Aberration calculation of microlens array using differential algebraic method.

Microlens array (MLA), through which all the sub-beams are focused, is widely used in multi-electron-beam systems. In this work, based on the differential algebraic (DA) method, we propose an approach in calculating the high-order aberrations for both axial and off-axial microlenses, considering the multipole fields that are introduced by the neighborhood structures in MLA, as well as the rotationally symmetric field. To perform the DA calculation, the electric fields of the microlenses are analyzed by using the azimuthal Fourier analysis and the Fourier-Bessel series Expansion. The resulting field components, including both rotationally symmetric field and the multipole fields, are transferred into DA arguments and operated as per DA methodology. Then, by developing and employing the DA theory and algorithm, the primary and high-order aberrations are calculated and obtained simultaneously for both the axial and off-axial microlenses by tracing only one reference ray. Finally, we calculate, analyze and discuss the primary and high-order aberrations of two example MLAs, for both axial and off-axial microlenses. The effects of the dodecapole fields on the aberrations are also analyzed.