration for Fourier ptychography.|thumb|Optical configuration for Fourier ptychography A conventional microscope is used with a relatively small numerical aperture objective lens. The specimen is illuminated from a series of different angles. Parallel beams coming out of the specimen are brought to a focus in the back focal plane of the objective lens, which is therefore a Fraunhofer diffraction pattern of the specimen exit wave (Abbe's theorem). Tilting the illumination has the effect of shifting the diffraction pattern across the objective aperture (which also lies in the back focal plane). Now the standard ptychographical shift invariance principle applies, except that the diffraction pattern is acting as the object and the back focal plane stop is acting like the illumination function in conventional ptychography. The image is in the Fraunhofer diffraction plane of these two functions (another consequence of Abbe's theorem), just like in conventional ptychography. The only difference is that the method reconstructs the diffraction pattern, which is much wider than the aperture stop limitation. A final Fourier transform must be undertaken to produce the high-resolution image. All the reconstruction algorithms used in conventional ptychography apply to Fourier ptychography, and indeed nearly all the diverse extensions of conventional ptychography have been used in Fourier ptychography.<ref name=":14" />
=== Imaging ptychography === alt=Diagram showing the optical configuration for imaging ptychography.|thumb|Optical configuration for imaging ptychography A lens is used to make a conventional image. An aperture in the image plane acts equivalently to the illumination in conventional ptychography, while the image corresponds to the specimen. The detector lies in the Fraunhofer or Fresnel diffraction plane downstream of the image and aperture.<ref>{{cite journal | vauthors = Maiden AM, Sarahan MC, Stagg MD, Schramm SM, Humphry MJ | title = Quantitative electron phase imaging with high sensitivity and an unlimited field of view | language = En | journal = Scientific Reports | volume = 5 | article-number = 14690 | date = October 2015 | pmid = 26423558 | pmc = 4589788 | doi = 10.1038/srep14690 | bibcode = 2015NatSR...514690M }}</ref>
=== Bragg ptychography or reflection ptychography === alt=Diagram showing the optical configuration for reflection or Bragg ptychography.|thumb|Optical configuration for reflection or Bragg ptychography This geometry can be used either to map surface features or to measure strain in crystalline specimens. Shifts in the specimen surface, or the atomic Bragg planes perpendicular to the surface, appear in the phase of the ptychographic image.<ref name=":15" /><ref name=":16" /><ref>{{Cite journal|last1=Godard|first1=P.|last2=Carbone|first2=G.|last3=Allain|first3=M.|last4=Mastropietro|first4=F.|last5=Chen|first5=G.|last6=Capello|first6=L.|last7=Diaz|first7=A.|last8=Metzger|first8=T.H.|last9=Stangl|first9=J.|last10=Chamard|first10=V.|date=2011|title=Three-dimensional high-resolution quantitative microscopy of extended crystals|journal=Nature Communications|language=en|volume=2|issue=1|page=568|doi=10.1038/ncomms1569|pmid=22127064|bibcode=2011NatCo...2..568G |issn=2041-1723|doi-access=free}}</ref>
=== Vectorial ptychography === Vectorial ptychography needs to be invoked when the multiplicative model of the interaction between the probe and the specimen cannot be described by scalar quantities.<ref>{{cite journal | vauthors = Ferrand P, Allain M, Chamard V | title = Ptychography in anisotropic media | language = EN | journal = Optics Letters | volume = 40 | issue = 22 | pages = 5144–5147 | date = November 2015 | pmid = 26565820 | doi = 10.1364/OL.40.005144 | url = https://hal.archives-ouvertes.fr/hal-01213942/file/Ferrand-OL-2015.pdf | bibcode = 2015OptL...40.5144F | s2cid = 11476364 }}</ref> This happens typically when polarized light probes an anisotropic specimen, and when this interaction modifies the state of polarization of light. In that case, the interaction needs to be described by the Jones formalism,<ref>{{Cite journal |vauthors = Jones RC |date=1 July 1941 |title=A New Calculus for the Treatment of Optical SystemsI. Description and Discussion of the Calculus |journal=JOSA |language=EN |volume=31 |issue=7 |pages=488–493 |doi=10.1364/JOSA.31.000488 |bibcode=1941JOSA...31..488J }}</ref> where field and object are described by a two-component complex vector and a 2×2 complex matrix respectively. The optical configuration for vectorial ptychography is similar to that of classical (scalar) ptychography, although a control of light polarization (before and after the specimen) needs to be implemented in the setup. Jones maps of the specimens can be retrieved, allowing the quantification of a wide range of optical properties (phase, birefringence, orientation of neutral axes, diattenuation, etc.).