Abstract

Recent progress in phase modulation using nanofabricated electron holograms has demonstrated how the phase of an electron beam can be controlled. In this paper, we apply this concept to the correction of spherical aberration in a scanning transmission electron microscope and demonstrate an improvement in spatial resolution. Such a holographic approach to spherical aberration correction is advantageous for its simplicity and cost-effectiveness.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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  2. O. Krivanek, N. Dellby, and A. Lupini, “Towards sub-Å electron beams,” Ultramicroscopy 78(1), 1–11 (1999).
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    [Crossref] [PubMed]
  7. B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
    [Crossref] [PubMed]
  8. V. Grillo, G. Carlo Gazzadi, E. Karimi, E. Mafakheri, R. W. Boyd, and S. Frabboni, “Highly efficient electron vortex beams generated by nanofabricated phase holograms,” Appl. Phys. Lett. 104(4), 043109 (2014).
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    [Crossref] [PubMed]
  11. N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature 494(7437), 331–335 (2013).
    [Crossref] [PubMed]
  12. V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of nondiffracting electron Bessel beams,” Phys. Rev. X 4(1), 011013 (2014).
  13. V. Grillo, A. H. Tavabi, F. Venturi, H. Larocque, R. Balboni, G. C. Gazzadi, S. Frabboni, P.-H. Lu, E. Mafakheri, F. Bouchard, R. E. Dunin-Borkowski, R. W. Boyd, M. P. J. Lavery, M. J. Padgett, and E. Karimi, “Measuring the orbital angular momentum spectrum of an electron beam,” Nat. Commun. 8, 15536 (2017).
    [Crossref] [PubMed]
  14. M. Linck, B. McMorran, J. Pierce, and P. Ercius, “Aberration-corrected stem by means of diffraction gratings,” Microscopy and Microanalysis 20(S3), 946–947 (2014).
    [Crossref]
  15. V. Grillo, J. S. Pierce, E. Karimi, T. R. Harvey, R. Balboni, G. C. Gazzadi, E. Mafakheri, F. Venturi, B. J. McMorran, S. Frabboni, and R. W. Boyd, “Structured electron beam illumination: a new control over the electron probe weird probes and new experiments,” Microscopy and Microanalysis 21(S3), 25–26 (2015).
    [Crossref]
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    [Crossref] [PubMed]
  18. O. L. Krivanek, J. Rusz, J.-C. Idrobo, T. J. Lovejoy, and N. Dellby, “Toward single mode, atomic size electron vortex beams,” Microscopy and Microanalysis 20(03), 832–836 (2014).
    [Crossref] [PubMed]
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    [Crossref]
  22. V. Grillo and F. Rossi, “Stem_cell: A software tool for electron microscopy. part 2 analysis of crystalline materials,” Ultramicroscopy 125, 112–129 (2013).
    [Crossref] [PubMed]
  23. E. Mafakheri, A. Tavabi, P.-H. Lu, R. Balboni, F. Venturi, C. Menozzi, G. Gazzadi, S. Frabboni, A. Sit, R. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
    [Crossref]
  24. J. Silcox, P. Xu, and R. F. Loane, “Resolution limits in annular dark field stem,” Ultramicroscopy 47(1–3), 173–186 (1992).
    [Crossref]
  25. P. Nellist and S. Pennycook, “Subangstrom resolution by underfocused incoherent transmission electron microscopy,” Phys. Rev. Lett. 81(19), 4156 (1998).
    [Crossref]
  26. V. Grillo and E. Carlino, “A novel method for focus assessment in atomic resolution STEM HAADF experiments,” Ultramicroscopy 106(7), 603–613 (2006).
    [Crossref]
  27. V. Grillo and E. Rotunno, “Stem_cell: A software tool for electron microscopy. part I simulations,” Ultramicroscopy 125, 97–111 (2013).
    [Crossref]
  28. J.M. Cowley, “Coherent interference in convergent-beam electron diffraction and shadow imaging,” Ultramicroscopy 4, 435–449 (1979).
    [Crossref]
  29. J.E. Barth and P. Kruit, “Addition of different contributions to the charged particle probe size,” Optik 101, 101–109 (1996).

