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

Abstract

A quantum state of an electron influences its electromagnetic field. If a spatial profile of the electron wave packet is not Gaussian, the particle may acquire additional intrinsic multipole moments, which alter its field, especially at small distances. Here the fields of a vortex electron with orbital angular momentum ℓ are obtained in a form of a multipole expansion with an electric quadrupole term kept by using the generalized (non-paraxial) Laguerre-Gaussian beams. The quadrupole contribution arises beyond a paraxial approximation, is linearly enhanced for highly twisted packets with |ℓ| ≫ 1, and can be important for the interactions of twisted beams with bulk matter and artificial structures. Moreover, this term results in an azimuthal asymmetry of the magnetic field in a rest frame of the electron, which appears thanks to the spreading of the packet with time. Thus, somewhat contrary to physical intuition, the spreading may enhance non-paraxial phenomena. For the available electron beams, this asymmetry can in principle be reliably detected, which would be the first experimental evidence of a non-paraxial effect with the vortex electrons.

The passage of a vortex electron over an inclined grating

“…Its electromagnetic fields are a sum of those of the charge e and of the electric quadrupole moment Q αβ , (5) (and of the magnetic moment µ in the case of vortex electrons). The fields in cylindrical coordinates in the rest frame were calculated in [11]. In our problem, we prefer to use the Cartesian coordinates:…”

“…In the works [9,10], the generalized LG packet was used to describe a vortex electron with an electric quadrupole moment [11,25] Q…”

“…To calculate the surface current, we need only the electric fields. In the Cartesian coordinates x , y , z aligned with the vector of the mean velocity of the wave packet, the electric field of the quadrupole moment can be calculated as follows [11] :…”

“…We consider the non-paraxial regime of emission since the corresponding corrections be-come important already at moderate values of orbital angular momentum (OAM) of vortex electrons [9,11]. A vortex electron carries an orbital angular momentum with respect to the propagation axis.…”

“…This allows us to apply the method of generalized surface currents [18][19][20], with incident fields obtained by planar rotation (see Fig. 1) of the electromagnetic fields [11]. Calculations of the radiation fields in the wave zone in section VI B and the spectral and angular distribution of the radiation in section III follow the line of [10].…”