<ref name=":18">{{cite journal | vauthors = Ferrand P, Baroni A, Allain M, Chamard V | title = Quantitative imaging of anisotropic material properties with vectorial ptychography | language = EN | journal = Optics Letters | volume = 43 | issue = 4 | pages = 763–766 | date = February 2018 | pmid = 29443988 | doi = 10.1364/OL.43.000763 | arxiv = 1712.00260 | s2cid = 3433117 | bibcode = 2018OptL...43..763F }}</ref> Similarly to scalar ptychography, the probes used for the measurement can be jointly estimated together with the specimen.<ref>{{cite journal | vauthors = Baroni A, Allain M, Li P, Chamard V, Ferrand P | title = Joint estimation of object and probes in vectorial ptychography | language = EN | journal = Optics Express | volume = 27 | issue = 6 | pages = 8143–8152 | date = March 2019 | pmid = 31052637 | doi = 10.1364/OE.27.008143 | url = https://hal-amu.archives-ouvertes.fr/hal-02059897/file/Baroni-oe-27-6-8143.pdf | bibcode = 2019OExpr..27.8143B | doi-access = free }}</ref> As a consequence, vectorial ptychography is also an elegant approach for quantitative imaging of coherent vectorial light beams (mixing wavefront and polarization features).<ref>{{cite journal | vauthors = Baroni A, Ferrand P | title = Reference-free quantitative microscopic imaging of coherent arbitrary vectorial light beams | journal = Optics Express | volume = 28 | issue = 23 | pages = 35339–35349 | date = November 2020 | pmid = 33182982 | doi = 10.1364/OE.408665 | bibcode = 2020OExpr..2835339B | url = https://www.osapublishing.org/abstract.cfm?URI=oe-28-23-35339 | doi-access = free }}</ref>
== Advantages ==
=== Lens insensitive === Ptychography can be undertaken without using any lenses at all,<ref name=":12">{{cite journal | vauthors = Rodenburg JM, Hurst AC, Cullis AG | title = Transmission microscopy without lenses for objects of unlimited size | journal = Ultramicroscopy | volume = 107 | issue = 2–3 | pages = 227–231 | date = February 2007 | pmid = 16959428 | doi = 10.1016/j.ultramic.2006.07.007 }}</ref><ref name=":13">{{cite journal | vauthors = Stockmar M, Cloetens P, Zanette I, Enders B, Dierolf M, Pfeiffer F, Thibault P | title = Near-field ptychography: phase retrieval for inline holography using a structured illumination | language = En | journal = Scientific Reports | volume = 3 | issue = 1 | article-number = 1927 | date = 31 May 2013 | pmid = 23722622 | pmc = 3668322 | doi = 10.1038/srep01927 | bibcode = 2013NatSR...3.1927S }}</ref> although most implementations use a lens of some type, if only to condense radiation onto the specimen. The detector can measure high angles of scatter, which do not need to pass through a lens. The resolution is therefore only limited by the maximal angle of scatter that reaches the detector, and so avoids the effects of diffraction broadening due to a lens of small numerical aperture or aberrations within the lens. This is key in X-ray, electron and EUV ptychography, where conventional lenses are difficult and expensive to make.
=== Image phase === Ptychography solves for the phase induced by the real part of the refractive index of the specimen, as well as absorption (the imaginary part of the refractive index). This is crucial for seeing transparent specimens that do not have significant natural absorption contrast, for example biological cells (at visible light wavelengths),<ref name=":7">{{cite journal | vauthors = Marrison J, Räty L, Marriott P, O'Toole P | title = Ptychography--a label free, high-contrast imaging technique for live cells using quantitative phase information | language = En | journal = Scientific Reports | volume = 3 | issue = 1 | article-number = 2369 | date = 6 August 2013 | pmid = 23917865 | pmc = 3734479 | doi = 10.1038/srep02369 | bibcode = 2013NatSR...3.2369M }}</ref> thin high-resolution electron microscopy specimens,<ref>{{cite journal | vauthors = Yang H, MacLaren I, Jones L, Martinez GT, Simson M, Huth M, Ryll H, Soltau H, Sagawa R, Kondo Y, Ophus C, Ercius P, Jin L, Kovács A, Nellist PD | display-authors = 6 | title = Electron ptychographic phase imaging of light elements in crystalline materials using Wigner distribution deconvolution | journal = Ultramicroscopy | volume = 180 | pages = 173–179 | date = September 2017 | pmid = 28434783 | doi = 10.1016/j.ultramic.2017.02.006 | doi-access = free }}</ref> and almost all materials at hard X-ray wavelengths. In the latter case, the (linear) phase signal is also ideal for high-resolution X-ray ptychographic tomography.