Relativistic Electron Vortex Beam (REVB)

• Relativistic electron vortex beams are fully relativistic spinor states characterized by a vortex phase, azimuthal current, and quantized angular momentum derived from the Dirac equation.
• Exact free-space constructions and field-dressed solutions using Dirac–Bessel and Laguerre–Gaussian formulations reveal intricate spin–orbit interactions and Berry-phase effects.
• These beams enable advanced applications in electron microscopy, scattering experiments, and strong-field physics by leveraging precise relativistic corrections and phase control.

A relativistic electron vortex beam (REVB) is a relativistic spin-$\tfrac12$ electron state described at the Dirac level and endowed with a vortex phase structure, azimuthal current density, and angular momentum about its propagation axis. In free space, REVBs admit exact Dirac–Bessel constructions with cylindrical symmetry; in broader packet form they also appear as Lorentz-covariant relativistic Laguerre–Gaussian states and as exact Bateman–Hillion Klein–Gordon vortex modes. In external fields, the same basic vortex structure extends to stationary states in homogeneous or inhomogeneous longitudinal magnetic fields and to Volkov–Bessel states in plane electromagnetic waves, including self-consistent field configurations generated by the beam current itself (Campos et al., 2020, Bliokh et al., 2011, Karlovets, 2018).

1. Exact free-space constructions

The canonical free-space starting point is the Dirac equation

$\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$

solved in cylindrical coordinates $(r,\phi,z)$ by states with definite longitudinal momentum and vortex structure. In the RDI formulation, one seeks spinors $\psi_{m,p_z}(r,\phi,z,t)$ carrying orbital angular momentum $\hbar m$ about the $z$-axis, and obtains positive-energy Bessel-beam spinors of the form

$$\psi_{m,p_z}(r,\phi,z,t) = N\,e^{i(m\phi+p_z z/\hbar-\epsilon t/\hbar)} \begin{pmatrix} (\epsilon+mc^2)J_m(\kappa r)/B\ i\kappa e^{i\phi}J_{m+1}(\kappa r)/B\ p_z J_m(\kappa r)/B\ i p_z\kappa e^{i\phi}J_{m+1}(\kappa r)/[B(\epsilon+mc^2)] \end{pmatrix},$$

with $B=2mc^2$, $\epsilon=\sqrt{p_z^2c^2+\kappa^2c^2+m^2c^4}$, and $\kappa=$ transverse momentum$\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$0. The $\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$1-dependence $\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$2 gives the OAM eigenvalue $\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$3, and the exact relativistic dispersion is $\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$4 (Campos et al., 2020).

A complementary free-particle construction is the exact Dirac–Bessel solution obtained by superposing plane-wave spinors on a momentum cone of opening angle $\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$5. In that formulation, the leading scalar-like term $\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$6 is accompanied by sidebands $\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$7, and the parameter

$\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$8

controls relativistic and nonparaxial corrections (Bliokh et al., 2011).

An exact cylindrical-coordinate treatment of the free Dirac equation further emphasizes that the rigorous conserved quantity is the total angular momentum operator

$\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$9

For the exact eigenspinor $(r,\phi,z)$0, one has

$(r,\phi,z)$1

so that the vortex charge $(r,\phi,z)$2 contributes the orbital part, while the spin contributes the half-unit completing the net quantization (Guo et al., 11 Jul 2025).

2. Angular momentum, Berry phase, and intrinsic spin–orbit structure

In Dirac vortex beams, orbital and spin angular momentum are generally not separately conserved, even though the total angular momentum is. After Foldy–Wouthuysen projection onto positive-energy states, the position operator acquires a Berry connection,

$(r,\phi,z)$3

and the corresponding OAM and SAM operators become $(r,\phi,z)$4 and $(r,\phi,z)$5, with $(r,\phi,z)$6. For a Dirac–Bessel beam,

$(r,\phi,z)$7

so the Berry-phase correction $(r,\phi,z)$8 quantifies spin-to-orbit angular-momentum conversion in free space (Bliokh et al., 2011).