2017 (3)

V. Grillo, A. H. Tavabi, F. Venturi, H. Larocque, R. Balboni, G. C. Gazzadi, S. Frabboni, P.-H. Lu, E. Mafakheri, F. Bouchard, R. E. Dunin-Borkowski, R. W. Boyd, M. P. J. Lavery, M. J. Padgett, and E. Karimi, “Measuring the orbital angular momentum spectrum of an electron beam,” Nat. Commun. 8, 15536 (2017).
[Crossref] [PubMed]

D. Pohl, S. Schneider, P. Zeiger, J. Rusz, P. Tiemeijer, S. Lazar, K. Nielsch, and B. Rellinghaus, “Atom size electron vortex beams with selectable orbital angular momentum,” Sci. Rep. 7, 934 (2017).
[Crossref] [PubMed]

E. Mafakheri, A. Tavabi, P.-H. Lu, R. Balboni, F. Venturi, C. Menozzi, G. Gazzadi, S. Frabboni, A. Sit, R. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
[Crossref]

2016 (2)

C. Boothroyd, A. Kovács, and K. Tillmann, “Fei titan g2 60–300 holo,” JLSRF 2, 44 (2016).
[Crossref]

R. Shiloh, R. Remez, and A. Arie, “Prospects for electron beam aberration correction using sculpted phase masks,” Ultramicroscopy 163, 69–74 (2016).
[Crossref] [PubMed]

2015 (1)

V. Grillo, J. S. Pierce, E. Karimi, T. R. Harvey, R. Balboni, G. C. Gazzadi, E. Mafakheri, F. Venturi, B. J. McMorran, S. Frabboni, and R. W. Boyd, “Structured electron beam illumination: a new control over the electron probe weird probes and new experiments,” Microscopy and Microanalysis 21(S3), 25–26 (2015).
[Crossref]

2014 (5)

V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of nondiffracting electron Bessel beams,” Phys. Rev. X 4(1), 011013 (2014).

O. L. Krivanek, J. Rusz, J.-C. Idrobo, T. J. Lovejoy, and N. Dellby, “Toward single mode, atomic size electron vortex beams,” Microscopy and Microanalysis 20(03), 832–836 (2014).
[Crossref] [PubMed]

M. Linck, B. McMorran, J. Pierce, and P. Ercius, “Aberration-corrected stem by means of diffraction gratings,” Microscopy and Microanalysis 20(S3), 946–947 (2014).
[Crossref]

R. Shiloh, Y. Lereah, Y. Lilach, and A. Arie, “Sculpturing the electron wave function using nanoscale phase masks,” Ultramicroscopy 144, 26–31 (2014).
[Crossref] [PubMed]

V. Grillo, G. Carlo Gazzadi, E. Karimi, E. Mafakheri, R. W. Boyd, and S. Frabboni, “Highly efficient electron vortex beams generated by nanofabricated phase holograms,” Appl. Phys. Lett. 104(4), 043109 (2014).
[Crossref]

2013 (3)

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature 494(7437), 331–335 (2013).
[Crossref] [PubMed]

V. Grillo and F. Rossi, “Stem_cell: A software tool for electron microscopy. part 2 analysis of crystalline materials,” Ultramicroscopy 125, 112–129 (2013).
[Crossref] [PubMed]

V. Grillo and E. Rotunno, “Stem_cell: A software tool for electron microscopy. part I simulations,” Ultramicroscopy 125, 97–111 (2013).
[Crossref]

2011 (1)

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref] [PubMed]

2010 (2)

M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature 464(7289), 737–739 (2010).
[Crossref] [PubMed]

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
[Crossref] [PubMed]

2009 (1)