<ref name=":8">{{cite journal | vauthors = Dierolf M, Menzel A, Thibault P, Schneider P, Kewish CM, Wepf R, Bunk O, Pfeiffer F | display-authors = 6 | title = Ptychographic X-ray computed tomography at the nanoscale | language = En | journal = Nature | volume = 467 | issue = 7314 | pages = 436–439 | date = September 2010 | pmid = 20864997 | doi = 10.1038/nature09419 | s2cid = 2449015 | bibcode = 2010Natur.467..436D }}</ref> The strength and contrast of the phase signal also means that far fewer photon or electron counts are needed to make an image: this is very important in electron ptychography, where damage to the specimen is a major issue that must be avoided at all costs.<ref name="Yi Jiang et al.">{{cite journal | vauthors = Jiang Y, Chen Z, Han Y, Deb P, Gao H, Xie S, Purohit P, Tate MW, Park J, Gruner SM, Elser V, Muller DA | display-authors = 6 | title = Electron ptychography of 2D materials to deep sub-ångström resolution | journal = Nature | volume = 559 | issue = 7714 | pages = 343–349 | date = July 2018 | pmid = 30022131 | doi = 10.1038/s41586-018-0298-5 | s2cid = 49865457 | bibcode = 2018Natur.559..343J | arxiv = 1801.04630 }}</ref>
=== Tolerance to incoherence === Unlike holography, ptychography uses the object itself as an interferometer. It does not require a reference beam. Although holography can solve the image phase problem, it is very difficult to implement in the electron microscope, where the reference beam is extremely sensitive to magnetic interference or other sources of instability. This is why ptychography is not limited by the conventional "information limit" in conventional electron imaging.<ref>{{Cite journal |vauthors = Nellist P, McCallum B, Rodenburg JM |date=April 1995 |title=Resolution beyond the 'information limit' in transmission electron microscopy |journal=Nature |volume=374 |issue=6523 |pages=630–632 |doi=10.1038/374630a0 |bibcode=1995Natur.374..630N |s2cid=4330017}}</ref> Furthermore, ptychographical data is sufficiently diverse to remove the effects of partial coherence that would otherwise affect the reconstructed image.<ref name=":2" /><ref name=":9">{{cite journal | vauthors = Thibault P, Menzel A | title = Reconstructing state mixtures from diffraction measurements | journal = Nature | volume = 494 | issue = 7435 | pages = 68–71 | date = February 2013 | pmid = 23389541 | doi = 10.1038/nature11806 | s2cid = 4424305 | bibcode = 2013Natur.494...68T }}</ref>
=== Self-calibration === The ptychographical data set can be posed as a blind deconvolution problem.<ref name=":0" /><ref name=":1" /><ref>{{Cite journal |vauthors = McCallum BC, Rodenburg JM |date=1 February 1993 |title=Simultaneous reconstruction of object and aperture functions from multiple far-field intensity measurements |journal=JOSA A |volume=10 |issue=2 |pages=231–239 |doi=10.1364/JOSAA.10.000231 |bibcode=1993JOSAA..10..231M}}</ref> It has sufficient diversity to solve for both the moving functions (illumination and object), which appear symmetrically in the mathematics of the inversion process. This is now routinely done in any ptychographical experiment, even if the illumination optics have been previously well characterised. Diversity can also be used to solve retrospectively for errors in the offsets of the two functions, blurring in the scan, detector faults, like missing pixels, etc.
=== Inversion of multiple scattering === In conventional imaging, multiple scattering in a thick sample can seriously complicate, or even entirely invalidate, simple interpretation of an image. This is especially true in electron imaging (where multiple scattering is called "dynamical scattering"). Conversely, ptychography generates estimates of hundreds or thousands of exit waves, each of which contains different scattering information. This can be used to retrospectively remove multiple scattering effects.<ref name="Maiden2012" />
=== Robustness to noise === The number counts required for a ptychography experiment is the same as for a conventional image, even though the counts are distributed over very many diffraction patterns. This is because the dose fractionation theorem applies to ptychography. Maximum-likelihood methods can be employed to reduce the effects of Poisson noise.<ref>{{Cite journal |vauthors = Thibault P, Guizar-Sicairos M |date=2012 |title=Maximum-likelihood refinement for coherent diffractive imaging |journal=New Journal of Physics |volume=14 |issue=6 |article-number=063004 |doi=10.1088/1367-2630/14/6/063004 |bibcode=2012NJPh...14f3004T |doi-access=free}}</ref>
== Applications == Applications of ptychography are diverse because it can be used with any type of radiation that can be prepared as a quasi-monochromatic propagating wave.
