A differential algebra computational framework for high-order aberration analysis of electromagnetic lenses.

A computational framework based on the differential algebra (DA) method is presented for calculating aberration coefficients up to fifth order in electromagnetic lenses. To bridge discrete numerical field calculations with high-order DA tracking, a local analytical field-fitting method is proposed. The electromagnetic field obtained from finite-element analysis is first decomposed by Fourier analysis, interpolated along the optical axis using linear Lagrange interpolation, and subsequently fitted to the corresponding analytical basis functions through least-squares fitting. The fitted field representations are validated against OPERA FEA results and an AFFs-based method, while the DA aberration calculation program is further verified against COSY INFINITY using ideal models. For practical finite-element models, the two field-representation methods yield generally consistent high-order aberration coefficients. These results demonstrate the applicability of the proposed framework to high-order aberration calculations based on practical finite-element fields.