P. Hawkes, “Aberration correction past and present,” Phil. Trans. R. Soc. A 367(1903), 3637–3664 (2009).
[Crossref] [PubMed]

2006 (1)

V. Grillo and E. Carlino, “A novel method for focus assessment in atomic resolution STEM HAADF experiments,” Ultramicroscopy 106(7), 603–613 (2006).
[Crossref]

1999 (1)

O. Krivanek, N. Dellby, and A. Lupini, “Towards sub-Å electron beams,” Ultramicroscopy 78(1), 1–11 (1999).
[Crossref]

1998 (2)

M. Haider, S. Uhlemann, E. Schwan, H. Rose, B. Kabius, and K. Urban, “Electron microscopy image enhanced,” Nature 392(6678), 768 (1998).
[Crossref]

P. Nellist and S. Pennycook, “Subangstrom resolution by underfocused incoherent transmission electron microscopy,” Phys. Rev. Lett. 81(19), 4156 (1998).
[Crossref]

1996 (1)

J.E. Barth and P. Kruit, “Addition of different contributions to the charged particle probe size,” Optik 101, 101–109 (1996).

1992 (1)

J. Silcox, P. Xu, and R. F. Loane, “Resolution limits in annular dark field stem,” Ultramicroscopy 47(1–3), 173–186 (1992).
[Crossref]

1979 (1)

J.M. Cowley, “Coherent interference in convergent-beam electron diffraction and shadow imaging,” Ultramicroscopy 4, 435–449 (1979).
[Crossref]

1936 (1)

O. Scherzer, “Über einige fehler von elektronenlinsen,” Zeitschrift für Physik A Hadrons and Nuclei 101(9), 593–603 (1936).

Agrawal, A.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref] [PubMed]

Allen, L.

L. Allen, R. Angel, J. D. Mangus, G. A. Rodney, R. R. Shannon, and C. P. Spoelhof, “The Hubble space telescope optical systems failure report,” NASA Report (1990).

Anderson, I. M.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref] [PubMed]

Angel, R.

L. Allen, R. Angel, J. D. Mangus, G. A. Rodney, R. R. Shannon, and C. P. Spoelhof, “The Hubble space telescope optical systems failure report,” NASA Report (1990).

Arie, A.

R. Shiloh, R. Remez, and A. Arie, “Prospects for electron beam aberration correction using sculpted phase masks,” Ultramicroscopy 163, 69–74 (2016).
[Crossref] [PubMed]

R. Shiloh, Y. Lereah, Y. Lilach, and A. Arie, “Sculpturing the electron wave function using nanoscale phase masks,” Ultramicroscopy 144, 26–31 (2014).
[Crossref] [PubMed]

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature 494(7437), 331–335 (2013).
[Crossref] [PubMed]

Balboni, R.

V. Grillo, A. H. Tavabi, F. Venturi, H. Larocque, R. Balboni, G. C. Gazzadi, S. Frabboni, P.-H. Lu, E. Mafakheri, F. Bouchard, R. E. Dunin-Borkowski, R. W. Boyd, M. P. J. Lavery, M. J. Padgett, and E. Karimi, “Measuring the orbital angular momentum spectrum of an electron beam,” Nat. Commun. 8, 15536 (2017).
[Crossref] [PubMed]

E. Mafakheri, A. Tavabi, P.-H. Lu, R. Balboni, F. Venturi, C. Menozzi, G. Gazzadi, S. Frabboni, A. Sit, R. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
[Crossref]

V. Grillo, J. S. Pierce, E. Karimi, T. R. Harvey, R. Balboni, G. C. Gazzadi, E. Mafakheri, F. Venturi, B. J. McMorran, S. Frabboni, and R. W. Boyd, “Structured electron beam illumination: a new control over the electron probe weird probes and new experiments,” Microscopy and Microanalysis 21(S3), 25–26 (2015).
[Crossref]

V. Grillo, E. Karimi, R. Balboni, G. C. Gazzadi, S. Frabboni, E. Mafakheri, and R. W. Boyd, “Toward holographic approach to spherical aberration correction in stem,” IMC conferenceIT-1-P-6140 (2014).

