Optical anapoles

The anapole, a non-radiating charge-current configuration, was recently observed in a variety of artificial materials and nanostructures. We provide a brief overview of this rapidly developing field and discuss implications for spectroscopy, energy materials, electromagnetics, as well as quantum and nonlinear optics.


Introduction
Toroidal electrodynamics, a new chapter of electromagnetic research is currently attracting considerable and growing attention 1,2,3,4 . It includes the study of toroidal multipoles and anapoles (see Fig. 1). The recent surge of interest in toroidal multipoles is driven by the emerging understanding that alongside the well-known electric and magnetic multipoles they are necessary for a complete characterization of the electromagnetic properties of matter 2 . Indeed, while electromagnetic fields in free-space can be fully characterized with transverse electric (TE) and transverse magnetic (TM) multipoles 5 , the characterization of current density requires three multipole series, the electric, magnetic and toroidal multipoles 6,7 (see Fig. 2). The distinctive role of toroidal multipoles is particularly apparent in the optical properties of matter containing large molecules or structural elements of toroidal symmetry and of size comparable to the electromagnetic wavelength. Dynamic toroidal response of metamaterials had been the subject of intense discussions since 2007 8,9 , but the first unambiguous experimental demonstration of dominant toroidal response in matter was recorded in a microwave metamaterial in 2010 1 (see Fig. 3). Subsequently, dynamic toroidal response has been observed in metallic 10,11,12,13,14,15 , plasmonic 16,17,18,19,20,21,22 and dielectric metamaterials 23,24 at frequencies ranging from microwave through to terahertz and up to nearinfrared/visible parts of the spectrum (see Fig. 3). The analysis of transmission, reflection 12 and polarization phenomena 13 in complex molecular systems and metamaterials is incomplete without account of the dynamic toroidal response. Toroidal resonances could play a role in nano-lasers 19 , sensors 25 and data storage devices 3,26 . We shall also note that static toroidal dipoles, also known as 'static anapoles' introduced by Ya. B. Zeldovich in the context of parity violation in nuclear physics 27 , have been observed in magnetism 26 and could be the only allowed electromagnetic form-factor for dark matter candidate particles 28 .
An electric dipole (a pair of oscillating charges) together with a toroidal dipole (oscillating poloidal current on a torus (see Fig. 1)) can form non-radiating charge current configuration, known as 'dynamic anapole' 29,30,31,32 . The anapole state emerges at a particular frequency of oscillations when the fields radiated by the co-located electric and toroidal dipoles cancel each other through destructive interference. Crucially, electric and toroidal dipoles have identical radiation patterns (see Fig. 2), thus the net emission of an anapole is zero. The dynamic anapole, a non-radiating energy "reservoir", has inspired a broad search for anapole excitations in matter (see Fig. 3). Perfect anapoles do not emit or absorb light and therefore cannot be detected by far-field observations. Anapole excitations can only be detected if they are (weakly) coupled to electromagnetic modes interacting with free-space radiation or if they are not perfectly balanced, i.e. the electric dipole emission does not precisely cancel out the toroidal dipole radiation. A slightly off-balance anapole will create a narrow peak in the scattering spectrum. Electromagnetic anapoles were first detected as narrow transmission peaks in the spectra of a microwave metamaterial in 2013 30 . Since then, a number of alternative nanostructures supporting anapoles were discussed theoretically 31,33,34 . Anapole excitations were observed in dielectric nanoparticles 35,36,37,38,39 and in metallic 40 as well as plasmonic metamaterials 41 .

Detection of anapoles
Recent detection of anapole modes in a diverse range of structures is a significant achievement in the field of toroidal electrodynamics that illustrates the independent physical nature of electric and toroidal dipoles despite them having identical far-field radiation pattern (see recent discussion of this subject in refs. 42,43 ). Indeed, although their far-field emission patterns are identical, the electric and toroidal dipoles correspond to entirely different charge and current distributions (see Fig. 1,2). Furthermore, the oscillating vector potential emitted by electric and toroidal dipoles differs in a way that is irremovable through gauge transforms 29 .
The independent physical significance of toroidal and electric dipoles also manifests itself in relativistic electrodynamics. Although fields emitted by electric and toroidal dipoles, when in inertial motion, are identical, linear acceleration of oscillating electric and toroidal dipoles changes the polarization properties of the respective radiated fields in a different way: the Observation of anapole mode in a silicon nano-disk 35 . The colour maps show the electric field distribution in the disk, as mapped experimentally and modelled numerically, indicating the excitation of an anapole mode. The field was mapped using scanning near-field optical microscopy, as shown in the schematic. (d) 2018: Plasmonic metamaterial supporting anapole mode of excitation 41 . Scanning electron microscope image shows a cross-section of the nanostructure. It consists of a dumbbell-perforated section of the gold film with an additional gold split ring resonator below it. The schematic shows the unit cell of the metamaterial, as well as the sketch of the resonant mode, simultaneously supporting electric and toroidal dipoles. absolute value of ellipticity of the toroidal dipole radiation becomes greater than that of electric dipole 44 . The difference in ellipticities Δ diverges along the dipole axis (see Fig. 4). Therefore, an ideal anapole, that is well-balanced to emit no radiation when at rest, will emit light and interact with light, when accelerated.
The effect described above requires extremely high accelerations, and is challenging to observe. However, there is a more accessible way of revealing the difference between the electric and toroidal dipoles. It exploits the difference in coupling of electric and toroidal dipoles to electromagnetic fields in ambient media, and can thus be seen as a form of solvatochromism, a phenomenon of changing the colour of a chemical substance depending on the host-solvent 45   Strongest effect due to ellipticity difference will be observable at small angle relative to dipole axis, where radiated power is still significant and ellipticity difference is large.

Future perspective
The study of anapoles promises some intriguing discoveries. It has been shown that dynamic anapole modes are supported in artificial metamaterials. Could the dynamic anapole be present in organic matter that is often built from molecules with elements of toroidal symmetry such as benzene rings? Indeed, some fullerenes support static anapoles 50 . Moreover, static anapoles have been recently theoretically identified in some cyclic molecules 51 , diatomic molecules 52 and chiral molecules 53 . Interactions between toroidal currents allegedly break reciprocity, which could have implications for energy and information transfer at the molecular level and for the dynamics of chemical and biochemical processes 54 . As anapoles are energy reservoirs with a long lifetime, they could be of considerable importance as qubits for quantum technologies 55 . High quality anapole-related resonances can be used in enhancing nonlinear electromagnetic properties of materials 38,56,57,58 and in sensor applications 34,41 . Matter with high density of anapoles could be an exotic energy storage material from which energy bursts could be released by sudden changes to ambient conditions. Spectroscopy of anapoles presents considerable challenges due to weak coupling to free-space electromagnetic waves, as explained above. However, the use of structured light, most notably space-time non-separable pulses with toroidal topology may help, as they are better suited to drive toroidal excitations than transverse pulses 59 . Alternatively, toroidal and electric dipole constituents of an anapole mode could be engaged with electron beam excitations 60 .