Electrons in free space can carry
quantized orbital
angular momentum (OAM) projected along the direction of propagation.[1] This orbital angular momentum corresponds to
helical wavefronts, or, equivalently, a
phase proportional to the
azimuthal angle.[2] Electron beams with quantized orbital angular momentum are also called electron vortex beams.
Theory
An electron in free space travelling at non-
relativistic speeds, follows the
Schrödinger equation for a
free particle, that is where is the reduced
Planck constant, is the single-electron
wave function, its mass, the position vector, and is time.
This equation is a type of
wave equation and when written in the
Cartesian coordinate system (,,), the solutions are given by a
linear combination of
plane waves, in the form of where is the
linear momentum and is the electron energy, given by the usual
dispersion relation. By measuring the momentum of the electron, its
wave function must collapse and give a particular value. If the energy of the electron beam is selected beforehand, the total momentum (not its directional components) of the electrons is fixed to a certain degree of precision.
When the Schrödinger equation is written in the
cylindrical coordinate system (,,), the solutions are no longer plane waves, but instead are given by
Bessel beams,[2] solutions that are a linear combination of that is, the product of three types of functions: a plane wave with momentum in the -direction, a radial component written as a
Bessel function of the first kind, where is the linear momentum in the radial direction, and finally an azimuthal component written as where (sometimes written ) is the
magnetic quantum number related to the angular momentum in the -direction. Thus, the dispersion relation reads . By azimuthal symmetry, the wave function has the property that is necessarily an
integer, thus is quantized. If a measurement of is performed on an electron with selected energy, as does not depend on , it can give any integer value. It is possible to experimentally
prepare states with non-zero by adding an azimuthal phase to an initial state with ; experimental techniques designed to
measure the orbital angular momentum of a single electron are under development. Simultaneous
measurement of electron energy and orbital angular momentum is allowed because the
Hamiltoniancommutes with the
angular momentum operator related to .
There are a variety of methods to prepare an electron in an orbital angular momentum state. All methods involve an interaction with an
optical element such that the electron acquires an azimuthal phase. The optical element can be material,[3][4][5] magnetostatic,[6] or electrostatic.[7] It is possible to either directly imprint an azimuthal phase, or to imprint an azimuthal phase with a holographic diffraction grating, where grating pattern is defined by the interference of the azimuthal phase and a planar[8] or spherical[9] carrier wave.
Applications
Electron vortex beams have a variety of proposed and demonstrated applications, including for
mapping magnetization,[4][10][11][12] studying chiral molecules and chiral plasmon resonances,[13] and identification of crystal chirality.[14]
Measurement
Interferometric methods borrowed from
light optics also work to determine the orbital angular momentum of free electrons in pure states. Interference with a planar reference wave,[5] diffractive filtering and self-interference[15][16][17] can serve to characterize a prepared electron orbital angular momentum state. In order to measure the orbital angular momentum of a superposition or of the mixed state that results from interaction with an atom or material, a non-interferometric method is necessary. Wavefront flattening,[17][18] transformation of an orbital angular momentum state into a planar wave,[19] or cylindrically symmetric Stern-Gerlach-like measurement[20] is necessary to measure the orbital angular momentum mixed or superposition state.