Efficient methods for wave propagation in electron microscopy

Abstract

Accurate wave-optical simulation in electron microscopy is severely constrained by the extreme sampling requirements imposed by short wavelengths and relatively large convergence angles. Conventional implementations of the angular spectrum method (ASM) rapidly become computationally intractable, often exceeding realistic memory and time limits. We present two numerical approaches -- the scaling angular spectrum method (SASM) and the no-lensing angular spectrum method (NLASM) -- that systematically reduce the sampling requirements while retaining the essential physics of wave propagation. SASM replaces the original optical system with a scaled equivalent in which lens-induced beam convergence or divergence is reduced, lowering memory usage and computational cost by approximately the square of the scaling factor. NLASM suppresses lensing effects altogether, enabling highly efficient propagation away from focal planes. Benchmarking against the Bluestein (chirp-z) transform reveals that the three methods are complementary and together enable wave-optical simulations of complex electron-optical systems previously considered infeasible. These results establish practical pathways toward routine wave-based modeling in electron microscope design.

Figures (3)

• Figure 1: A relation between the propagation of a beam function $b_\mathrm{j}(\mathbf{R})$ and $u_\mathrm{j}(\mathbf{R})=b_\mathrm{j}(\mathbf{R})\exp\{{\rm i} k\mathbf{R}^2/(2\varrho_\mathrm{j})\}$. It turns out that $u_\mathrm{j}(\mathbf{R})$ being propagated on a distance $\Delta z_\mathrm{j}$ is completely equivalent to the beam $b_\mathrm{j}(M_{\rm j}\mathbf{R})$ being propagated on a distance $S_{\rm j}\Delta z_\mathrm{j}$, and then rescaled by the factor $M_\mathrm{j}=\sqrt{|\varrho_{\rm j}/(\Delta z_{\rm j}+\varrho_{\rm j})|}$ and multiplied by the phase factor corresponding to the parabolic wavefront.
• Figure 2: Numerical simulations of a wavefront propagated through an optical system consisting of one lens. (a--c) Schemes of original, scaled, and no-lensing optical systems. The model parameters are listed in Table \ref{['tab:SingleLens']}. wd is the working distance in the original system (1 mm), which is here equal to the focal distance of the lens, thanks to the collimated incoming beam. $\bar{\text{wd}}$ is the scaled working distance. (d--f) The beam intensity profile in planes 0 nm, $-200$ nm, and $-10\,\upmu$m from the working plane, respectively. The beam intensity is expressed as a probability density. The intensity profiles are calculated using various methods: the scaling angular spectrum method (SASM), the no-lensing angular spectrum method (NLASM), and the Bluestein (BS) method. For comparison, we also show profiles calculated by the abTEM code (AT) Susi2020 by direct calculation of the angular spectra in the plane of interest, without propagation from the focusing lens. We zoom a detailed view on the edge of the beam in (f), which is an important part of the beam profile carrying information about the numerical aperture and defining the focused spot size. (g, h) The minimal necessary number of pixels in one dimension of the initial plane ($z=0$) in dependence on the $z$ coordinate of the plane of interest. ASM stands for the "pure" angular spectrum method. Discretely placed symbols in (h) show the sampling used to calculate the beam profiles shown in (d--f). The green marks stand for AT.
• Figure 3: Numerical example of wave propagation through a complex optical system with multiple elements. (a, b) Schemes of original and no-lensing/scaled optical system. The model parameters are listed in Table \ref{['tab:PP']}. wd$_l$ is a local working distance, wd is the final working distance, L$_1$ and L$_2$ are the first and second lenses, and PP is a phase plate. $\bar{\text{wd}}$ is the scaled final working distance. The beam undergoes a crossover in the section between L$_1$ and PP. The crossover in the scaled system is expressed as a reversed propagation direction in the L$_1$-PP section. (c) The beam intensity profile in the working plane, calculated by a combination of NLASM and SASM (black), and by BS (blue). (d) The beam intensity profile 50 nm behind the working plane, calculated by a combination of NLASM and SASM (black), only by NLASM (orange), and by BS (blue). The beam intensity is expressed as a probability density. The simulations carried out by NL/SASM, and NLSASM used 2$^{10}\times 2^{10} =$ 2$^{20}$ pixels. The simulations carried out by BS used $2^{13}\times2^{13}=2^{26}$ pixels.