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A review on toroidal excitations, from static toroidal moments in condensed matter, to dynamic toroidal multipoles demonstrated experimentally with metamaterials. 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 charge-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 excitations can have important implications for the fields of photonics, sensing, energy and information.

Toroidal multipoles are fundamental electromagnetic excitations different from those associated with the familiar charge and magnetic multipoles. They have been held responsible for parity violation in nuclear and particle physics, but direct evidence of their existence in classical electrodynamics has remained elusive. We report on the observation of a resonant electromagnetic response in an artificially engineered medium, or metamaterial, that cannot be attributed to magnetic or charge multipoles and can only be explained by the existence of a toroidal dipole. Our direct experimental evidence of the toroidal response brings attention to the often ignored electromagnetic interactions involving toroidal multipoles, which could be present in naturally occurring systems, especially at the macromolecule level, where toroidal symmetry is ubiquitous.

Using a quasi-static model of a toroidal metamaterial we demonstrate analytically that simultaneous excitation of the magnetic and toroidal dipoles in an array of subwavelength toroidal solenoids results in gyrotropic behaviour resembling conventional optical activity. We derive the polarization eigenstates of this uniaxial chiral toroidal metamaterial and show that such a medium is reciprocal, while the eigenstates are represented by two counter-rotating ellipses, one of which can be used for probing changes of the host permittivity in a manner exclusive to the toroidal metamaterial. We also show that the mechanism of permittivity sensing involving resonant toroidal response is fundamentally different from that, which has been exploited so far under the term ‘toroidal’.

A new definition of the derivative of variable order is given based on the interpolation of derivatives of natural order. For the joint interpolation of a function and its derivative of variable order, interpolation operators of Hermite–Fejér type are constructed in the one-dimensional and multidimensional cases. Upper bounds for the norms of these operators in the one-dimensional and multidimensional periodic Sobolev spaces are obtained. It is shown that, in the one-dimensional case, the norm of this operator is bounded. In the multidimensional case, the upper bound depends on the ratio of the number of nodes for each coordinate.

Now there are more than 30 different definitions of fractional order derivatives, and the number is growing. Some of them are just “mind games”, but others are introduced to solve some serious problems. In this article a new definition of a fractional order derivative is given, which generalizes the formula for differentiating Jacobi polynomials. This makes it possible to build a scale of systems of orthogonal polynomials, the closures of which are Sobolev spaces. Using these derivatives, a fractional order Cauchy singular integro-differential equation is stated. Its unique solvability is proven, and a Galerkin method for its approximate solution is justified: the convergence of the method is proven, and the error estimation is obtained.

Abstract We discuss the difference Schrödinger operator on a line with a periodic, possibly complex, potential. We show that this operator has no eigenvalues. The proof is based on the use of the notion of Bloch solutions introduced by Buslaev and Fedotov for difference equations on a line.

The time derivative of a charge density is linked to a current density by the continuity equation. However, it features only the longitudinal part of a current density, which is known to produce no radiation. This fact usually remains unnoticed and may appear puzzling at first, suggesting that the temporal variation of a charge density should be also irrelevant to radiation. We alleviate the apparent contradiction by showing that the effective longitudinal currents are not spatially confined, even when the time-dependent radiating charge density that generates them is. This enforces the co-existence of the complementary, i.e. transverse, part of the current, which, in turn, gives rise to radiation. We illustrate the necessarily delocalized nature and relevance of longitudinal currents to the emission of electromagnetic waves by a dynamic electric dipole, discussing the practical implications of that for radation in partially conducting condensed matter. More generally, we show how the connection between the longitudinal and transverse currents shapes the structure of the conventional multipole expansion and fuels the ongoing confusion surrounding the charge and toroidal multipoles.

In 2003 Bergman and Stockman introduced the spaser, a quantum amplifier of surface plasmons by stimulated emission of radiation 1 . They argued that by exploiting a metal/dielectric composite medium it should be possible to construct a nanodevice, where a strong coherent field is built up in a spatial region much smaller than the wavelength 1 , 2 . V-shaped metallic structures, combined with semiconductor quantum dots, were discussed as a possible realization of the spaser 1 . Here we introduce a further development of the spaser concept. We show that by combining the metamaterial and spaser ideas one can create a narrow-diversion coherent source of electromagnetic radiation that is fuelled by plasmonic oscillations. We argue that a two-dimensional array of a certain class of plasmonic resonators supporting coherent current excitations with high quality factor can act as a planar source of spatially and temporally coherent radiation, which we term a ‘lasing spaser.’ The ‘spaser’ (surface plasmon amplification by stimulated emission of radiation) is a relatively new and exciting concept analogous to the laser. It involves amplifying specific surface plasmon modes using a nanoscale device. Zheludev and co-workers extend this concept by suggesting that metamaterials could be used to create a lasing spaser, that is, a spaser that can emit light with high spatial coherence.

AbstractA new definition of the fractional derivative based on the interpolation of natural-order derivatives is given. The main advantage of the new definition is the locality of such derivatives. In other words, the value of the derivative at a point does not depend on the domain of the function, in contrast to the cases of Riemann–Liouville and Caputo derivatives. This enables one to construct and justify simple computational methods for solving equations containing such derivatives. Moreover, this definition allows one to generalize the concept of a derivative to the case of differentiation of variable order. The paper consideres a class of equations containing the introduced derivatives. The unique solvability of the initial equations is proved, and the quadrature-difference method for solving them is substantiated. Effective error estimates for approximate solutions are obtained. Theoretical conclusions are confirmed by a numerical solution of a model problem.

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.