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Abstract
The optical anapole state resulting from interference of the electric and toroidal moments is of much interest due to its nonradiating nature. Interference of optical modes supported by a diverse range of Mie-resonant structures has found many applications, such as in biosensors and optical communication. This review provides an overview of the recent progress of anapole states in photonics. After a brief historical background, a complete mathematical description is presented. It allows one to clearly demonstrate and identify the existence of anapole states and highlight their fundamental properties. Then, we focus on the excitation of anapoles in photonics and discuss the relation to other states, such as bound states in the continuum. Finally, we discuss a series of advances that uncover the anapole potential in various applications, from nonlinear photonics and lasing to optical communication and sensing.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Classical electrodynamics is based on Maxwell’s equations, which to date are capable of describing all sorts of phenomena from radio waves to the nanoscale. They are robust working tools allowing to design of reliable devices in numerous applications. It is still remarkable how accurate they describe electromagnetic wave propagation in various media and a broad range of boundary constraints. It includes conventional natural media and artificial ones like metamaterials and metasurfaces with unusual optical properties, which are becoming "a new normal" nowadays and have found more and more practical applications recently. Maxwell’s equations can accurately describe even the optical properties of atomically thin 2D-type materials. For the past 150 years, there is no doubt in their validity in a wide range of scales.
Nevertheless, recently there was some spark of interest in the existence of so-called nonradiating sources of electromagnetic radiations. It has a long history and is even linked to the origin of quantum mechanics. Indeed, in the classical picture, described by Maxwell’s equations, electromagnetic waves are generated by oscillating (i.e. accelerating / decelerating) electric charges. It implies that a moving electric charge confined in a finite volume should produce electromagnetic radiation. The stability of atoms, where it is assumed that electrons are moving in closed orbits, forced Niels Bohr to introduce his model, which somehow contradicts Maxwell’s equations. Apparently, it sets a lower limit of the validity of Maxwell’s equations. Since then, there have been numerous attempts to explore various solutions of Maxwell’s equations to demonstrate the existence of nonradiative sources. Remarkably, such attempts were successful. One of the simplest examples is the radial oscillation of a charged spherical shell. It produces longitudinal excitations, which do not produce any far-field radiation. Nevertheless, there are many other interesting examples, which can be found in Refs. [1–7]. Importantly, Devaney and Wolf [1] have derived a set of general theorems about the nonraridating sources. One of the essential characteristics of such nonradiating current sources is the vanishing Fourier transform of the transverse part of the current on the light cone since only that part of the current contributes to the generated far-field electromagnetic radiation. The existence of nonradiating sources is germane to the inverse source problem, trying to find the sources given the observed radiated field. Thus, the solution cannot be unique since adding any nonradiating source will lead to the same radiated field. Uniqueness may be achieved by requiring the solution to have a minimum energy [7].
Recently, it was theoretically suggested and experimentally demonstrated that subwavelength structures can support electromagnetic excitations producing zero scattering to the far-field. Although it might inspire one to design an "invisibility cloak", it is still far from reality [8]. But the physics behind it is quite intriguing. The details will be presented in the next section. It is helpful to analyse radiation/scattering properties by using multipole decomposition. Each multipole can be considered as a channel for energy to flow in and out of the structure. In general, they differ from eigenstates and offer alternative descriptions of the light-matter interaction. Different multipoles can be classified based on their symmetry properties, identifying each separate channel uniquely. The lowest order multipoles are electric and magnetic dipoles. In the long-wavelength limit, they correspond to the Rayleigh regime.
