Browsing by Author "Uhlemann, Stephan"

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Charge Particle Optics Simulation Utilizing Hamiltonian Mechanics Perturbation Expansion and Boundary Elements Field Computation
(Technische Universität Dresden, 2026-05-26) Lubk, Axel; Houdellier, Florent; Müller, Heiko; Uhlemann, Stephan
Background incl. aims Advanced charge particle optics (CPO) requires fast and accurate computational tools for calculating (relativistic) particle trajectories including aberrations that, ideally, handle both arbitrary electrics and magnetic field sources, handle arbitrarily bend optical axis, allow incorporation of symmetries (e.g., rotational or mirror), incorporate optimization of design parameters such as pole piece diameter or pole distances. Ideally such tools should also be available under open source licenses in order to facilitate widespread use as well as distributed and sustainable development. Despite the enormous level of development and usefulness of commercial (e.g. Simion, EOD, Comsol) and open source packages, they often lack a subset of the above functionalities, somewhat hampering a wide spread development and use of advanced CPO for, e.g., Transmission Electron Microscopy, Secondary Ion Mass Spectroscopy, Photo Electron Spectroscopy, in academia, industry, and also teaching. The CPO software development described below intends to address that need. Methods Here we report on the development of an open source computational CPO framework incorporating the following principles to allow for an accurate, fast and flexible trajectory calculation: (A) We use boundary element method (BEM) computation of electric and magnetic fields, yielding smooth and accurate potentials, fields and higher-order derivatives at optical axis at arbitrary sampling, while reducing the meshing effort to surfaces (e.g., electrodes and pole pieces) of the CPO device. Herein, single layer representations of both electric and magnetic scalar potential are most efficient, while Green’s representation with Calderon preconditioning allows stable single step solution of magnetic field distributions in the presence of high µr materials. (B) We employ semianalytical hierarchical solution of perturbation series of Hamiltonian equations of motion around an optical axis[1] in order to provide computationally effective, fast and accurate built-up of aberrations along particle trajectories. While not implemented yet the Hamiltonian perturbation expansion also facilitates straight forward extension to curved axis and the eikonal representation of aberrations. (C) We integrate the fast field and particle trajectory computation with non-linear optimization routines facilitating automatic optimization of design parameters (e.g., multipole sizes, pole piece gap) with respect to certain target functionalities. This tool chain is written in Python and makes use of advanced open source libraries (namely OpenCascade for CAD, gmsh for meshing, BEMPP for BEM field computation, sympy for semianalytic Hamiltonian mechanics perturbation expansion including automatic code generation, scipy for solving equations of motion, nlopt for geometry optimization) in a modular way, which are partly adapted to the specifics of CPO. Notably, BEMPP was extended by parallel just-in-time compiled numba and opencl kernels for field derivative computations on optical axis as required for computation of paraxial trajectories and aberrations. Results We demonstrate and discuss the above tool chain with the help several electrostatic and magnetostatic CPO / building blocks of CPO, notably electrostatic Einzellens, electrostatic quadrupole – round aperture assembly (see Fig. 1) and magnetostatic quadrupole, touching implementation (e.g., CAD import, defining boundary conditions, vector potential gauge), numerical (e.g., mesh size, precision of paraxial solution) and CPO (e.g., chromatic and geometric aberrations) aspects. Conclusion A modular combination of adapted BEM field computations and semianalytical perturbation series expansion of Hamiltonian equations of motion admits a computationally efficient modeling of CPO utilizing a combination of powerful and freely available open source software packages. Further development aims at incorporation of curved optical axis and general enhancement of functionality and user friendliness in order to support development of advanced CPO across the community. Reference: [1] Kern, F., Krehl, J., Thampi, A., Lubk, A. “A Hamiltonian mechanics framework for charge particle optics in straight and curved systems”, Optik, 2021, 242, 167242 [2] Tamura, K., Okayama, S., Shimizu, R. “Third-order spherical aberration correction using multistage self-aligned quadrupole correction-lens systems”, Journal of Electron Microscopy, 2010, 59, 197 [3] We acknowledge financial support by the European Union's Horizon Europe framework program for research and innovation under grant agreement n. 101094299 (IMPRESS project).