Relativistic Electron Vortex Beams: Angular Momentum and Spin-Orbit Interaction

Motivated by the recent discovery of electron vortex beams carrying orbital angular momentum (AM), we construct exact Bessel-beam solutions of the Dirac equation. They describe relativistic and nonparaxial corrections to the scalar electron beams. We describe the spin and orbital AM of the electron with Berry-phase corrections and predict the intrinsic spin-orbit coupling in free space. This can be observed as a spin-dependent probability distribution of the focused electron vortex beams. Moreover, the magnetic moment is calculated, which shows different $g$ factors for spin and orbital AM and also contains the Berry-phase correction.

Figure 1

(a) The Bessel-beam spectrum (5) forms a cone of plane waves with a fixed polar angle $\theta ={\theta }_0$ and color-coded azimuthal phase $e^{i\ell \varphi }$ on the monoenergetic sphere $p=\mathrm{const}$. (b) An example of the probability density and azimuthal current distributions for the scalar Bessel beam with $\ell =1$ (the dashed line shows the zero of the current).

Figure 2

Spin-dependent distributions of the probability density (9) and azimuthal current (10) for the nonparaxial Bessel beams with $\ell =3$ (a) and $\ell =1$ (b). The $s=\pm 1/2$ spin states are indicated by “$+$” and “$-$” signs, and the SOI parameter is $\Delta =0.3$, which corresponds to $p\simeq 2.4m$ and ${\theta }_0=\pi /4$. The drastic difference between ${\rho }_1^{+}(0)$ and ${\rho }_1^{-}(0)$ should be experimentally observable cf. 9, 20.