Ptychographic imaging, along with advances in detectors and computing, has resulted in the development of X-ray microscopes.<ref>{{cite journal | vauthors = Chapman HN | title = Microscopy: A new phase for X-ray imaging | journal = Nature | volume = 467 | issue = 7314 | pages = 409–410 | date = September 2010 | pmid = 20864990 | doi = 10.1038/467409a | bibcode = 2010Natur.467..409C | s2cid = 205058970 }}</ref><ref name="Stanford">{{cite web |url=https://www-ssrl.slac.stanford.edu/wekergroup/ptychography |title=Ptychography |website=www6.slac.stanford.edu |access-date=29 July 2018}}</ref> Coherent beams are required in order to obtain far-field diffraction patterns with speckle patterns. Coherent X-ray beams can be produced by modern synchrotron radiation sources, free-electron lasers and high-harmonic sources. In terms of routine analysis, X-ray ptychotomography<ref name=":8" /> is today the most commonly used technique. It has been applied to many materials problems including, for example, the study of paint,<ref>{{cite journal | vauthors = Chen B, Guizar-Sicairos M, Xiong G, Shemilt L, Diaz A, Nutter J, Burdet N, Huo S, Mancuso J, Monteith A, Vergeer F, Burgess A, Robinson I | display-authors = 6 | title = Three-dimensional structure analysis and percolation properties of a barrier marine coating | language = En | journal = Scientific Reports | volume = 3 | issue = 1 | article-number = 1177 | date = 31 January 2013 | pmid = 23378910 | pmc = 3558722 | doi = 10.1038/srep01177 | bibcode = 2013NatSR...3.1177C }}</ref> imaging battery chemistry,<ref>{{Cite journal | vauthors = Shapiro DA, Yu YS, Tyliszczak T, Cabana J, Celestre R, Chao W, Kaznatcheev K, Kilcoyne AD, Maia F, Marchesini S, Meng YS | display-authors = 6 |date=7 September 2014 |title=Chemical composition mapping with nanometre resolution by soft X-ray microscopy |journal=Nature Photonics |volume=8 |issue=10 |pages=765–769 |doi=10.1038/nphoton.2014.207 |issn=1749-4885 |bibcode=2014NaPho...8..765S| s2cid = 35874797 }}</ref> imaging stacked layers of tandem solar cells,<ref>{{cite journal | vauthors = Pedersen EB, Angmo D, Dam HF, Thydén KT, Andersen TR, Skjønsfjell ET, Krebs FC, Holler M, Diaz A, Guizar-Sicairos M, Breiby DW, Andreasen JW | display-authors = 6 | title = Improving organic tandem solar cells based on water-processed nanoparticles by quantitative 3D nanoimaging | journal = Nanoscale | volume = 7 | issue = 32 | pages = 13765–13774 | date = August 2015 | pmid = 26220159 | doi = 10.1039/C5NR02824H | bibcode = 2015Nanos...713765P }}</ref> and the dynamics of fracture.<ref>{{Cite journal| vauthors = Bo Flyostad J, Skjnsfjell ET, GuizarSicairos M, Hydalsvik K, He J, Andreasen JW, Zhang Z, Breiby DW | display-authors = 6 |date=10 February 2015|title=Quantitative 3D X-ray Imaging of Densification, Delamination and Fracture in a Micro-Composite under Compression |journal=Advanced Engineering Materials |language=en |volume=17 |issue=4 |pages=545–553 |doi=10.1002/adem.201400443 | s2cid = 22356243 |issn=1438-1656 |url=https://backend.orbit.dtu.dk/ws/files/119895493/Quantitative_3D_X_ray_Imaging_of_Densification_postprint.pdf |type=Submitted manuscript}}</ref> In the X-ray regime, ptychography has also been used to obtain a 3D mapping of the disordered structure in the white ''Cyphochilus'' beetle,<ref>{{cite journal | vauthors = Wilts BD, Sheng X, Holler M, Diaz A, Guizar-Sicairos M, Raabe J, Hoppe R, Liu SH, Langford R, Onelli OD, Chen D, Torquato S, Steiner U, Schroer CG, Vignolini S, Sepe A | display-authors = 6 | title = Evolutionary-Optimized Photonic Network Structure in White Beetle Wing Scales | journal = Advanced Materials | volume = 30 | issue = 19 | article-number = e1702057 | date = May 2018 | pmid = 28640543 | doi = 10.1002/adma.201702057 | bibcode = 2018AdM....30Q2057W | doi-access = free }}</ref> and a 2D imaging of the domain structure in a bulk heterojunction for polymer solar cells.<ref>{{cite journal | vauthors = Patil N, Skjønsfjell ET, Van den Brande N, Chavez Panduro EA, Claessens R, Guizar-Sicairos M, Van Mele B, Breiby DW | display-authors = 6 | title = X-Ray Nanoscopy of a Bulk Heterojunction | journal = PLOS ONE | volume = 11 | issue = 7 | article-number = e0158345 | date = July 2016 | pmid = 27367796 | pmc = 4930208 | doi = 10.1371/journal.pone.0158345 | bibcode = 2016PLoSO..1158345P | doi-access = free }}</ref>
