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Electron microscopes have achieved remarkable performance in spatial, temporal, and energy resolution, and their development continues to advance. Modern instruments increasingly incorporate automated procedures, multiple detectors, and aberration correctors to produce smaller and more precise electron probes. However, such correctors are often complex and costly.
A promising alternative approach is the control of electron beams using shaped laser fields via the ponderomotive interaction in free space. This concept can be interpreted as the implementation of an effective phase plate positioned along the electron beam trajectory, where the transversal phase profile imprinted onto the electrons is proportional to the transversal laser intensity profile [1]. Advances in laser beam shaping provide substantial flexibility in tailoring these intensity profiles. Appropriately designed laser-induced phase plates could therefore be used to correct aberrations [2], focus electron beams, or generate structured electron wavefunctions.
A key challenge of this approach lies in determining the appropriate phase modulation applied by spatial light modulators to produce the desired intensity distributions. This constitutes a highly nonlinear problem with many degrees of freedom. Although numerous algorithms have been proposed to address this task [3,4,5], our analysis shows that commonly used methods fail to generate smooth, speckle-free intensity profiles, which are essential for electron beam manipulation which does not introduce unintended distortions.
In this work, we investigate strategies for optimal phase mask computation tailored to our application and present a custom algorithm capable of producing suitable intensity profiles. Our approach combines a physics-based initial estimate with a subsequent refinement procedure. We choose Zernike polynomials as an example target intensity profiles as they could be used to correct electron aberrations in electron microscopes.
References:
[1] F. J. García De Abajo and A. Konečná, “Optical Modulation of Electron Beams in Free Space,” Phys. Rev. Lett., vol. 126, no. 12, p. 123901, Mar. 2021.
[2] M. C. C. Mihaila and M. Kozák, “Design for light-based spherical aberration correction of ultrafast electron microscopes,” Opt. Express, vol. 33, no. 1, pp. 758–775, Jan. 2025.
[3] R. W. Gerchberg, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik, vol. 35, pp. 237–246, 1972.
[4] P. Schroff, A. La Rooij, E. Haller, and S. Kuhr, “Accurate holographic light potentials using pixel crosstalk modelling,” Sci Rep, vol. 13, no. 1, p. 3252, Feb. 2023.
[5] K. Shi and G. Zhang, “Design and Application of Phase-Only Diffractive Optical Element Based on Non-Iterative Method,” Photonics, vol. 9, no. 11, p. 874, Nov. 2022.