A geometric-phase formulation recasts the same structure in terms of a vortex line and an effective monopole charge $(r,\phi,z)$9. For a closed loop subtending solid angle $\psi_{m,p_z}(r,\phi,z,t)$0, the Berry phase is

$\psi_{m,p_z}(r,\phi,z,t)$1

In the paraxial regime, with vortex line parallel to $\psi_{m,p_z}(r,\phi,z,t)$2, $\psi_{m,p_z}(r,\phi,z,t)$3 and $\psi_{m,p_z}(r,\phi,z,t)$4. In the non-paraxial regime, with vortex line orthogonal to $\psi_{m,p_z}(r,\phi,z,t)$5, $\psi_{m,p_z}(r,\phi,z,t)$6 and $\psi_{m,p_z}(r,\phi,z,t)$7; for $\psi_{m,p_z}(r,\phi,z,t)$8, this gives $\psi_{m,p_z}(r,\phi,z,t)$9. In the same picture, non-paraxial beams incorporate spin–orbit interaction, and one finds

$\hbar m$0

so that only $\hbar m$1 remains a good quantum number (Bandyopadhyay et al., 2013).

Recent exact cylindrical Dirac solutions express the intrinsic spin–orbit coupling strength through a Bessel-integral shift $\hbar m$2, with

$\hbar m$3

The same analysis identifies a helicity anomaly: because vortex states are not momentum eigenstates in the transverse plane, the expectation of the transverse helicity component becomes imaginary, while the real part

$\hbar m$4

remains a genuine observable and depends on the vortex charge through the same Bessel integrals (Guo et al., 11 Jul 2025).

3. Stationary REVBs in longitudinal magnetic fields

Stationary vortex spinors in magnetic fields can be obtained from the free-space form by minimal coupling $\hbar m$5, with $\hbar m$6 chosen so that circular orbits result. In a homogeneous longitudinal field $\hbar m$7, one uses

$\hbar m$8

with $\hbar m$9, and the vector potential in symmetric gauge

$z$0

The energy eigenvalues are the relativistic Landau levels

$z$1

(Campos et al., 2020).

For inhomogeneous longitudinal fields $z$2, the RDI construction yields closed-form radial amplitudes built from associated Laguerre polynomials and an energy spectrum

$z$3

In both homogeneous and inhomogeneous cases, the density and current satisfy

$z$4

with $z$5 perpendicular to $z$6; stable circular motion requires vanishing radial current, $z$7, so the Lorentz force is purely centripetal (Campos et al., 2020).

A fully relativistic Dirac treatment of the homogeneous-field problem shows that the current and spin textures can be substantially more intricate than in scalar vortex models. The exact azimuthal current $z$8 contains strong corotating rings and weak counter-rotating rings, and the local spin density obeys

$z$9

A distinguished family of states with negative spin and negative kinetic OAM, $$\psi_{m,p_z}(r,\phi,z,t) = N\,e^{i(m\phi+p_z z/\hbar-\epsilon t/\hbar)} \begin{pmatrix} (\epsilon+mc^2)J_m(\kappa r)/B\ i\kappa e^{i\phi}J_{m+1}(\kappa r)/B\ p_z J_m(\kappa r)/B\ i p_z\kappa e^{i\phi}J_{m+1}(\kappa r)/[B(\epsilon+mc^2)] \end{pmatrix},$$0 and $$\psi_{m,p_z}(r,\phi,z,t) = N\,e^{i(m\phi+p_z z/\hbar-\epsilon t/\hbar)} \begin{pmatrix} (\epsilon+mc^2)J_m(\kappa r)/B\ i\kappa e^{i\phi}J_{m+1}(\kappa r)/B\ p_z J_m(\kappa r)/B\ i p_z\kappa e^{i\phi}J_{m+1}(\kappa r)/[B(\epsilon+mc^2)] \end{pmatrix},$$1, exhibits exactly zero spin–orbit mixing when $$\psi_{m,p_z}(r,\phi,z,t) = N\,e^{i(m\phi+p_z z/\hbar-\epsilon t/\hbar)} \begin{pmatrix} (\epsilon+mc^2)J_m(\kappa r)/B\ i\kappa e^{i\phi}J_{m+1}(\kappa r)/B\ p_z J_m(\kappa r)/B\ i p_z\kappa e^{i\phi}J_{m+1}(\kappa r)/[B(\epsilon+mc^2)] \end{pmatrix},$$2; in that protected ground state,