Visible-light ptychography has been used for imaging live biological cells and studying their growth, reproduction and motility.<ref>{{cite journal | vauthors = Kasprowicz R, Suman R, O'Toole P | title = Characterising live cell behaviour: Traditional label-free and quantitative phase imaging approaches | journal = The International Journal of Biochemistry & Cell Biology | volume = 84 | pages = 89–95 | date = March 2017 | pmid = 28111333 | doi = 10.1016/j.biocel.2017.01.004 | doi-access = free }}</ref> In its vectorial version, it can also be used for mapping quantitative optical properties of anisotropic materials such as biominerals<ref name=":18" /> or metasurfaces<ref>{{cite journal | vauthors = Song Q, Baroni A, Sawant R, Ni P, Brandli V, Chenot S, Vézian S, Damilano B, de Mierry P, Khadir S, Ferrand P, Genevet P | display-authors = 6 | title = Ptychography retrieval of fully polarized holograms from geometric-phase metasurfaces | journal = Nature Communications | volume = 11 | issue = 1 | article-number = 2651 | date = May 2020 | pmid = 32461637 | pmc = 7253437 | doi = 10.1038/s41467-020-16437-9 | bibcode = 2020NatCo..11.2651S | url = }}</ref>
Electron ptychography is uniquely (amongst other electron imaging modes) sensitive to both heavy and light atoms simultaneously. It has been used, for example, in the study of nanostructure drug-delivery mechanisms by looking at drug molecules stained by heavy atoms within light carbon nanotubes cages.<ref name=":17" /> With electron beams, shorter-wavelength, higher-energy electrons used for higher-resolution imaging can cause damage to the sample by ionising it and breaking bonds, but electron-beam ptychography has now produced record-breaking images of molybdenum disulphide with a resolution of 0.039 nm using a lower-energy electron beam and detectors that are able to detect single electrons, so atoms can be located with more precision.<ref name="Yi Jiang et al." /><ref>{{cite journal |url=https://physicsworld.com/a/electron-images-achieve-record-breaking-resolution/ |title=Electron images achieve record-breaking resolution |vauthors = Wogan T |journal=Physics World |volume=31 |issue=9 |page=5 |access-date=27 July 2018 |name-list-style=vanc |date=26 July 2018 |bibcode=2018PhyW...31i...5W |doi=10.1088/2058-7058/31/9/8|s2cid=125423491 |url-access=subscription }}</ref>
Ptychography has several applications in the semiconductor industry, including imaging their surfaces using EUV,<ref>{{cite journal | vauthors = Zhang B, Gardner DF, Seaberg MD, Shanblatt ER, Kapteyn HC, Murnane MM, Adams DE | title = High contrast 3D imaging of surfaces near the wavelength limit using tabletop EUV ptychography | journal = Ultramicroscopy | volume = 158 | pages = 98–104 | date = November 2015 | pmid = 26233823 | doi = 10.1016/j.ultramic.2015.07.006 | doi-access = free }}</ref> their 3D bulk structure using X-rays,<ref>{{cite journal | vauthors = Holler M, Guizar-Sicairos M, Tsai EH, Dinapoli R, Müller E, Bunk O, Raabe J, Aeppli G | display-authors = 6 | title = High-resolution non-destructive three-dimensional imaging of integrated circuits | language = En | journal = Nature | volume = 543 | issue = 7645 | pages = 402–406 | date = March 2017 | pmid = 28300088 | doi = 10.1038/nature21698 | s2cid = 4448836 | bibcode = 2017Natur.543..402H | url = http://infoscience.epfl.ch/record/227719 }}</ref> and mapping strain fields by Bragg ptychography, for example, in nanowires.<ref>{{cite journal | vauthors = Hill MO, Calvo-Almazan I, Allain M, Holt MV, Ulvestad A, Treu J, Koblmüller G, Huang C, Huang X, Yan H, Nazaretski E, Chu YS, Stephenson GB, Chamard V, Lauhon LJ, Hruszkewycz SO | display-authors = 6 | title = Measuring Three-Dimensional Strain and Structural Defects in a Single InGaAs Nanowire Using Coherent X-ray Multiangle Bragg Projection Ptychography | language = EN | journal = Nano Letters | volume = 18 | issue = 2 | pages = 811–819 | date = February 2018 | pmid = 29345956 | doi = 10.1021/acs.nanolett.7b04024 | url = https://hal.archives-ouvertes.fr/hal-01687989/file/Hill_maBPP_final.pdf | bibcode = 2018NanoL..18..811H }}</ref>
{{Gallery |title=Typical ptychographic images |width=220 |height=120 |align=center |footer=
|File:X-ray_diffraction_pattern_3clpro.jpg |alt2=X-ray diffraction pattern. |The diffraction pattern of a beam of x-rays passing through a stationary crystal. The dots are areas of constructive interference; the crystal's atomic structure can be worked out from the pattern. In ptychography, a sample (which does not need to be crystalline) is moved sequentially through the beam, creating a range of diffraction patterns.
