Toroidal Moments Probed by Electron Beams

  • Nahid TalebiEmail author
Part of the Springer Series in Optical Sciences book series (SSOS, volume 228)


Dipole selection rules underpin much of our understanding of the characterization of matter and its interaction with external radiation. However, there are several examples where these selection rules simply break down and a more sophisticated knowledge of matter becomes necessary. An example, which is becoming increasingly more fascinating, is macroscopic toroidization (density of toroidal dipoles), which is a direct consequence of retardation. In fact, unlike the classical family of electric and magnetic multipoles that are outcomes of the Taylor expansion of the electromagnetic potentials and sources, toroidal dipoles are obtained by the decomposition of moment tensors. This chapter aims to discuss the fundamental and practical aspects of toroidal multipolar moments in electrodynamics, from their emergence in the expansion set, the electromagnetic field associated with them, and the unique characteristics of their interaction with external radiation and other moments to the recent attempts to realize pronounced toroidal resonances in smart configurations of meta-molecules (Talebi et al. in Nanophotonics 7:93, 2018, [1]). In particular, we outline our work in designing oligomeric meta-molecules that purely support toroidal moments within a frequency range using the Babinet’s principle and duality of electromagnetics theory. We further discuss the radiation and coupling of toroidal moments in individual and merged oligomeric systems and experimentally probe those moments using electron beams.


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Authors and Affiliations

  1. 1.Stuttgart Center for Electron Microscopy (StEM)Max Planck Institute for Solid State ResearchStuttgartGermany
  2. 2.Institute of Experimental and Applied PhysicsChristian-Albrechts University in KielKielGermany

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