$$\psi_{m,p_z}(r,\phi,z,t) = N\,e^{i(m\phi+p_z z/\hbar-\epsilon t/\hbar)} \begin{pmatrix} (\epsilon+mc^2)J_m(\kappa r)/B\ i\kappa e^{i\phi}J_{m+1}(\kappa r)/B\ p_z J_m(\kappa r)/B\ i p_z\kappa e^{i\phi}J_{m+1}(\kappa r)/[B(\epsilon+mc^2)] \end{pmatrix},$$3

and the spinor factorizes into a scalar-like vortex profile with phase singularity $$\psi_{m,p_z}(r,\phi,z,t) = N\,e^{i(m\phi+p_z z/\hbar-\epsilon t/\hbar)} \begin{pmatrix} (\epsilon+mc^2)J_m(\kappa r)/B\ i\kappa e^{i\phi}J_{m+1}(\kappa r)/B\ p_z J_m(\kappa r)/B\ i p_z\kappa e^{i\phi}J_{m+1}(\kappa r)/[B(\epsilon+mc^2)] \end{pmatrix},$$4 protected by the Zeeman effect (Kruining et al., 2017).

4. Lorentz-covariant packet descriptions, non-paraxiality, and Gouy phase

Beyond ideal Bessel modes, REVBs admit localized Lorentz-covariant packet descriptions. A relativistic vortex packet with OAM $$\psi_{m,p_z}(r,\phi,z,t) = N\,e^{i(m\phi+p_z z/\hbar-\epsilon t/\hbar)} \begin{pmatrix} (\epsilon+mc^2)J_m(\kappa r)/B\ i\kappa e^{i\phi}J_{m+1}(\kappa r)/B\ p_z J_m(\kappa r)/B\ i p_z\kappa e^{i\phi}J_{m+1}(\kappa r)/[B(\epsilon+mc^2)] \end{pmatrix},$$5 and mean four-momentum $$\psi_{m,p_z}(r,\phi,z,t) = N\,e^{i(m\phi+p_z z/\hbar-\epsilon t/\hbar)} \begin{pmatrix} (\epsilon+mc^2)J_m(\kappa r)/B\ i\kappa e^{i\phi}J_{m+1}(\kappa r)/B\ p_z J_m(\kappa r)/B\ i p_z\kappa e^{i\phi}J_{m+1}(\kappa r)/[B(\epsilon+mc^2)] \end{pmatrix},$$6 can be defined in momentum space by

$$\psi_{m,p_z}(r,\phi,z,t) = N\,e^{i(m\phi+p_z z/\hbar-\epsilon t/\hbar)} \begin{pmatrix} (\epsilon+mc^2)J_m(\kappa r)/B\ i\kappa e^{i\phi}J_{m+1}(\kappa r)/B\ p_z J_m(\kappa r)/B\ i p_z\kappa e^{i\phi}J_{m+1}(\kappa r)/[B(\epsilon+mc^2)] \end{pmatrix},$$7