|File:Ptychography experiment with visible light in a laboratory.jpg |alt3=Ptychography experiment with visible light in a laboratory. |A visible-light ptychograph of a USAF optical resolution target, made using a pinhole aperture in a piece of cardboard. In the graphs, the hue represents the phase, and the modulus represents the luminance. (a) shows a single image with complex diffraction detail. (b) shows the computer-processed version of (a). (c) shows the result from combined computer-processed diffraction data after the whole sample was scanned.<ref name=":4"/>
|File:X-ray ptychograph of a zone plate.jpg |alt4=X-ray ptychograph of part of a zone plate. |X-ray ptychography at a small-angle scattering beamline of a synchrotron. This x-ray ptychograph of a zone plate shows the luminosity data in image (a) and the phase data in image (b). Insets I, II and III from (b) are shown in (i), (j) and (k) respectively as processed in 2015; they show a clear improvement in resolution over the algorithms used in 2008 shown in (l), (m) and (n). }}Ptychography has also been applied to ultrashort pulse reconstruction, applied to different temporal characterization techniques, such as FROG or amplitude swing.<ref>{{Cite journal |last=Barbero |first=Cristian |last2=Sola |first2=Íñigo J. |last3=Alonso |first3=Benjamín |date=2025-10-01 |title=Ptychographic retrieval for complete ultrashort pulse amplitude swing reconstruction |url=https://www.sciencedirect.com/science/article/pii/S0030399225005304 |journal=Optics & Laser Technology |volume=188 |article-number=112939 |doi=10.1016/j.optlastec.2025.112939 |issn=0030-3992}}</ref>
== History ==
=== Origins in crystallography === The name "ptychography" was coined by Hegerl and Hoppe in 1970<ref>{{Cite journal |vauthors = Hegerl R, Hoppe W |date=November 1970 |title=Dynamische Theorie der Kristallstrukturanalyse durch Elektronenbeugung im inhomogenen Primärstrahlwellenfeld |journal=Berichte der Bunsengesellschaft für Physikalische Chemie |language=de |volume=74 |issue=11 |pages=1148–1154 |doi=10.1002/bbpc.19700741112 |issn=0005-9021}}</ref> to describe a solution to the crystallographic phase problem first suggested by Hoppe in 1969.<ref name="Hoppe_1969">{{cite journal |vauthors = Hoppe W |year=1969 |title=Beugung im inhomogenen Primärstrahlwellenfeld. I. Prinzip einer Phasenmessung von Elektronenbeungungsinterferenzen |language=de |journal=Acta Crystallographica Section A |volume=25 |issue=4 |pages=495–501 |bibcode=1969AcCrA..25..495H |doi=10.1107/S0567739469001045}}</ref> The idea required the specimen to be highly ordered (a crystal) and to be illuminated by a precisely engineered wave so that only two pairs of diffraction peaks interfere with one another at a time. A shift in the illumination changes the interference condition (by the Fourier shift theorem). The two measurements can be used to solve for the relative phase between the two diffraction peaks by breaking a complex-conjugate ambiguity that would otherwise exist.<ref name=":5">{{cite book | vauthors = Rodenburg JM | chapter = Ptychography and Related Diffractive Imaging Methods |date=2008 | title =Advances in Imaging and Electron Physics| volume = 150 |pages=87–184 |publisher=Elsevier |doi=10.1016/s1076-5670(07)00003-1 |isbn=978-0-12-374217-9 }}</ref> Although the idea encapsulates the underlying concept of interference by convolution (ptycho) and translational invariance, crystalline ptychography cannot be used for imaging of continuous objects, because very many (up to millions) of beams interfere at the same time, and so the phase differences are inseparable. Hoppe abandoned his concept of ptychography in 1973.
=== Algorithmic and Experimental Developments (1989–1998) ===
Decades later, Rodenburg and collaborators extended Hoppe's concept by developing non-iterative algorithms such as Wigner Distribution Deconvolution (WDD)<ref name=":2" /> and Single-Side-Band (SSB)<ref name=":2" /> SSB,<ref name=":6">{{Cite journal |vauthors = Rodenburg JM, McCallum BC, Nellist PD |date=March 1993|title=Experimental tests on double-resolution coherent imaging via STEM |journal=Ultramicroscopy |volume=48 |issue=3 |pages=304–314 |doi=10.1016/0304-3991(93)90105-7 |issn=0304-3991}}</ref> analysis. In the early to mid-1990s, Nellist, McCallum, and Rodenburg demonstrated the first experimental electron ptychography, achieving resolution beyond the traditional information limit.<ref>{{cite journal |last1=Nellist |first1=P. D. |last2=McCallum |first2=B. C. |last3=Rodenburg |first3=J. M. |year=1995 |title=Resolution beyond the 'information limit' in transmission electron microscopy |journal=Nature |volume=374 |issue=6523 |pages=630–632 |doi=10.1038/374630a0 |bibcode=1995Natur.374..630N }}</ref> Around the same time, Chapman implemented the first X-ray ptychography using the WDD formalism.