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

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 $\ell$ are obtained in a form of a multipole expansion with an electric quadrupole term kept by using the generalized (nonparaxial) Laguerre-Gaussian beams. The quadrupole contribution arises beyond a paraxial approximation, is linearly enhanced for highly twisted packets with $\left|\ell \right|\gg 1$, 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 nonparaxial phenomena. For the available electron beams, this asymmetry can in principle be reliably detected, which would be experimental evidence of a nonparaxial effect with the vortex electrons.