The toroidal dipole is a localized electromagnetic excitation, distinct from the magnetic and electric dipoles. While the electric dipole can be understood as a pair of opposite charges and the magnetic dipole as a current loop, the toroidal dipole corresponds to currents flowing on the surface of a torus. Toroidal dipoles provide physically significant contributions to the basic characteristics of matter including absorption, dispersion, and optical activity. Toroidal excitations also exist in free-space as spatially and temporally localized electromagnetic pulses propagating at the speed of light and interacting with matter. We review recent experimental observations of resonant toroidal dipole excitations in metamaterials and the discovery of anapoles, non-radiating current configurations involving toroidal dipoles. While certain fundamental and practical aspects of toroidal electrodynamics remain open for the moment, we envision that exploitation of toroidal response can have important implications for the fields of photonics, sensing, energy and information. IntroductionThe interactions of electromagnetic radiation with matter underpin some of the most important technologies today -from telecommunications to information processing and data storage; from spectroscopy and imaging to light-assisted manufacturing. Our understanding and description of the electromagnetic properties of matter traditionally involves the concept of electric and magnetic dipoles, as well as their more complex combinations, known as multipoles. Introduced by Maxwell and Lorentz and later refined by Jackson and Landau, this framework, termed the multipole expansion, is central in physics and is being routinely applied in the study of optical, condensed matter, atomic, nuclear phenomena and beyond 1 . Within this framework, electromagnetic media can be represented by a set of point-like multipole sources [2][3][4][5] . The commonly used set of multipoles comprises the electric and magnetic families, which can be represented by oscillating charges and loop currents respectively. Dynamic toroidal multipoles constitute a third independent family of elementary electromagnetic sources, rather than an alternative multipole expansion or higher-order corrections to the conventional electric and magnetic multipoles (see tutorial inset I).Introduced in 1958 by Y. B. Zeldovich, toroidal moments have been considered in systems of toroidal topology (see Fig. 1) and studied in the context of nuclear 6 , atomic 7 , and molecular physics 8 , classical electrodynamics 9,10 , and solid state physics 3 . In the field of electromagnetism, in particular, a number of works have led to the development of a complete theoretical framework for toroidal electrodynamics [11][12][13][14][15] and the prediction of exotic effects, including dynamic non-radiating charge current configurations 10 and non-reciprocal interactions 9 . Following recent experimental observations of toroidal contributions in the response of materials across the electromagnetic spectrum, dyn...

Engaging strongly resonant interactions allows dramatic enhancement of functionalities of many electromagnetic devices. However, resonances can be dampened by Joule and radiation losses. While in many cases Joule losses may be minimized by the choice of constituting materials, controlling radiation losses is often a bigger problem. Recent solutions include the use of coupled radiant and sub-radiant modes yielding narrow asymmetric Fano resonances in a wide range of systems, from defect states in photonic crystals and optical waveguides with mesoscopic ring resonators to nanoscale plasmonic and metamaterial systems exhibiting interference effects akin to electromagnetically-induced transparency. Here we demonstrate theoretically and confirm experimentally a new mechanism of resonant electromagnetic transparency, which yields very narrow isolated symmetric Lorentzian transmission lines in toroidal metamaterials. It exploits the long sought non-trivial non-radiating charge-current excitation based on interfering electric and toroidal dipoles that was first proposed by Afanasiev and Stepanovsky in [J. Phys. A Math. Gen. 28, 4565 (1995)].

The toroidal dipole is a peculiar electromagnetic excitation that can not be presented in terms of standard electric and magnetic multipoles. A static toroidal dipole has been shown to lead to violation of parity in atomic spectra and many other unusual electromagnetic phenomena. The existence of electromagnetic resonances of toroidal nature was experimentally demonstrated only recently, first in the microwave metamaterials, and then at optical frequencies, where they could be important in spectroscopy analysis of a wide class of media with constituents of toroidal symmetry, such as complex organic molecules, fullerenes, bacteriophages, etc. Despite the experimental progress in studying toroidal resonances, no direct link has yet been established between microscopic toroidal excitations and macroscopic scattering characteristics of the medium. To address this essential gap in the electromagnetic theory, we have developed an analytical approach for calculating the transmissivity and reflectivity of thin slabs of materials that exhibit toroidal dipolar excitations.