with invariant normalization $$\psi_{m,p_z}(r,\phi,z,t) = N\,e^{i(m\phi+p_z z/\hbar-\epsilon t/\hbar)} \begin{pmatrix} (\epsilon+mc^2)J_m(\kappa r)/B\ i\kappa e^{i\phi}J_{m+1}(\kappa r)/B\ p_z J_m(\kappa r)/B\ i p_z\kappa e^{i\phi}J_{m+1}(\kappa r)/[B(\epsilon+mc^2)] \end{pmatrix},$$8. The transverse width is $$\psi_{m,p_z}(r,\phi,z,t) = N\,e^{i(m\phi+p_z z/\hbar-\epsilon t/\hbar)} \begin{pmatrix} (\epsilon+mc^2)J_m(\kappa r)/B\ i\kappa e^{i\phi}J_{m+1}(\kappa r)/B\ p_z J_m(\kappa r)/B\ i p_z\kappa e^{i\phi}J_{m+1}(\kappa r)/[B(\epsilon+mc^2)] \end{pmatrix},$$9, so the paraxial condition becomes the invariant statement

$B=2mc^2$0

In such packets the basic non-paraxial parameter is $B=2mc^2$1, but for large $B=2mc^2$2 the corrections are enhanced by $B=2mc^2$3, leading to the invariant measure

$B=2mc^2$4

For $B=2mc^2$5 and $B=2mc^2$6, this quantity can reach $B=2mc^2$7, and the corresponding mean invariant-mass and magnetic-moment corrections are also of order $B=2mc^2$8 (Karlovets, 2018).

The Gouy phase of relativistic vortex electrons is likewise a covariant quantity. In the packet formulation,

$B=2mc^2$9

so the phase depends on time rather than on $\epsilon=\sqrt{p_z^2c^2+\kappa^2c^2+m^2c^4}$0. In the exact Bateman–Hillion Klein–Gordon construction, an $\epsilon=\sqrt{p_z^2c^2+\kappa^2c^2+m^2c^4}$1 vortex mode has

$\epsilon=\sqrt{p_z^2c^2+\kappa^2c^2+m^2c^4}$2

and both the wavefunction and the Gouy phase are form-invariant under a longitudinal Lorentz boost (Ducharme et al., 2015, Karlovets, 2018).

An FW-based paraxial Laguerre–Gauss treatment adds a centroid interpretation: the hidden transverse motion yields a discrete centroid velocity and a discrete kinematic mass labeled by $\epsilon=\sqrt{p_z^2c^2+\kappa^2c^2+m^2c^4}$3, while the orbital magnetic moment depends only on the azimuthal index and is independent of the radial index. This suggests that even paraxial twisted electrons retain specifically relativistic structured-wave invariants beyond the scalar paraxial approximation (1904.02434).

5. External electromagnetic fields, Volkov–Bessel states, and self-consistent dynamics

A central extension of REVB theory concerns exact non-stationary solutions in background electromagnetic fields. In the RDI framework, a “null rotation” of the tetrad superposes any plane electromagnetic wave $\epsilon=\sqrt{p_z^2c^2+\kappa^2c^2+m^2c^4}$4, with $\epsilon=\sqrt{p_z^2c^2+\kappa^2c^2+m^2c^4}$5, onto stationary longitudinal-field solutions. For the free beam one obtains a Volkov–Bessel state

$\epsilon=\sqrt{p_z^2c^2+\kappa^2c^2+m^2c^4}$6

where $\epsilon=\sqrt{p_z^2c^2+\kappa^2c^2+m^2c^4}$7. The resulting spinor agrees with the Volkov solution modulated by $\epsilon=\sqrt{p_z^2c^2+\kappa^2c^2+m^2c^4}$8 of the transverse coordinate $\epsilon=\sqrt{p_z^2c^2+\kappa^2c^2+m^2c^4}$9. Applying the same transformation to the uniform-$\kappa=$0 state yields the exact Redmond solution in a circularly polarized plane wave plus uniform longitudinal magnetic field; starting from the inhomogeneous longitudinal-field state gives a new closed-form Dirac spinor with exact combined fields

$\kappa=$1

where the induced radiation fields $\kappa=$2 are generated self-consistently by the vortex current (Campos et al., 2020).

