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  • Open Access

Crossed ponderomotive lenses for spherical aberration correction in electron optics

Yuuki Uesugi* and Yuichi Kozawa

  • Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Katahira 2-1-1, Aoba-ku, Sendai 980-8577, Japan

  • *Contact author: uesugi@tohoku.ac.jp
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Phys. Rev. A 112, 013507 – Published 7 July, 2025

DOI: https://doi.org/10.1103/jk8d-dn5z

Abstract

This article evaluates the lens characteristics of a nonrotationally symmetric electron lens based on a ponderomotive potential, i.e., a ponderomotive lens, formed by intersecting one or more optical beams perpendicular to an electron beam. Based on geometric optics, design formulas are derived for the focal length and general spherical aberration coefficients of specifically crossed ponderomotive lenses. Numerical calculations demonstrate that a pair of these crossed ponderomotive lenses can effectively correct spherical aberration in the objective lens of an electron microscope. Unlike rotationally symmetric ponderomotive lenses, which require the optical beam to be coaxially aligned with the electron beam, the crossed ponderomotive lens avoids the need to place optical mirrors and lenses directly on the beam axis. Thus, it offers practical advantages in designing and building electron optical instruments and contributes to system miniaturization. With lens properties similar to multipole lenses, the proposed crossed ponderomotive lens is expected to facilitate diverse developments in electron optical systems incorporating ponderomotive potentials.

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References (50)

