Efficient methods for wave propagation in electron microscopy.

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 towards routine wave-based modelling in electron microscope design.