A directly dynamical Volkov–Bessel formulation shows that the center of a relativistic vortex electron follows the classical point-charge trajectory in the plane wave while the internal transverse Bessel structure is maintained. For a circularly polarized pulse,

$\kappa=$3

so the beam center rotates; for a linearly polarized pulse, the beam center undergoes a lateral shift along the polarization direction. In a two-mode field combining LP and CP components, the classical center motion is the sum of the two displacements and produces a twisted spiral pattern, while each harmonic mode remains an eigenstate of $\kappa=$4 with eigenvalue $\kappa=$5 and photon exchange simply shifts $\kappa=$6 (Ababekri et al., 2024).

The same field-coupled framework also closes the Maxwell–Dirac loop. The RDI extraction formula

$\kappa=$7

guarantees that the vector potential satisfies Maxwell’s equations with source $\kappa=$8. In the combined plane-wave plus longitudinal-$\kappa=$9 case, the induced fields $\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$00 superpose with the applied laser and static field, with

$\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$01

Physically, these self-fields lead to small beam broadening or focusing corrections and become necessary when the vortex charge/current density is large (Campos et al., 2020).

A complementary Berry-curvature analysis of laser-driven vortex electrons attributes the transverse center shift in paraxial beams to an orbital Hall effect and the corresponding shift in non-paraxial beams to a spin Hall effect involving spin–orbit interaction, with the paraxial shift predicted to be larger than the non-paraxial one (Bandyopadhyay et al., 2015).

6. Generation, diagnosis, applications, and operator-level controversies

Experimentally, the free-electron vortex-state literature identifies several generation routes that have been demonstrated in TEMs at $\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$02–$\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$03: spiral phase plates, fork-holograms, phase (blazed) holograms, magnetic monopole needle tips, aberration tuning, and astigmatic mode conversion. These methods create free-electron vortex states that can then be analyzed in terms of Dirac–Bessel, Laguerre–Gaussian, or Volkov–Bessel models depending on the propagation and field environment (Bliokh et al., 2017).

For highly relativistic beams, atomic scattering provides a concrete diagnostic protocol. Scattering by a very wide target can be used to probe the electron transverse momentum when its values are larger than $\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$04. Scattering by a target of a width comparable to that of the incident beam allows one to obtain information about the electron OAM. By varying target sizes in the range from couple to hundreds of nanometers, one can in principle distinguish OAM values from several units of $\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$05 up to thousands and more (Ivanov et al., 2023).

The broader application space includes spin–orbit phenomena in relativistic vortex beams, scattering of focused packets by atomic targets, collision processes in particle and nuclear physics, Vavilov–Cherenkov radiation, transition radiation, magnetic dichroism at the nanoscale, plasmon excitation, and ultrafast electron microscopy. Several of these proposals rely explicitly on the relativistic corrections that scale as $\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$06, or on the preservation and manipulation of Bessel-phase structure in external fields (Karlovets, 2018, Bliokh et al., 2017, Ababekri et al., 2024).

A persistent conceptual controversy concerns the proper relativistic observables for vortex electrons. One operator set, based on the parity-extended Poincaré group and the associated particle spin and OAM, predicts the same singular circulation as in the nonrelativistic case,

$\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$07

independent of spin orientation. The usual Dirac spin and OAM operators instead predict a spin-orientation-dependent circulation,

$\gamma^\mu(i\hbar\partial_\mu)\psi = mc\,\psi,$08

which can vanish for a relativistic paraxial electron beam with spin parallel to the propagation direction. This contradistinction has been proposed as an experimentally testable way to address the long-standing question of what the proper relativistic observables are for electron vortex beams (Han et al., 2019).

In that sense, REVB theory spans several mutually connected levels: exact Dirac eigensolutions, covariant packet dynamics, Berry-phase and monopole descriptions of angular momentum transfer, field-dressed Volkov–Bessel propagation, and experimentally accessible observables in microscopy, scattering, and strong-field settings. The common structure across these formulations is a relativistic spinorial vortex state whose angular-momentum content, current distribution, and field response cannot in general be reduced to a scalar vortex with a relativistic dispersion.