<ref>{{cite journal |last1=Chapman |first1=H. N. |year=1996 |title=Phase-retrieval X-ray microscopy by Wigner-distribution deconvolution |journal=Ultramicroscopy |volume=66 |pages=153–172 |doi=10.1016/0304-3991(96)00084-8 |doi-broken-date=18 November 2025 |bibcode=1996UlMic..66..153C }}</ref> However, progress was limited at the time because ptychographic reconstruction generally required prior knowledge of the probe function and a scanning step size approximately half the target resolution.<ref>{{cite journal |last1=Rodenburg |first1=J. M. |last2=Bates |first2=R. H. T. |year=1992 |title=The theory of super-resolution electron microscopy via Wigner-distribution deconvolution |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |volume=339 |issue=1655 |pages=521–553 |doi=10.1098/rsta.1992.0050 |bibcode=1992RSPTA.339..521R }}</ref>
=== Extending crystallography to non-crystalline samples (1999) ===
A milestone came in 1999, when Miao and collaborators demonstrated the first experimental extension of crystallography to non-crystalline specimens using coherent diffraction and iterative phase retrieval.<ref name="Miao1999" /> This experiment replaced physical lenses with computational algorithms, enabling direct structure determination from coherent diffraction patterns of isolated non-crystalline samples. The 1999 CDI experiment unified the principles of microscopy and crystallography, marking the beginning of computational microscopy.<ref name="Miao2025" /> It also led to the establishment of an ongoing international conference series on coherent scattering and phase retrieval, first held in 2001.<ref>{{cite journal |last1=Spence |first1=J. C. H. |last2=Howells |first2=M. |last3=Marks |first3=L. D. |last4=Miao |first4=J. |year=2001 |title=Lensless imaging: a workshop on "new approaches to the phase problem for non-periodic objects" |journal=Ultramicroscopy |volume=90 |pages=1–6 |doi=10.1016/S0304-3991(01)00076-7 |doi-broken-date=18 November 2025 |bibcode=2001UlMic..90....1S }}</ref> Building on this computational foundation, research in the 2000s extended these concepts to scanning implementations, leading to the development of modern iterative ptychography.
=== Modern ptychography and general uptake (2004–present) ===
In 2004, Faulkner and Rodenburg introduced the Ptychographical Iterative Engine (PIE) algorithm,<ref name=":3" /> which enabled iterative phase retrieval from overlapping diffraction patterns and laid the foundation for later refinements such as the extended PIE (ePIE) algorithm.<ref name=":1" /> Widespread interest in ptychography followed the first experimental demonstration of iterative X-ray ptychography in 2007 at the Swiss Light Source (SLS), where overlapping diffraction data were used for lensless imaging.<ref name="Rodenburg2007" /> In 2008, Thibault and collaborators further advanced the field by applying the difference-map (DM) iterative inversion algorithm<ref name=":0" /> to ptychographic data, allowing the simultaneous reconstruction of both the probe and the object functions.<ref>{{cite journal |last1=Thibault |first1=Pierre |last2=Dierolf |first2=Martin |last3=Menzel |first3=Andreas |last4=Bunk |first4=Oliver |last5=David |first5=Christian |last6=Pfeiffer |first6=Franz |year=2008 |title=High-resolution scanning X-ray diffraction microscopy |journal=Science |volume=321 |issue=5887 |pages=379–382 |doi=10.1126/science.1158573 |pmid=18635805 |pmc=5304454 |bibcode=2008Sci...321..379T |url=http://infoscience.epfl.ch/record/125719 }}</ref> Later, mixed-state ptychography was introduced to account for partial coherence and experimental instabilities, thereby enhancing the accuracy and robustness of reconstructions.<ref name=":9" />
These developments triggered rapid progress at X-ray wavelengths and established modern iterative ptychography as a robust and quantitative imaging technique. By 2010, the SLS had developed X-ray ptychotomography,<ref name=":8" /> now a major application of the technique. Since 2010, several groups have developed the capabilities of ptychography to characterize and improve reflective<ref>{{cite journal | vauthors = Kewish CM, Thibault P, Dierolf M, Bunk O, Menzel A, Vila-Comamala J, Jefimovs K, Pfeiffer F | display-authors = 6 | title = Ptychographic characterization of the wavefield in the focus of reflective hard X-ray optics | journal = Ultramicroscopy | volume = 110 | issue = 4 | pages = 325–329 | date = March 2010 | pmid = 20116927 | doi = 10.1016/j.ultramic.2010.01.004 }}</ref> and refractive X-ray optics.