  Focused beams of electrons, which act as both sources, and sensors of
electric fields, can be used to characterise the electric response of complex
photonic systems by locally probing the induced optical near fields. This
functionality can be complemented by embracing the recently developed vortex
electron beams (VEBs), made up of electrons with orbital angular momentum,
which could, in addition, probe induced magnetic near fields. In this work, we
revisit the theoretical description of this technique, dubbed vortex Electron
Energy-Loss Spectroscopy (v-EELS). We map the fundamental, quantum-mechanical
picture of the scattering of the VEB electrons to the intuitive classical
models, which treat the electron beams as a superposition of linear electric
and magnetic currents. We then apply this formalism to characterise the optical
response of dielectric nanoantennas with v-EELS. Our calculations reveal that
VEB electrons probe electric or magnetic modes with different efficiency, which
can be adjusted by changing either beam vorticity or acceleration voltage to
determine the nature of the probed excitations. We also study a
chirally-arranged nanostructure, which in the interaction with electron
vortices produces dichroism in electron energy loss spectra. Our theoretical
work establishes VEBs as versatile probes that could provide information on
optical excitations otherwise inaccessible with conventional electron beams.
