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

TL;DR

This paper compares the eigenstructures of Bloch wave and multislice formulations in transmission electron microscopy, revealing their theoretical equivalence and demonstrating practical applications like estimating the mean inner potential.

Contribution

It reformulates the multislice method into a transmission matrix and establishes its eigenstructure equivalence with the Bloch wave scattering matrix.

Findings

Eigenvectors are related by a 2D Fourier matrix.

Eigenvalue angles differ by multiples of 2π.

Application to estimate mean inner potential.

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.