<ref>{{Cite journal| vauthors = Schropp A, Boye P, Feldkamp JM, Hoppe R, Patommel J, Samberg D, Stephan S, Giewekemeyer K, Wilke RN, Salditt T, Gulden J | display-authors = 6 |date=March 2010 |title=Hard x-ray nanobeam characterization by coherent diffraction microscopy |journal=Applied Physics Letters |language=en |volume=96 |issue=9 |article-number=091102 |doi=10.1063/1.3332591 |issn=0003-6951 |bibcode=2010ApPhL..96i1102S| s2cid = 121069070 | url = https://www.dora.lib4ri.ch/psi/islandora/object/psi%3A15159 }}</ref><ref>{{Cite journal |vauthors = Guizar-Sicairos M, Narayanan S, Stein A, Metzler M, Sandy AR, Fienup JR, Evans-Lutterodt K |date=March 2011 |title=Measurement of hard x-ray lens wavefront aberrations using phase retrieval |journal=Applied Physics Letters |language=en |volume=98 |issue=11 |article-number=111108 |doi=10.1063/1.3558914 |bibcode=2011ApPhL..98k1108G |s2cid=120543549 |issn=0003-6951 |url=https://www.dora.lib4ri.ch/psi/islandora/object/psi%3A14057}}</ref> Bragg ptychography, for measuring strain in crystals, was demonstrated by Hruszkewycz in 2012.<ref name=":15">{{cite journal | vauthors = Hruszkewycz SO, Holt MV, Murray CE, Bruley J, Holt J, Tripathi A, Shpyrko OG, McNulty I, Highland MJ, Fuoss PH | display-authors = 6 | title = Quantitative nanoscale imaging of lattice distortions in epitaxial semiconductor heterostructures using nanofocused X-ray Bragg projection ptychography | language = EN | journal = Nano Letters | volume = 12 | issue = 10 | pages = 5148–5154 | date = October 2012 | pmid = 22998744 | doi = 10.1021/nl303201w | bibcode = 2012NanoL..12.5148H }}</ref> In 2012 it was also shown that electron ptychography could improve on the resolution of an electron lens by a factor of five,<ref name=":10">{{cite journal | vauthors = Humphry MJ, Kraus B, Hurst AC, Maiden AM, Rodenburg JM | title = Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging | journal = Nature Communications | volume = 3 | article-number = 730 | date = March 2012 | pmid = 22395621 | pmc = 3316878 | doi = 10.1038/ncomms1733 | bibcode = 2012NatCo...3..730H | number = 370 }}</ref> a method which was used in 2018 to provide the highest-resolution transmission image ever obtained<ref name="Yi Jiang et al." /> earning a Guinness world record,<ref>{{Cite web|title=Highest resolution microscope|url=https://www.guinnessworldrecords.com/world-records/highest-resolution-microscope|access-date=18 July 2021|website=Guinness World Records|language=en-GB}}</ref> and once again in 2021 to achieve an even better resolution.<ref name="Chen2021" /><ref>{{Cite web|last=Blaustein|first=Anna|title=See the Highest-Resolution Atomic Image Ever Captured|url=https://www.scientificamerican.com/article/see-the-highest-resolution-atomic-image-ever-captured/|access-date=18 July 2021|website=Scientific American|language=en}}</ref><ref>{{Cite web|title=Atomic Dodgeball|url=https://www.scientificamerican.com/index.cfm/_api/render/file/?method=inline&fileID=2E4CCFB6-8C40-4213-AC2868D1C863EEB5|access-date=27 August 2021|page=16|website=Scientific American|language=en}}</ref> Real-space light ptychography became available in a [https://www.phasefocus.com/ commercial system] for live-cell imaging in 2013.<ref name=":7" /> Fourier ptychography using iterative methods was also demonstrated by Zheng et al.<ref name=":14">{{cite journal | vauthors = Zheng G, Horstmeyer R, Yang C | title = Wide-field, high-resolution Fourier ptychographic microscopy | language = En | journal = Nature Photonics | volume = 7 | issue = 9 | pages = 739–745 | date = September 2013 | pmid = 25243016 | pmc = 4169052 | doi = 10.1038/nphoton.2013.187 | arxiv = 1405.0226 | bibcode = 2013NaPho...7..739Z }}</ref> in 2013, a field which is growing rapidly. The group of Margaret Murnane and Henry Kapteyn at JILA, CU Boulder demonstrated EUV reflection ptychographic imaging in 2014.<ref name=":16">{{Cite journal |vauthors = Seaberg MD, Zhang B, Gardner DF, Shanblatt ER, Murnane MM, Kapteyn HC, Adams DE |date=22 July 2014 |title=Tabletop nanometer extreme ultraviolet imaging in an extended reflection mode using coherent Fresnel ptychography |journal=Optica |language=EN |volume=1 |issue=1 |pages=39–44 |doi=10.1364/OPTICA.1.000039 |issn=2334-2536 |arxiv=1312.2049 |bibcode=2014Optic...1...39S |s2cid=10577107}}</ref>
== See also == *Coherent diffractive imaging (CDI) *Phase retrieval *Computational imaging *Fourier ptychography
== References == {{reflist}}
==External links== *[https://news.cornell.edu/stories/2021/05/cornell-researchers-see-atoms-record-resolution Cornell researchers see atoms at record resolution], cornell.edu at 20 May 2021
{{X-ray science}}
Category:Microscopy Category:Laboratory techniques in condensed matter physics