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

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 $2πn$ (integer $n$). We numerically verify our findings as well as demonstrate the application of the eigenstructure for the estimation of the mean inner potential.

💡 Research Summary

This paper presents a rigorous comparative study of the two dominant computational frameworks used to model dynamical electron scattering in transmission electron microscopy (TEM): the Bloch‑wave method and the multislice method. The authors first reformulate the traditional multislice algorithm, which proceeds by sequentially applying thin‑slice projected potentials and free‑space Fresnel propagation, into a matrix product that explicitly separates the specimen’s transmission function from the propagation operator. By defining the real‑space transmission matrix

 A = ∏_{m=1}^{M} G_m O_m

where O_m is a diagonal matrix containing the projected potential of slice m and G_m is the Fresnel propagator, they obtain a “transmission matrix” in reciprocal space

 \hat S = F_{2D} A F_{2D}^{-1}.

Here F_{2D} denotes the two‑dimensional discrete Fourier transform. This construction mirrors the scattering matrix S that naturally arises from the Bloch‑wave formulation, where

 S = C Λ C^{-1}

with C containing the Bloch‑wave Fourier coefficients (eigenvectors) and Λ a diagonal matrix of eigenvalues of unit modulus that encode specimen thickness‑dependent phase shifts.

The central theoretical result is that the eigenvectors of \hat S and S are related by the 2‑D Fourier matrix:

 W = F_{2D} V,

where V are the eigenvectors of the real‑space transmission matrix A, and that the eigenvalues of the two matrices coincide up to an additive multiple of 2π (i.e., their phases differ only by 2π n for integer n). Consequently, the two formulations are mathematically equivalent at the level of eigenstructure, despite their historically distinct algorithmic implementations.

To validate the theory, the authors generate synthetic crystals of GaAs, SrTiO₃, and Au. For each material they compute both S (Bloch‑wave) and \hat S (multislice‑derived) over a range of specimen thicknesses, then perform eigendecompositions. Frobenius‑norm comparisons reveal that the eigenvalue phases are nearly linear and differ by at most ~64 mrad, while eigenvector discrepancies remain on the order of 10⁻², independent of thickness. The overall matrix difference is also around 10⁻², confirming high numerical agreement.

The paper proceeds to demonstrate practical implications of this equivalence. Diffraction patterns generated by multiplying either matrix with a plane‑wave illumination are virtually indistinguishable, reproducing the Pendellösung oscillations of Bragg beam intensities with matching maxima and minima. Projected electrostatic potentials are recovered from the transmission matrix (directly from A) and, for the Bloch‑wave case, via the approximation A ≈ F_{2D}^{-1} S F_{2D}. The resulting potentials for all three crystals show excellent visual and quantitative agreement, with only minor pixel‑wise differences.

A novel application is introduced: estimating the mean inner potential (MIP) of a specimen from the determinant of the transmission matrix. Because det(\hat S) = det(A) ≈ exp