Unified Approach towards the Dynamics of Optical and Electron Vortex Beams

Abstract

We have proposed a unified framework towards the dynamics of optical and electron vortex beams from the perspective of the geometric phase and the associated Hall effects. The unification is attributed to the notion that the spin degrees of freedom of a relativistic particle, either massive or massless, are associated with a vortex line. Based on a cylindrical coordinate formulation, which leads to a local vortex structure related to orbital angular momentum (OAM), it can be shown that, when electron vortex beams (EVBs) move in an external electric field, paraxial beams give rise to an OAM Hall effect, and nonparaxial beams with tilted vortices initiate a spin Hall effect in free space as well as in an external field. A similar analysis reveals that the paraxial optical vortex beams (OVBs) in an inhomogeneous medium induce an OAM Hall effect, whereas nonparaxial beams with tilted vortices drive the spin Hall effect. Moreover, both OVBs and EVBs with tilted vortices give rise to OAM states with an arbitrary fractional value.

Geometric phase and fractional orbital-angular-momentum states in electron vortex beams

“…It has been shown in some earlier papers [12,18,19] that the presence of the spin vector elevates the localization region of a relativistic massive as well as a massless particle from S 2 to S 3 . Indeed noting that S 3 is equivalent to SU (2), we can write S 2 = SU (2) U (1) so that S 3 can be constructed from S 2 by Hopf fibration.…”

“…It has been observed that optical and electron vortex beams carrying OAM share much similarities in their behaviour. Indeed, just like EVBs, the dynamics of the optical vortex beams(OVBs) can also be studied from the perspective of geometric phase acquired by the beam field orbiting around the vortex line [12]. In this note, we shall study the situation when EVBs carry fractional OAM.…”

“…In this note, we shall study the situation when EVBs carry fractional OAM. When the scalar electron in an EVB orbiting around the vortex line, acquires the Berry phase, which involves quantized monopole charge, we have screw or edge dislocation [11,12]. The screw dislocation essentially corresponds to paraxial beams, where the vortex line is parallel to the wave front propagation direction, whereas for edge dislocation, the vortex line is orthogonal to the wave front propagation direction.…”

Berry curvature and various thermal Hall effects

“…The Berry phase effect [16][17][18] is essential to understand the underlying mechanism of various quantum, spin, or anomalous Hall effects [19][20][21]. And there is a crucial role of the Berry phase and the Berry curvature in understanding the orbital angular momentum Hall effect and spin Hall effect associated with electron and optical vortex beams [22][23][24][25]. Recently Berry curvatures were also found in both of the phonon Hall effect [10,13] and magnon Hall effect [14] through different methods.…”

Robust and adjustable C-shaped electron vortex beams

“…It has been previously shown that fractional vortex beams can be synthesized to only show rotation but maintain the shape of the intensity as the beam propagates. This was achieved by the use of a radial quantum number such that the Gouy phases of individual contributions compensate to produce a uniform Gouy phase [27,37]. There is the potential that application of such an approach could provide additional stability to the C-shaped beam structure, beyond that bestowed by the threading vortex lines; however, this is beyond the scope of the current paper.…”