Ptychography

{{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 comprehensive solution to the so-called "phase problem". 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){{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}} and iterative methods.{{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}}{{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 }}{{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 }}{{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 }}{{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 }} The difference map algorithm developed by Thibault and co-workers is available in a downloadable package called [https://ptycho.github.io/ptypy/index.html PtyPy].{{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 | pages = 20160640 | date = December 2016 | pmid = 28119552 | pmc = 5247528 | doi = 10.1098/rspa.2016.0640 | bibcode = 2016RSPSA.47260640E }}

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.

File:Ptychography setup using a single aperture.png

This is conceptually the simplest ptychographical arrangement. 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.

File:Ptychography setup using a focussed probe.png

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),{{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 | pages = 12532 | date = August 2016 | pmid = 27561914 | pmc = 5007440 | doi = 10.1038/ncomms12532 | bibcode = 2016NatCo...712532Y }} 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.

File:Optical configuration for near-field ptychography.png

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. In the electron microscope, a lens can be used to map the magnified Fresnel image onto the detector.

{{See also|Fourier ptychography}}

File:Optical setup for Fourier ptychography.png

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.

File:Optical configuration for imaging ptychography.png

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.{{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 | pages = 14690 | date = October 2015 | pmid = 26423558 | pmc = 4589788 | doi = 10.1038/srep14690 | bibcode = 2015NatSR...514690M }}

File:Optical configuration for reflection or Bragg ptychography.png

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.{{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|pages=568|doi=10.1038/ncomms1569|pmid=22127064|issn=2041-1723|doi-access=free}}

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.{{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 }} 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,{{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}} 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.).{{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 }} Similarly to scalar ptychography, the probes used for the measurement can be jointly estimated together with the specimen.{{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 }} As a consequence, vectorial ptychography is also an elegant approach for quantitative imaging of coherent vectorial light beams (mixing wavefront and polarization features).{{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 }}

Ptychography can be undertaken without using any lenses at all,{{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 }}{{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 | pages = 1927 | date = 31 May 2013 | pmid = 23722622 | pmc = 3668322 | doi = 10.1038/srep01927 | bibcode = 2013NatSR...3E1927S }} 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.

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),{{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 | pages = 2369 | date = 6 August 2013 | pmid = 23917865 | pmc = 3734479 | doi = 10.1038/srep02369 | bibcode = 2013NatSR...3E2369M }} thin high-resolution electron microscopy specimens,{{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 }} 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.{{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 }} 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.{{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 }}

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.{{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}} Furthermore, ptychographical data is sufficiently diverse to remove the effects of partial coherence that would otherwise affect the reconstructed image.{{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 }}

The ptychographical data set can be posed as a blind deconvolution problem.{{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}} 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.

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.{{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 }}

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 dose fractionation applies to ptychography. Maximum-likelihood methods can be employed to reduce the effects of Poisson noise.{{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 |pages=063004 |doi=10.1088/1367-2630/14/6/063004 |bibcode=2012NJPh...14f3004T |doi-access=free}}

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.{{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 }}{{cite web |url=https://www-ssrl.slac.stanford.edu/wekergroup/ptychography |title=Ptychography |website=www6.slac.stanford.edu |access-date=29 July 2018}} 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 is today the most commonly used technique. It has been applied to many materials problems including, for example, the study of paint,{{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 | pages = 1177 | date = 31 January 2013 | pmid = 23378910 | pmc = 3558722 | doi = 10.1038/srep01177 | bibcode = 2013NatSR...3E1177C }} imaging battery chemistry,{{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 }} imaging stacked layers of tandem solar cells,{{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 }} and the dynamics of fracture.{{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}} In the X-ray regime, ptychography has also been used to obtain a 3D mapping of the disordered structure in the white Cyphochilus beetle,{{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 | pages = e1702057 | date = May 2018 | pmid = 28640543 | doi = 10.1002/adma.201702057 | doi-access = free }} and a 2D imaging of the domain structure in a bulk heterojunction for polymer solar cells.{{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 | pages = e0158345 | date = July 2016 | pmid = 27367796 | pmc = 4930208 | doi = 10.1371/journal.pone.0158345 | bibcode = 2016PLoSO..1158345P | doi-access = free }}