Barth, J.E.

J.E. Barth and P. Kruit, “Addition of different contributions to the charged particle probe size,” Optik 101, 101–109 (1996).

Boothroyd, C.

C. Boothroyd, A. Kovács, and K. Tillmann, “Fei titan g2 60–300 holo,” JLSRF 2, 44 (2016).
[Crossref]

Bouchard, F.

V. Grillo, A. H. Tavabi, F. Venturi, H. Larocque, R. Balboni, G. C. Gazzadi, S. Frabboni, P.-H. Lu, E. Mafakheri, F. Bouchard, R. E. Dunin-Borkowski, R. W. Boyd, M. P. J. Lavery, M. J. Padgett, and E. Karimi, “Measuring the orbital angular momentum spectrum of an electron beam,” Nat. Commun. 8, 15536 (2017).
[Crossref] [PubMed]

Boyd, R. W.

V. Grillo, A. H. Tavabi, F. Venturi, H. Larocque, R. Balboni, G. C. Gazzadi, S. Frabboni, P.-H. Lu, E. Mafakheri, F. Bouchard, R. E. Dunin-Borkowski, R. W. Boyd, M. P. J. Lavery, M. J. Padgett, and E. Karimi, “Measuring the orbital angular momentum spectrum of an electron beam,” Nat. Commun. 8, 15536 (2017).
[Crossref] [PubMed]

V. Grillo, J. S. Pierce, E. Karimi, T. R. Harvey, R. Balboni, G. C. Gazzadi, E. Mafakheri, F. Venturi, B. J. McMorran, S. Frabboni, and R. W. Boyd, “Structured electron beam illumination: a new control over the electron probe weird probes and new experiments,” Microscopy and Microanalysis 21(S3), 25–26 (2015).
[Crossref]

V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of nondiffracting electron Bessel beams,” Phys. Rev. X 4(1), 011013 (2014).

V. Grillo, G. Carlo Gazzadi, E. Karimi, E. Mafakheri, R. W. Boyd, and S. Frabboni, “Highly efficient electron vortex beams generated by nanofabricated phase holograms,” Appl. Phys. Lett. 104(4), 043109 (2014).
[Crossref]

V. Grillo, E. Karimi, R. Balboni, G. C. Gazzadi, S. Frabboni, E. Mafakheri, and R. W. Boyd, “Toward holographic approach to spherical aberration correction in stem,” IMC conferenceIT-1-P-6140 (2014).

Carlino, E.

V. Grillo and E. Carlino, “A novel method for focus assessment in atomic resolution STEM HAADF experiments,” Ultramicroscopy 106(7), 603–613 (2006).
[Crossref]

Carlo Gazzadi, G.

V. Grillo, G. Carlo Gazzadi, E. Karimi, E. Mafakheri, R. W. Boyd, and S. Frabboni, “Highly efficient electron vortex beams generated by nanofabricated phase holograms,” Appl. Phys. Lett. 104(4), 043109 (2014).
[Crossref]

Cowley, J.M.

J.M. Cowley, “Coherent interference in convergent-beam electron diffraction and shadow imaging,” Ultramicroscopy 4, 435–449 (1979).
[Crossref]

Dellby, N.

O. L. Krivanek, J. Rusz, J.-C. Idrobo, T. J. Lovejoy, and N. Dellby, “Toward single mode, atomic size electron vortex beams,” Microscopy and Microanalysis 20(03), 832–836 (2014).
[Crossref] [PubMed]

O. Krivanek, N. Dellby, and A. Lupini, “Towards sub-Å electron beams,” Ultramicroscopy 78(1), 1–11 (1999).
[Crossref]

Dennis, M. R.

