A T-matrix scattering formalism for electron-beam spectroscopy

TL;DR

The paper addresses the need for robust, efficient tools to predict fast-electron interactions with structured nanophotonic materials. It introduces a general T-matrix framework that describes fast-electron fields in a cylindrical-wave basis and their scattering by ensembles of objects, yielding unified CL and EELS observables. Key contributions include explicit field representations in the cylindrical basis, local and global T-matrix formalisms for single and multiple scatterers, extensions to periodic lattices, closed-form CL and EEL expressions, and an open-source implementation in treams_ebeam. This framework enables accurate, scalable design and interpretation of free-electron–driven nanoscale light–matter interactions, with practical utility for complex photonic platforms.

Abstract

Advanced computational tools that describe the interaction of electrons with structured nanophotonic materials are crucial for theoretical predictions, specific design tasks, and the interpretation of experimental results. These tools open the door to systematic exploration of free-electron-driven nanophotonic light sources, among others. Here, we report on the implementation of electron-beam spectroscopy in a T-matrix-based scattering formulation. Such a framework is quite versatile in predicting the electromagnetic response of complex photonic materials composed of periodically or aperiodically arranged individual scatterers. By extending this formalism to describe interactions with fast electrons, we provide a fast and accurate numerical tool for simulating cathodoluminescence (CL) and electron energy-loss spectroscopy (EELS) measurements. The desired functionalities are implemented into the existing software suite treams for electromagnetic scattering computations, and the extended code treams_ebeam is available online at https://github.com/tfp-photonics/treams_ebeam. We demonstrate the implementation details on a carefully selected set of problems, including single scatterers of various shapes and materials, a periodic chain of elliptical nanodisks, and a finite cluster of nanospheres arranged in a two-dimensional (2D) lattice. By uniting fast-electron physics with advanced scattering theory, our framework unlocks new possibilities for designing, understanding, and engineering next-generation nanoscale light-matter interactions.

Figures (4)

• Figure 1: Examples of scatterer configurations excited by an electron beam, along with the validity domains of the relevant field expansions in the SWB. (a) For a scatterer of arbitrary shape the validity domain is outside the circumscribing sphere enclosing the scatterer, outlined with the dashed black circle. (b) Cluster with three scatterers. In a local description, the fields are expanded at the position ${\mathbf{r}}_i$ of the $i$th scatterer, and are valid outside the corresponding circumscribing sphere (black dashed circle). The red dotted circle indicates the validity domain of the global description, which lies outside the circumscribing sphere enclosing all scatterers. (c) Infinite one-dimensional (1D) array with pitch ${\mathbf{\Lambda}} = \Lambda { \bm{\hat{\mathbf{z}}} }$ with a complex unit cell, composed of two scatterers. The red dotted circles mark the domain of validity for a global T-matrix, describing the scatterers of the unit cell as a combined object. This description might be invalid due to overlapping circumscribing spheres, in which case only the local description (black dashed circles) can be employed.
• Figure 2: EEL (blue solid line) and CL probability (red dashed line) of (a) a dielectric nanosphere of radius $R=50$ nm, (b) a metallic nanowire of radius $R=50$ nm and infinite length, and (c) an amorphous silicon elliptical nanodisk of height $h=90$ nm, long axis $r_\mathrm{l}=286$ nm and short axis $r_\mathrm{s}=96$ nm. In all panels, we consider $R-b=10$ nm and $\beta=0.7$.
• Figure 3: (a) Schematic of the system under study: a chain of the amorphous silicon elliptical nanodisks excited by an electron beam with parameters $\beta=0.7$ and $b-R=7$ nm. (b) EEL and (c) CL probabilities per nanoparticle for chains with $N =1,3,5,9,15,\infty$ nanoparticles. Consecutive curves are vertically offset by $3\cdot 10^{-3}$ eV$^{-1}$ for readability. (d) CL probability per nanoparticle versus azimuthal angle $\theta$ for the different cases of the finite chain. The dashed green lines trace the angles at which Smith-Purcell radiation is emitted in the first diffraction order.
• Figure 4: (a) Schematic of the system under study: a finite 2D array of aluminum nanospheres is excited by an electron beam that passes transversely through its center with reduced velocity $\beta=0.7$. (b) EEL and (c) CL probabilities for arrays consisting of $N = 2, 4, 6, 8$ nanoparticles per side. The black, dashed curves show the respective spectra for a single aluminum nanosphere positioned at the same distance from the electron trajectory as the nearest nanoparticle in the corresponding array.