Visible-light ptychography has been used for imaging live biological cells and studying their growth, reproduction and motility.{{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 }} In its vectorial version, it can also be used for mapping quantitative optical properties of anisotropic materials such as biominerals or metasurfaces{{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 | pages = 2651 | date = May 2020 | pmid = 32461637 | pmc = 7253437 | doi = 10.1038/s41467-020-16437-9 | bibcode = 2020NatCo..11.2651S | url = }}

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. 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.{{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 |pages=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 }}

Ptychography has several applications in the semiconductor industry, including imaging their surfaces using EUV,{{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 }} their 3D bulk structure using X-rays,{{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 }} and mapping strain fields by Bragg ptychography, for example, in nanowires.{{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 }}

{{Gallery

|title=Typical ptychographic images

|width=220 |height=120

|align=center

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|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.{{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 | pages = 20160640 | date = December 2016 | pmid = 28119552 | pmc = 5247528 | doi = 10.1098/rspa.2016.0640 | bibcode = 2016RSPSA.47260640E }}

|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).

}}

The name "ptychography" was coined by Hegerl and Hoppe in 1970{{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}} to describe a solution to the crystallographic phase problem first suggested by Hoppe in 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}} 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.{{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=9780123742179 }} 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.

Between 1989 and 2007 Rodenburg and co-workers developed various inversion methods for the general imaging ptychographic phase problem, including Wigner-distribution deconvolution (WDD), SSB,{{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}} the "PIE" iterative method (a precursor of the "ePIE" algorithm), demonstrating proof-of-principles at various wavelengths.{{Cite journal |vauthors = Friedman SL, Rodenburg JM |date=1992 |title=Optical demonstration of a new principle of far-field microscopy |journal=Journal of Physics D: Applied Physics |language=en |volume=25 |issue=2 |pages=147–154 |doi=10.1088/0022-3727/25/2/003 |issn=0022-3727 |bibcode=1992JPhD...25..147F|s2cid=250816583 }}{{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 | pages = 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 }} Chapman used the WDD inversion method to demonstrate the first implementation of X-ray ptychography in 1996.{{Cite journal| vauthors = Chapman HN |date=December 1996 |title=Phase-retrieval X-ray microscopy by Wigner-distribution deconvolution |journal=Ultramicroscopy |volume=66 |issue=3–4 |pages=153–172 |doi=10.1016/s0304-3991(96)00084-8 |issn=0304-3991}} The smallness of computers and poor quality of detectors at that time may account for the fact that ptychography was not at first taken up by other workers.

Widespread interest in ptychography only started after the first demonstration of iterative phase-retrieval X-ray ptychography in 2007 at the Swiss Light Source (SLS). The conceptual framework of modern ptychography builds on coherent diffractive imaging (CDI), which was first experimentally demonstrated in 1999 by Miao and colleagues using synchrotron X-rays and iterative phase retrieval. Progress at X-ray wavelengths was then quick. By 2010, the SLS had developed X-ray ptychotomography, now a major application of the technique. Thibault, also working at the SLS, developed the difference-map (DM) iterative inversion algorithm and mixed-state ptychography. Since 2010, several groups have developed the capabilities of ptychography to characterize and improve reflective{{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 }} and refractive X-ray optics.{{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 |pages=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 }}{{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 |pages=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}} Bragg ptychography, for measuring strain in crystals, was demonstrated by Hruszkewycz in 2012.{{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 }} In 2012 it was also shown that electron ptychography could improve on the resolution of an electron lens by a factor of five,{{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 | pages = 730 | date = March 2012 | pmid = 22395621 | pmc = 3316878 | doi = 10.1038/ncomms1733 | bibcode = 2012NatCo...3..730H | number = 370 }} a method which was used in 2018 to provide the highest-resolution transmission image ever obtained earning a Guinness world record,{{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}} and once again in 2021 to achieve an even better resolution.{{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|s2cid=230435950}}{{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}}{{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|pages=16|website=Scientific American|language=en}} Real-space light ptychography became available in a [https://www.phasefocus.com/ commercial system] for live-cell imaging in 2013. Fourier ptychography using iterative methods was also demonstrated by Zheng et al.{{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 }} 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.{{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}}

• Jianwei (John) Miao – led the first experimental demonstration of coherent diffractive imaging (CDI) and contributed to its evolution into computational microscopy techniques such as atomic electron tomography (AET) and ptychographic AET (pAET).
• Coherent diffractive imaging (CDI)
• Phase retrieval
• Computational imaging
• Fourier ptychography

{{reflist}}

• [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