Toroidal Moments Probed by Electron Beams

Toroidal moments have recently attracted attentions as an individual class of moments, by expanding the vector potential and decomposing the third component into toroidal moments and magnetic quadrupoles. Toroidal moments sustain peculiar time-space symmetry relations and are known as hybrid modes, which can be only understood by including retardation effects. Here, we outline our recent attempts to realize toroidal moments in void oligomeric meta-molecules. These structures are investigated experimentally using electron energy-loss spectroscopy and numerically using finite-difference time-domain method. We discuss about various toroidal moments supported by these structures, the role of symmetry and topology in excitation of these moments, the radiation from electron-induced toroidal moments resolved by cathodoluminescence spectroscopy, and finally hybridization of toroidal moments in coupled oligomers. Our findings outline the necessity of including toroidal moments within the classes of electromagnetic moments, despite the current discrepancy about their significance.