An Approach for the aberration calculation of multi-electron-beam systems

Conventional approaches for calculating aberrations in multi‑electron‑beam systems (MEBS) are typically with reference to the global principal optical axis. For sub-beams located far off-axis, a large number of higher-order aberration terms arise, making conventional methods increasingly inadequate. Here, we propose an approach in which aberration coefficients are calculated with reference to the central reference ray of each sub-beam, rather than the principal optical axis, using the differential algebraic (DA) method implemented in MEBS. First, the multi-source unit consisting of a Schottky emitter and two pre-lens electrodes is characterized by calculating its transfer coefficients via the DA method. The virtual source position for each sub-beam is then determined. Second, for off-axis aperture lenses, the local multipole field components are extracted from three-dimensional finite element field solutions via azimuthal Fourier analysis. These components are then expressed analytically in the global coordinate system. For the other electromagnetic lenses (e.g., the objective lens), the global field components are used directly. Third, starting from each virtual source position, the central electron trajectory of each sub-beam is traced through the entire electromagnetic field distribution using the DA method. Finally, the aberration coefficients up to third-order with reference to each sub‑beam's actual optical axis are obtained. The proposed method is applied to a single-source single-column MEBS. The results show that, with our approach, the spot size is dominated by the first-order geometric aberrations for all sub-beams, regardless of their off-axis distance from the global principal optical axis.