Theory and applications of toroidal moments in electrodynamics: their emergence, characteristics, and technological relevance

Abstract

Dipole selection rules underpin much of our understanding in characterization of matter and its interaction with external radiation. However, there are several examples where these selection rules simply break down, for which a more sophisticated knowledge of matter becomes necessary. An example, which is increasingly becoming more fascinating, is macroscopic toroidization (density of toroidal dipoles), which is a direct consequence of retardation. In fact, dissimilar to the classical family of electric and magnetic multipoles, which are outcomes of the Taylor expansion of the electromagnetic potentials and sources, toroidal dipoles are obtained by the decomposition of the moment tensors. This review aims to discuss the fundamental and practical aspects of the toroidal multipolar moments in electrodynamics, from its emergence in the expansion set and the electromagnetic field associated with it, the unique characteristics of their interaction with external radiations and other moments, to the recent attempts to realize pronounced toroidal resonances in smart configurations of meta-molecules. Toroidal moments not only exhibit unique features in theory but also have promising technologically relevant applications, such as data storage, electromagnetic-induced transparency, unique magnetic responses and dichroism.

Experimental Observation of Toroidal Dipole Modes in All‐Dielectric Metasurfaces

“…Remarkably, in this case the toroidal response can be engineered to be sufficiently strong. Traditionally, the toroidal response is sought in metamaterial slabs made of clusters consisting of four or more subwavelength particles, and it is commonly believed that the lateral excitation is optimal for producing a toroidal response . In particular, an appearance of a polar toroidal response under the lateral excitation has been demonstrated for the first time for the microwaves in a metamaterial constructed from an array of clusters possessing four electrically disconnected metallic‐wire loops .…”

“…The practical importance of metasurfaces supporting toroidal dipole moments was discussed extensively over the last decades. It is expected that they can serve as a platform for the efficient light–matter interaction with prospects in applications for data storage and sensing . In particular, it can be realized via suppression of the total scattering due to excitation of nonradiating anapole states .…”

Toroidal Dipole Resonances in All‐Dielectric Oligomer Metasurfaces

“…[2] It is characterized by poloidal currents flowing on the surface of a torus along its meridians. [4] When it comes to the dynamic case, the toroidal dipole, as well as other higherorder toroidal multipoles are neglected in standard multipole expansion, where only electric and magnetic multipoles are considered. [3] The magnetoelectric effects in the context of condensed matter physics associated with toroidal responses may find application potentials in advanced data storage devices.…”

All‐Dielectric Silicon Metasurface with Strong Subterahertz Toroidal Dipole Resonance

“…Since decades ago, toroidal modes were employed in physics, for instance, to explain the parity violation of the weak interactions in atomic nuclei, to propose models of stable atoms, or to describe dark matter . In recent years, though, the interest on electromagnetic toroidal moments is rapidly increasing as, apart from the fundamental physical insight, they can be harnessed in numerous applications . Toroidal modes in individual dielectric particles or small clusters have been theoretically proposed for the excitation of toroidal response, cloaking, enhanced absorption, and nanolasing or experimentally demonstrated as nonradiating sources and for the enhancement of nonlinear effects .…”