V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of nondiffracting electron Bessel beams,” Phys. Rev. X 4(1), 011013 (2014).

Dunin-Borkowski, R.

E. Mafakheri, A. Tavabi, P.-H. Lu, R. Balboni, F. Venturi, C. Menozzi, G. Gazzadi, S. Frabboni, A. Sit, R. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
[Crossref]

Dunin-Borkowski, R. E.

V. Grillo, A. H. Tavabi, F. Venturi, H. Larocque, R. Balboni, G. C. Gazzadi, S. Frabboni, P.-H. Lu, E. Mafakheri, F. Bouchard, R. E. Dunin-Borkowski, R. W. Boyd, M. P. J. Lavery, M. J. Padgett, and E. Karimi, “Measuring the orbital angular momentum spectrum of an electron beam,” Nat. Commun. 8, 15536 (2017).
[Crossref] [PubMed]

Ercius, P.

M. Linck, B. McMorran, J. Pierce, and P. Ercius, “Aberration-corrected stem by means of diffraction gratings,” Microscopy and Microanalysis 20(S3), 946–947 (2014).
[Crossref]

Frabboni, S.

V. Grillo, A. H. Tavabi, F. Venturi, H. Larocque, R. Balboni, G. C. Gazzadi, S. Frabboni, P.-H. Lu, E. Mafakheri, F. Bouchard, R. E. Dunin-Borkowski, R. W. Boyd, M. P. J. Lavery, M. J. Padgett, and E. Karimi, “Measuring the orbital angular momentum spectrum of an electron beam,” Nat. Commun. 8, 15536 (2017).
[Crossref] [PubMed]

E. Mafakheri, A. Tavabi, P.-H. Lu, R. Balboni, F. Venturi, C. Menozzi, G. Gazzadi, S. Frabboni, A. Sit, R. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
[Crossref]

V. Grillo, J. S. Pierce, E. Karimi, T. R. Harvey, R. Balboni, G. C. Gazzadi, E. Mafakheri, F. Venturi, B. J. McMorran, S. Frabboni, and R. W. Boyd, “Structured electron beam illumination: a new control over the electron probe weird probes and new experiments,” Microscopy and Microanalysis 21(S3), 25–26 (2015).
[Crossref]

V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of nondiffracting electron Bessel beams,” Phys. Rev. X 4(1), 011013 (2014).

V. Grillo, G. Carlo Gazzadi, E. Karimi, E. Mafakheri, R. W. Boyd, and S. Frabboni, “Highly efficient electron vortex beams generated by nanofabricated phase holograms,” Appl. Phys. Lett. 104(4), 043109 (2014).
[Crossref]

V. Grillo, E. Karimi, R. Balboni, G. C. Gazzadi, S. Frabboni, E. Mafakheri, and R. W. Boyd, “Toward holographic approach to spherical aberration correction in stem,” IMC conferenceIT-1-P-6140 (2014).

Gazzadi, G.

E. Mafakheri, A. Tavabi, P.-H. Lu, R. Balboni, F. Venturi, C. Menozzi, G. Gazzadi, S. Frabboni, A. Sit, R. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
[Crossref]

Gazzadi, G. C.

V. Grillo, A. H. Tavabi, F. Venturi, H. Larocque, R. Balboni, G. C. Gazzadi, S. Frabboni, P.-H. Lu, E. Mafakheri, F. Bouchard, R. E. Dunin-Borkowski, R. W. Boyd, M. P. J. Lavery, M. J. Padgett, and E. Karimi, “Measuring the orbital angular momentum spectrum of an electron beam,” Nat. Commun. 8, 15536 (2017).
[Crossref] [PubMed]

V. Grillo, J. S. Pierce, E. Karimi, T. R. Harvey, R. Balboni, G. C. Gazzadi, E. Mafakheri, F. Venturi, B. J. McMorran, S. Frabboni, and R. W. Boyd, “Structured electron beam illumination: a new control over the electron probe weird probes and new experiments,” Microscopy and Microanalysis 21(S3), 25–26 (2015).
[Crossref]

V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of nondiffracting electron Bessel beams,” Phys. Rev. X 4(1), 011013 (2014).

