Gouy Phase and Fractional Orbital Angular Momentum in Relativistic Electron Vortex Beams

A Bateman-Hillion solution to the Dirac equation for a 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 momentum are fractional. Our results for these properties aligns with earlier work on Bessel beams that also showed fractional angular momenta can be parameterized in terms of a Berry phase. Laguerre-Gaussian beams contain Gouy phase that Bessel beams do not. We show that Gouy phase shift from far field to far field in a Gaussian beam can also be parameterized in terms of Berry phase indicating that these two fundamental phases are unexpectedly related to each other.