Depending on the geometry, both dipoles can be resonantly enhanced in plasmonic or dielectric structures. Alternatively, it was shown, that arbitrary multiple order scattering can be suppressed, producing zero energy in a given channel. Such states became known as anapoles, which can be translated from Greek as "no poles" for obvious reasons. Originally, they were introduced by Ya.B. Zeldovich in nuclear physics to explain the parity violation in the atomic nucleus in weak interactions [9]. The peculiarity of anapole states is that despite the absence of the scattering in a given channel, there is "trapped" excitation inside the structure. Thus, anapoles correspond to a particular class of nonradiating sources. The existence of anapoles can be explained based on the superposition principle, where two nontrivial near-field distributions produce a contribution to the same scattering channel and annihilate each other in the far-field, see Fig. 1. In the case of vanishing electric dipole scattering, the corresponding anapole state was explicitly shown to be a superposition of the Cartesian electric dipole and out-of-phase toroidal dipole [10] contributions, see Fig. 1. The latter corresponds to the poloidal currents configurations having the same symmetry as an electric dipole in the far-field, allowing for the destructive interference in all directions (see next Section for more details).
Fig. 1. (a) Schematic illustration of nonradiating anapole state as out-of-phase superposition of electric and toroidal dipoles. Reprinted with permission from Springer Nature [20] CC BY; (b) Spherical and Cartesian multipole decomposition of a dielectric sphere made with refractive index $n=4$ showing vanishing electric dipole scattering coefficient, that corresponds to anapole state. Reprinted with permission from Springer Nature [23] CC BY.
Recently, there was much interest in the research of anapole states in various nanoscale structures due to their unique optical properties. It includes selective modes excitation, enhanced harmonics generation in nonlinear structures, excitation of hybrid [11–13] and higher-order anapoles [14,15], etc. There are several excellent reviews covering their basic properties, for example, Refs [16–22]. In this review, we aim to cover gaps and focus on some specific aspects and applications of anapoles. It includes the relation of anapoles to bound-states-in-the-continuum (BICs), light emission, and nonlinear phenomena.
2. Multipole expansion and anapole (non-radiating source)
For the introduction of the anapole concept, let’s consider the electric field ${\mathbf E}$ and magnetic induction ${\mathbf B}$ generated by a time-harmonic electric current with density ${\mathbf j}({\mathbf r})\exp (-i\omega t)$ (where $\omega$ is the field angular frequency) locally distributed in free space. The spatial parts of generated electromagnetic field satisfy the macroscopic Maxwell equations [24]
In the framework of Green tensor formalism [24], the solution of the system (1)-(4) outside the source region can be presented in the integral form [24,27]:
The far-field radiated power ${\rm d}P_{\rm rad}$ into the solid angle ${\rm d}\Omega =\sin \theta \:{\rm d}\varphi {\rm d}\theta$ is determined by the time-averaged Poynting vector [30] so that
Assume that contributions of all multipole terms, except the first electric term in (20), are negligibly small, then we can write
where ${\mathbf D}$ is determined by the superposition of the ED and TD which at certain conditions can result in the excitation of anapole state [23] for whichNote that the radiated fields and power for the case when a current source is located in homogeneous surrounding medium with relative dielectric constant $\varepsilon _s$ have been considered elsewhere [15,25].
3. "Zoo" of anapole states
3.1 Nonradiating states in a single particle
The previous Section, based on the Cartesian multipole decomposition, showed how higher-order moments of the induced current inside an object could contribute to the lower-order multipole radiation channels. Importantly, it allows us to explicitly show the origin of the vanishing radiation in a given channel due to the destructive interference of several contributions. In the case of the electric dipole anapole, these contributions are electric and toroidal dipoles ones, which have an identical amplitude but out-of-phase radiation pattern.
It turns out that similar analysis can be generalised beyond the electric dipole mode and is equally applicable to magnetic dipole and even higher-order multipoles. Although, it becomes harder to explicitly show nontrivial contributions leading to radiation cancellation in a given channel. It requires taking into account higher-order current moments, and the math immediately becomes quite cumbersome. Recently, there were some examples to explicitly derive them, see, for example, Ref. [15]. Thus, we come to the question - are there other alternatives to find and demonstrate the existence of anapoles in nanophotonics?
