# Dynamical enhancement of nonparaxial effects in the electromagnetic field of a vortex electron

## TL;DR

This paper investigates how nonparaxial effects, especially quadrupole moments, are enhanced in vortex electrons with high orbital angular momentum, leading to observable asymmetries in their electromagnetic fields.

## Contribution

It derives a multipole expansion of vortex electron fields including quadrupole terms, highlighting the linear enhancement of nonparaxial effects for highly twisted electron packets.

## Key findings

- Quadrupole contributions become significant beyond paraxial approximation.
- High angular momentum enhances nonparaxial effects linearly.
- Azimuthal magnetic field asymmetry can be experimentally detected.

## Abstract

A quantum state of an electron influences its electromagnetic field. If a spatial profile of the electron wave packet is not Gaussian, the particle may acquire additional intrinsic multipole moments, which alter its field, especially at small distances. Here the fields of a vortex electron with orbital angular momentum $\ell$ are obtained in a form of a multipole expansion with an electric quadrupole term kept by using the generalized (non-paraxial) Laguerre-Gaussian beams. The quadrupole contribution arises beyond a paraxial approximation, is linearly enhanced for highly twisted packets with $|\ell| \gg 1$, and can be important for the interactions of twisted beams with bulk matter and artificial structures. Moreover, this term results in an azimuthal asymmetry of the magnetic field in a rest frame of the electron, which appears thanks to the spreading of the packet with time. Thus, somewhat contrary to physical intuition, the spreading may enhance non-paraxial phenomena. For the available electron beams, this asymmetry can in principle be reliably detected, which would be experimental evidence of a non-paraxial effect with the vortex electrons.