V. Grillo, E. Karimi, R. Balboni, G. C. Gazzadi, S. Frabboni, E. Mafakheri, and R. W. Boyd, “Toward holographic approach to spherical aberration correction in stem,” IMC conferenceIT-1-P-6140 (2014).

Gover, A.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature 494(7437), 331–335 (2013).
[Crossref] [PubMed]

Grillo, V.

V. Grillo, A. H. Tavabi, F. Venturi, H. Larocque, R. Balboni, G. C. Gazzadi, S. Frabboni, P.-H. Lu, E. Mafakheri, F. Bouchard, R. E. Dunin-Borkowski, R. W. Boyd, M. P. J. Lavery, M. J. Padgett, and E. Karimi, “Measuring the orbital angular momentum spectrum of an electron beam,” Nat. Commun. 8, 15536 (2017).
[Crossref] [PubMed]

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Appl. Phys. Lett. (2)

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Figures (5)

Fig. 1
Fig. 1 a) Electron microscope configuration used to isolate a single holographically-corrected electron beam. b) Scanning electron micrograph of a hologram that was placed in the C2 aperture plane. c) Magnified image of b). d) Images obtained and formed by selecting a single diffraction order from the grating. These images are analogous to “dark field” and “bright field” images obtained in diffraction contrast TEM. As further elaborated in the Appendix, the visible rings are points of phase discontinuity that arise from the hologram’s piecewise phase correction.
Fig. 2
Fig. 2 a) Raw STEM image recoded using a single spot n = 1 (main image) and a filtered image of the lattice (inset). b) Fourier transform of the experimental STEM image, showing a faint but visible spot corresponding to the (4,0,0) periodicity in Si. c) Intensities of the peaks in the Fourier transform plotted as a function of spatial frequency. An example of a simulated CTF with a “similar” trend is also shown. We did not try to fit the (2,0,0) intensity, as it should be almost completely inhibited.
Fig. 3
Fig. 3 a) Raw STEM image recorded under Scherzer working conditions and a filtered image of the lattice (inset). b) Fourier transform of the experimental STEM image, only showing higher order periodicities corresponding to (111), (200), and (022). Note that the 200 reflection is a forbidden reflection, that is, it can only be excited through multiple scattering.
Fig. 4
Fig. 4 Simulated output obtained from an effective discrete step hologram. (a) Discrete step hologram used in the simulations. The first discrete jump in the pattern of the hologram is highlighted in red. (b) Diffraction pattern obtained from the hologram. The first diffraction order that is selected by the microscope’s aperture is circled in blue. (c) The demodulated image of the aperture plane which is characterized by dark lines corresponding to the discontinuities of the hologram. A dark ring in the profile of the beam resulting from the discrete phase jumps of the hologram is highlighted in red.
Fig. 5
Fig. 5 Ronchigram of amorphous carbon using the n = 1 beam. The beam clearly shows a “circle of infinite magnification” whose presence is caused by a substantial spherical aberration.

Equations (4)

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χ = π Δ f λ k 2 + π 2 λ 3 C s k 4 + π A 2 λ k 2 cos ( 2 ( θ θ 2 A ) ) + 2 π 3 λ 2 A 3 k 3 cos ( 3 ( θ θ 3 A ) ) + 2 π 3 λ 2 B 2 k 3 cos ( θ θ B ) + ,
φ ( ρ , ϕ ) = φ 0 F [ Q ρ sin ( ϕ ) 2 π λ ( 1 4 C s ρ 4 1 2 Δ f ρ 2 ) ] ,
CTF ( k ) = A ( k ) * A ( k ) ,
O ( k ) = i δ ( k k i ) exp ( k 2 σ 2 )

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