# Fractional Angular Momenta, Gouy and Berry phases in Relativistic Bateman-Hillion-Gaussian Beams of Electrons

## TL;DR

This paper presents a relativistic Gaussian electron beam solution to the Dirac equation, revealing fractional angular momenta linked to Berry and Gouy phases, with implications for spin-orbit coupling and beam focusing.

## Contribution

It introduces a new relativistic Gaussian beam solution accounting for beam waist position, highlighting fractional angular momenta and their relation to Berry and Gouy phases beyond the paraxial limit.

## Key findings

- Fractional angular momenta are related to Berry phase.
- Gouy phase is connected to Berry phase and affects beam focusing.
- Differences between Laguerre-Gaussian and Bessel beams in phase properties.

## Abstract

A new Bateman-Hillion solution to the Dirac equation for a relativistic Gaussian electron beam taking explicit account of the $4$-position of the beam waist is presented. This solution has a pure Gaussian form in the paraxial limit but beyond it contains higher order Laguerre-Gaussian components attributable to the tighter focusing. One implication of the mixed mode nature of strongly diffracting beams is that the expectation values for spin and orbital angular momenta are fractional and are interrelated to each other by \textit{intrinsic spin-orbit coupling}. Our results for these properties align with earlier work on Bessel beams [Bliokh \textit{et al.} Phys. Rev. Lett. \textbf{107}, 174802 (2011)] and show that fractional angular momenta can be expressed by means of a Berry phase. The most significant difference arises, though, due to the fact that Laguerre-Gaussian beams naturally contain Gouy phase, while Bessel beams do not. We show that Gouy phase is also related to Berry phase and that Gouy phase fronts that are flat in the paraxial limit become curved beyond it.