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

A new Bateman-Hillion solution to the Dirac equation for a relativistic Gaussian electron beam taking explicit account of the four-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 intrinsic spin-orbit coupling. Our results for these properties align with earlier work on Bessel beams [Bliokh et al., Phys. Rev. Lett. 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.

Figure 1

A comparison of paraxial (flat) and beyond paraxial (curved) Gouy phase fronts for a 100 keV electron beam. The beam radius of 5 pm is set smaller than occurs in practice to accentuate the difference.

Figure 2

Fractional angular momenta (left-hand panels) and the total Gouy phase shift and Berry phase (right-hand panels) as functions of the divergence angle for spin-up electrons with a kinetic energy of 0.5 MeV (upper panels) and of 0.1 MeV (lower panels).