  1. O. Scherzer, Uber einige Fehler von Elektronenlinsen, Z. Phys. 101, 593 (1936).
  2. T. Kawasaki, T. Ishida, Y. Takai, Y. Ogawa, M. Tomita, T. Matsutani, T. Kodama, and T. Ikuta, Development of electrostatic spherical aberration corrector using annular and circular electrodes, Surf. Interface Anal. 48, 1160 (2016).
  3. M. Linck, P. A. Ercius, J. S. Pierce, and B. J. McMorran, Aberration corrected STEM by means of diffraction gratings, Ultramicroscopy 182, 36 (2017).
  4. A. Konečná and F. J. García de Abajo, Electron beam aberration correction using optical near fields, Phys. Rev. Lett. 125, 030801 (2020).
  5. M. T. Schreiber, C. Cassidy, M. Saidani, and M. Wolf, Electromagnetic lensing using the Aharonov-Bohm effect, New J. Phys. 26, 043012 (2024).
  6. E. D. Courant, M. S. Livingston, and H. S. Snyder, The strong-focusing synchrotron—A new high energy accelerator, Phys. Rev. 88, 1190 (1952).
  7. O. Scherzer, Sphärische und chromatische Korrektur von Elektronenlinsen, Optik (Stuttgart) 2, 114 (1947).
  8. P. W. Hawkes, Aberration correction past and present, Trans. R. Soc. A 367, 3637 (2009).
  9. M. Haider, G. Braunshausen, and E. Schwan, Correction of the spherical aberration of a 200kV TEM by means of a hexapole corrector, Optik (Stuttgart) 99, 167 (1995).
  10. O. L. Krivanek, N. Dellby, and A. R. Lupini, Towards sub-Å electron beams, Ultramicroscopy 78, 1 (1999).
  11. P. E. Batson, N. Dellby, and O. L. Krivanek, Sub-ångstrom resolution using aberration corrected electron optics, Nature (London) 418, 617 (2002).
  12. P. D. Nellist, M. F. Chisholm, N. Dellby, O. L. Krivanek, M. F. Murfitt, Z. S. Szilagyi, A. R. Lupini, A. Borisevich, W. H. Sides, Jr., and S. J. Pennycook, Direct sub-angstrom imaging of a crystal lattice, Science 305, 1741 (2004).
  13. P. L. Kapitza and P. A. M. Dirac, The reflection of electrons from standing light waves, Math. Proc. Cambridge Philos. Soc. 29, 297 (1933).
  14. B. Herts, Charged particles in a non-uniform radio-frequency field, Nature (London) 30, 1187 (1957).
  15. H. Schwarz, H. A. Tourtellotte, and W. W. Gaertner, Direct observation of nonlinear Thomson scattering, Phys. Lett. 19, 202 (1965).
  16. L. S. Bartell, H. B. Thompson, and R. R. Roskos, Observation of stimulated Compton scattering of electrons by laser beam, Phys. Rev. Lett. 14, 851 (1965).
  17. M. V. Fedorov, The Kapitza-Dirac effect in a strong radiation field, Sov. Phys. JETP 25, 952 (1967).
  18. Y. Takeda and I. Matsui, Electron reflection by standing wave of giant pulse laser, J. Phys. Soc. Jpn. 25, 1202 (1968).
  19. L. S. Bartell, R. R. Roskos, and H. B. Thompson, Reflection of electrons by standing light waves: Experimental study, Phys. Rev. 166, 1494 (1968).
  20. H. Ezawa and H. Namaizawa, Theory of the Kapitza-Dirac effect, J. Phys. Soc. Jpn. 26, 458 (1969).
  21. Y. W. Chan and W. L. Tsui, Classical theory of scattering of an electron beam by a laser standing wave, Phys. Rev. A 20, 294 (1979).
  22. P. H. Bucksbaum, D. W. Schumacher, and M. Bashkansky, High-intensity Kapitza-Dirac effect, Phys. Rev. Lett. 61, 1182 (1988).
  23. M. V. Fedorov, S. P. Goreslavsky, and V. S. Letokhov, Relativistic ponderomotive effects in the field of a standing light wave, Phys. Rev. E 55, 1015 (1997).
  24. K. Aflatooni, D. L. Freimund, and H. Batelaan, Observation of the Kapitza-Dirac effect, Nature (London) 413, 142 (2001).
  25. D. L. Freimund and H. Batelaan, Bragg scattering of free electrons using the Kapitza-Dirac effect, Phys. Rev. Lett. 89, 283602 (2002).
  26. H. Batelaan, Colloquium: Illuminating the Kapitza-Dirac effect with electron matter optics, Rev. Mod. Phys. 79, 929 (2007).
  27. M. Kozák, T. Eckstein, N. Schönenberger, and P. Hommelhoff, Inelastic ponderomotive scattering of electrons at a high-intensity optical travelling wave in vacuum, Nat. Phys. 14, 121 (2018).
  28. O. Schwartz, J. J. Axelrod, S. L. Campbell, C. Turnbaugh, R. M. Glaeser, and H. Müller, Laser phase plate for transmission electron microscopy, Nat. Methods 16, 1016 (2019).
  29. J. J. Axelrod, S. L. Campbell, O. Schwartz, C. Turnbaugh, R. M. Glaeser, and H. Müller, Observation of the relativistic reversal of the ponderomotive potential, Phys. Rev. Lett. 124, 174801 (2020).
  30. V. G. Minogin, Electron beam deflection, focusing, and collimation by a femtosecond laser lens, Quantum Electron. 39, 1095 (2009).
  31. A. Szczepkowicz, Application of transfer matrix and transfer function analysis to grating-type dielectric laser accelerators: Ponderomotive focusing of electrons, Phys. Rev. Accel. Beams 20, 081302 (2017).
  32. F. J. García de Abajo and A. Konečná, Optical modulation of electron beams in free space, Phys. Rev. Lett. 126, 123901 (2021).
  33. Y. Uesugi, Y. Kozawa, and S. Sato, Electron round lenses with negative spherical aberration by a tightly focused cylindrically polarized light beam, Phys. Rev. Appl. 16, L011002 (2021).
  34. M. C. C. Mihaila and M. Kozák, Design for light-based spherical aberration correction of ultrafast electron microscopes, Opt. Express 33, 758 (2025).
  35. M. C. C. Mihaila, P. Weber, M. Schneller, L. Grandits, S. Nimmrichter, and T. Juffmann, Transverse electron-beam shaping with light, Phys. Rev. X 12, 031043 (2022).
  36. M. C. C. Mihaila, P. Koutenský, K. Moriová, and M. Kozák, Light-based electron aberration corrector, arXiv:2504.18661.
  37. Y. Uesugi, Y. Kozawa, and S. Sato, Properties of electron lenses produced by ponderomotive potential with Bessel and Laguerre–Gaussian beams, J. Opt. 24, 054013 (2022).
  38. Y. Uesugi, The properties of ponderomotive lenses, Adv. Imag. Electron Phys. 228, 1 (2023).
  39. P. W. Hawkes and E. Kasper, Principles of Electron Optics, 2nd ed. (Academic Press, London, 2018), Vol. 1.
  40. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University Press, New York, 2007).
  41. S. Yuan and J. Sasian, Aberrations of anamorphic optical systems. I: The first-order foundation and method for deriving the anamorphic primary aberration coefficients, Appl. Opt. 48, 2574 (2009).
  42. C. Turnbaugh, J. J. Axelrod, S. L. Campbell, J. Y. Dioquino, P. N. Petrov, J. Remis, O. Schwartz, Z. Yu, Y. Cheng, R. M. Glaeser, and H. Müller, High-power near-concentric Fabry–Perot cavity for phase contrast electron microscopy, Rev. Sci. Instrum. 92, 053005 (2021).
  43. Y. Uesugi, Y. Hosaka, Y. Honda, A. Kosuge, K. Sakaue, T. Omori, T. Takahashi, J. Urakawa, and M. Washio, Feedback-free optical cavity with self-resonating mechanism, APL Photon. 1, 026103 (2016).
  44. S. Yuan and J. Sasian, Aberrations of anamorphic optical systems. II. Primary aberration theory for cylindrical anamorphic systems, Appl. Opt. 48, 2836 (2009).
  45. S. Yuan and J. Sasian, Orthonormal aberration polynomials for anamorphic optical imaging systems, Appl. Opt. 49, 6802 (2010).
  46. K. Mori, Y. Hayasaki, and K. Araki, Fundamental ray aberration analysis: Extension of ray matrix analysis to high-order aberrations, Appl. Opt. 59, 4466 (2020).
  47. K. Mori, Y. Hayasaki, and K. Araki, Fundamental-ray aberration analysis of off-axial optical systems: Analytical formulae of first-order aberrations, Appl. Opt. 60, 9012 (2021).
  48. D. Zhuridov, Optical systems with translational invariance, Appl. Opt. 63, 1506 (2024).
  49. D. Zhuridov, Anamorphic catadioptric telescope and generalized spherical aberration, arXiv:2401.10762.
  50. Y. Uesugi and Y. Kozawa, Dataset for “Crossed ponderomotive lenses for spherical aberration correction in electron optics,” figshare, Dataset, https://doi.org/10.6084/m9.figshare.29345843.

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Phys. Rev. A 112, 013507– Published 7 July, 2025
  • Received 9 April 2025
  • Accepted 17 June 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International 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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