Toroidal multipoles are the terms missing in the standard multipole expansion; they are usually overlooked due to their relatively weak coupling to the electromagnetic fields. Here, we propose and theoretically study all-dielectric metamaterials of a special class that represent a simple electromagnetic system supporting toroidal dipolar excitations in the THz part of the spectrum. We show that resonant transmission and reflection of such metamaterials is dominated by toroidal dipole scattering, the neglect of which would result in a misunderstanding interpretation of the metamaterials' macroscopic response. Because of the unique field configuration of the toroidal mode, the proposed metamaterials could serve as a platform for sensing or enhancement of light absorption and optical nonlinearities.

Plasmonics is a rapidly developing field at the boundary of physical optics and condensed matter physics. It studies phenomena induced by and associated with surface plasmons-elementary polar excitations bound to surfaces and interfaces of nanostructured good metals. This Roadmap is written collectively by prominent researchers in the field of plasmonics. It encompasses selected aspects of nanoplasmonics. Among them are fundamental aspects such as quantum plasmonics based on quantum-mechanical properties of both underlying materials and plasmons themselves (such as their quantum generator, spaser), plasmonics in novel materials, ultrafast (attosecond) nanoplasmonics, etc. Selected applications of nanoplasmonics are also reflected in this Roadmap, in particular, plasmonic waveguiding, practical applications of plasmonics enabled by novel materials, thermo-plasmonics, plasmonic-induced photochemistry and photo-catalysis. This Roadmap is a concise but authoritative overview of modern plasmonics. It will be of interest to a wide audience of both fundamental physicists and chemists and applied scientists and engineers.

The toroidal dipole is a localized electromagnetic excitation independent from the familiar magnetic and electric dipoles. It corresponds to currents flowing along minor loops of a torus. Interference of radiating induced toroidal and electric dipoles leads to anapole, a nonradiating charge-current configuration. Interactions of induced toroidal dipoles with electromagnetic waves have recently been observed in artificial media at microwave, terahertz, and optical frequencies. Here, we demonstrate a quasi-planar plasmonic metamaterial, a combination of dumbbell aperture and vertical split-ring resonator, that exhibits transverse toroidal moment and resonant anapole behavior in the optical part of the spectrum upon excitation with a normally incident electromagnetic wave. Our results prove experimentally that toroidal modes and anapole modes can provide distinct and physically significant contributions to the absorption and dispersion of slabs of matter in the optical part of the spectrum in conventional transmission and reflection experiments.

Toroidal multipoles are the subject of growing interest because of their unusual electromagnetic properties different from the electric and magnetic multipoles. In this paper, we present two new related classes of plasmonic metamaterial composed of purposely arranged of four U-shaped split ring resonators (SRRs) that show profound resonant toroidal responses at optical frequencies. The toroidal and magnetic responses were investigated by the finite-element simulations. A phenomenon of reversed toroidal responses at higher and lower resonant frequencies has also been reported between this two related metamaterials which results from the electric and magnetic dipoles interaction. Finally, we propose a physical model based on coupled LC circuits to quantitatively analyze the coupled system of the plasmonic toroidal metamaterials.

A toroidal dipole in metasurfaces provides an alternate approach for the excitation of high-Q resonances. In contrast to conventional multipoles, the toroidal dipole interaction strength depends on the time derivative of the surrounding electric field. A characteristic feature of toroidal dipoles is tightly confined loops of oscillating magnetic field that curl around the fictitious arrow of the toroidal dipole vector.