Relativistic vortex electrons: Paraxial versus nonparaxial regimes

Abstract

A plane-wave approximation in particle physics implies that a width of a massive wave packet σ ⊥ is much larger than its Compton wavelength λc = /mc. For Gaussian packets or for those with the non-singular phases (say, the Airy beams), corrections to this approximation are attenuated as λ 2 c /σ 2 ⊥ ≪ 1 and usually negligible. Here we show that this situation drastically changes for particles with the phase vortices associated with an orbital angular momentum ℓ . For highly twisted beams with |ℓ| ≫ 1, the non-paraxial corrections get |ℓ| times enhanced and |ℓ| can already be as large as 10 3 . We describe the relativistic wave packets, both for vortex bosons and fermions, which transform correctly under the Lorentz boosts, are localized in a 3D space, and represent a non-paraxial generalization of the massive Laguerre-Gaussian beams. We compare such states with their paraxial counterpart paying specific attention to relativistic effects and to the differences from the twisted photons. In particular, a Gouy phase is found to be Lorentz invariant and it generally depends on time rather than on a distance z. By calculating the electron packet's mean invariant mass, magnetic moment, etc., we demonstrate that the non-paraxial corrections can already reach the relative values of 10 −3 . These states and the non-paraxial effects can be relevant for the proper description of the spin-orbit phenomena in relativistic vortex beams, of scattering of the focused packets by atomic targets, of collision processes in particle and nuclear physics, and so forth.

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

“…The non-paraxial phenomena can be enhanced for highly twisted electrons with |ℓ| ≫ 1 [8], and it turns out that, somewhat contrary to intuition, the spreading may further enhance some of them. The direct detection of the above azimuthal asymmetry is feasible with the already available electron beams, which would be the first observation of a non-paraxial effect with the vortex elec-trons.…”

“…[8], these beams can actually be used only for relativistic electrons but hardly for those with ε c ∼ 300 keV. The generalized Laguerre-Gaussian beams, proposed in [8], can be used beyond the paraxial regime and, in particular, stay valid in the rest frame. They represent an exact solution to the Dirac equation in relativistic case and to the Schrödinger equation for non-relativistic energies.…”

Probability of radiation of twisted photons by cold relativistic particle bunches

“…The expression (79) has the form (39) and tends to zero at infinity as an exponent (for the wave packets with such an asymptote, see, e.g., [74,75]). When λ → 0, the exponential profile considered above is reproduced.…”

Effects of the transverse coherence length in relativistic collisions

“…These packets represent a superposition of the Bessel beams with a Gaussian envelope. Although the Bessel beam is just a special case of the generalized Laguerre-Gaussian state [35], the model of Ref. [9] may predict larger corrections to the plane-wave cross sections, which is yet to be explored in detail.…”

“…where p ⊥ ∼ (0.1 − 100) |ℓ| keV for the twisted leptons and hadrons. In contrast to the previous calculations of the single-twisted scattering with the Bessel beams, here we imploy a generalized Laguerre-Gaussian state ψ ℓ,n=0 [35], which is a more general model of the relativistic vortex packet. While for the Bessel beam the cross section is generally insensitive to the OAM in the single-twisted scenario, this is not the case for the Laguerre-Gaussian packet, whose mean transverse momentum grows as |ℓ|.…”