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 (non-paraxial) Laguerre-Gaussian beams. The quadrupole contribution arises beyond a paraxial approximation, is linearly enhanced for highly twisted packets with $|\ell| \gg 1$, 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 non-paraxial phenomena. For the available electron beams, this asymmetry can in principle be reliably detected, which would be experimental evidence of a non-paraxial effect with the vortex electrons.Comment: 8 pages; minor improvement

Gaussian and Airy wave packets of massive particles with orbital angular momentum

While wave-packet solutions for relativistic wave equations are oftentimes thought to be approximate (paraxial), we demonstrate that there is a family of such solutions, which are exact, by employing a null-plane (light-cone) variables formalism. A scalar Gaussian wave-packet in transverse plane is generalized so that it acquires a well-defined z-component of the orbital angular momentum (OAM), while may not acquire a typical "doughnut" spatial profile. Such quantum states and beams, in contrast to the Bessel ones, may have an azimuthal-angle-dependent probability density and finite quantum uncertainty of the OAM, which is determined by the packet's width. We construct a well-normalized Airy wave-packet, which can be interpreted as a one-particle state for relativistic massive boson, show that its center moves along the same quasi-classical straight path and, what is more important, spreads with time and distance exactly as a Gaussian wave-packet does, in accordance with the uncertainty principle. It is explained that this fact does not contradict the well-known "non-spreading" feature of the Airy beams. While the effective OAM for such states is zero, its quantum uncertainty (or the beam's OAM bandwidth) is found to be finite, and it depends on the packet's parameters. A link between exact solutions for the Klein-Gordon equation in the null-plane-variables formalism and the approximate ones in the usual approach is indicated, generalizations of these states for a boson in external field of a plane electromagnetic wave are also presented.Comment: 16 pages, 4 figures; some additional comments have been adde

Probing phase of a scattering amplitude beyond the plane-wave approximation

Within a plane-wave approach, a number of scattering events in a collision is insensitive to a general phase of a transition amplitude, although this phase is extremely important for a number of problems, especially in hadronic physics. In reality the particles are better described as wave packets, and here we show that the observables grow dependent upon this phase if one lays aside the simplified plane-wave model. We discuss two methods for probing how the Coulomb- and hadronic phases change with a transferred momentum $t$, either by colliding two beams at a non-vanishing impact-parameter or by employing such novel states as the vortex particles carrying orbital angular momentum or the Airy beams. For electron-electron collision, the phase contribution to a cross section can reach the values higher than $10^{-4} -- 10^{-3}$ for well-focused beams with energies of hundreds of keV.Comment: 7 pages, 1 figure; the effect persists for relativistic energie

Relativistic vortex electrons: paraxial versus non-paraxial regimes

A plane-wave approximation in particle physics implies that a width of a massive wave packet $\sigma_{\perp}$ is much larger than its Compton wavelength $\lambda_c = \hbar/mc$. For Gaussian beams or for packets with the non-singular phases (say, the Airy beams), corrections to this approximation are attenuated as $\lambda_c^2/\sigma_{\perp}^2 \ll 1$ and usually negligible. Here we show that this situation drastically changes for particles with the phase vortices associated with an orbital angular momentum $\ell\hbar$. For highly twisted beams with $|\ell| \gg 1$, the non-paraxial corrections get $|\ell|$ times enhanced and $|\ell|$ 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 the 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.Comment: Minor changes compared to v

Scattering of wave packets with phases

A general problem of 2→Nf scattering is addressed with all the states being wave packets with arbitrary phases. Depending on these phases, one deals with coherent states in (3+1) D, vortex particles with orbital angular momentum, the Airy beams, and their generalizations. A method is developed in which a number of events represents a functional of the Wigner functions of such states. Using width of a packet σp/⟨p⟩ as a small parameter, the Wigner functions, the number of events, and a cross section are represented as power series in this parameter, the first non-vanishing corrections to their plane-wave expressions are derived, and generalizations for beams are made. Although in this regime the Wigner functions turn out to be everywhere positive, the cross section develops new specifically quantum features, inaccessible in the plane-wave approximation. Among them is dependence on an impact parameter between the beams, on phases of the incoming states, and on a phase of the scattering amplitude. A model-independent analysis of these effects is made. Two ways of measuring how a Coulomb phase and a hadronic one change with a transferred momentum t are discussed

Diffraction radiation from a screen of finite conductivity

An exact solution has been found for the problem of diffraction radiation appearing when a charged particle moves perpendicularly to a thin finite screen having arbitrary conductivity and frequency dispersion. Expressions describing the Diffraction and Cherenkov emission mechanisms have been obtained for the spectral-angular forward and backward radiation densities.Comment: 6 pages, 4 figure