Eigenstructure Analysis of Bloch Wave and Multislice Formulations for Dynamical Scattering in Transmission Electron Microscopy

Abstract

We investigate the eigenstructure of matrix formulations used for modeling scattering processes within materials in transmission electron microscopy. Dynamical scattering is crucial for describing the interaction between an electron wave and the material under investigation. Unlike the Bloch wave formulation, which defines the transmission function via the scattering matrix, the traditional multislice method is lacking a pure transmission function due to the entanglement of electron waves with the propagation function. To address this, we reformulate the multislice method into a matrix framework, which we refer to as the transmission matrix. This allows a direct comparison to the scattering matrix derived from Bloch waves in terms of their eigenstructures. Through theory, we demonstrate their equivalence with eigenvectors related by a two-dimensional Fourier matrix, given that the eigenvalue angles differ by modulo $2\pi n$ (integer $n$). We numerically verify our findings as well as demonstrate the application of the eigenstructure for the estimation of the mean inner potential.