Fortunately, there is another robust method. It is based on using spherical multipole decomposition. Since spherical multipoles (multipole fields) form an orthogonal and complete set, they can be used to uniquely decompose an arbitrary vector field. It is also known as T-matrix method in application to the scattered field. Spherical multipoles moments (and associated scattering coefficients) can be of electric or magnetic origin [29]. Importantly, due to mutual orthogonality, each spherical multipole corresponds to an independent radiation channel for energy flow. In this case, each channel is defined by a specific spatial symmetry property of the electric and magnetic fields. One of the successful applications of the spherical multipoles is light scattering by spherical objects, which allows for the exact analytical solution and is known as Mie theory [30,33], because explicit analytical treatment provides a deeper insight into the origin of the anapoles.
The natural question arises - what is the relation between the Cartesian and spherical multipoles? Each spherical multipole can be expanded in a series of Cartesian multipoles, which produce the far-field scattering profiles of the same symmetry. As was mentioned above, it implies that even higher-order Cartesian moments of current can contribute to the lower order spherical multipoles [36–38]. The more rigorous explanation is based on irreducible tensor representation [15], but it is beyond the scope of this review.
There are robust methods of how to calculate spherical multipole decomposition based on either scattered field or the induced current inside the object, which produce the same results [29,36]. And now, different types of anapoles correspond to the vanishing spherical scattering multipoles. In other words, we can define anapoles as zero spherical scattering coefficients. In the case of the spherical particles, they can be found analytically, based on the Mie theory. For arbitrarily shaped objects, we should rely on the numerical approach. Although the spherical multipoles allow us to find various types of anapoles, they do not provide the explicit origin of the destructive interference of the nontrivial contributions. That is where the Cartesian multipoles should be used. At the same time, it opens new possibilities to construct various novel types of nonradiative sources, which are yet to be discovered.
Anapole states were observed during the last decade in several plasmonic and dielectric structures. It appears that dielectric particles allow finding anapole states and associated vanishing scattering in a particular radiation channel more easily. In particular, electric, magnetic, hybrid, and even higher-order anapoles were predicted and experimentally observed in a single particle from visible to microwave frequency range [11,14,23,39]. It demonstrates the versatility and omnipresence of anapole states in nanophotonics.
Anapole excitation in a simple metal configuration, however, has been recently realized through coupling an individual gold nanoparticle to a gold film that allows the excitation of electric dipole and higher-order modes [40]. A dielectric $\text {Al}_{2}\text {O}_{3}$ spacer mediates the coupling between higher modes of gold nanoparticle (in forms of circular or hexagonal nanoantenna) and its mirror-induced plasmon modes in the gold film to produce plasmon-induced toroidal resonances. The anapole states can also be achieved in metallic metamolecules via increasing the coupling strength between Cartesian electric dipole and toroidal dipole moments of the system, owing to introducing of a gain material for compensation of ohmic losses [41].
From theoretical analysis and fabrication perspectives, a finite cylinder, made of high refractive index materials, appears to be one of the most appealing geometry. By tuning the height to radius aspect ratio, it is possible to find various types of anapole states with electromagnetic fields confined inside the particle, see Fig. 2(a).
Fig. 2. (a) Anapole excitation of a dielectric disk showing multipole decomposition and electric and magnetic field distributions. Reprinted with permission from Optica Publishing Group [34] CC BY; (b) Near-field interaction of two anapole states in dielectric disks and various optical waveguides for light routing. Reprinted with permission from MDPI [35] CC BY.
The peculiarity of a cylindrical structure is that anapole mode can become dominant in the scattering profile. It allows observing strong suppression of the total scattering, rendering objects almost "invisible". At the same time, there is strong field enhancement inside the object. The reactive part of the electromagnetic energy is confined inside and in the near-field of the particle. It allows inducing near-field interaction between two anapole states, see Fig. 2(b). This principle was used to design strongly confined waveguide structures for light routing [35,42] and highly efficient wireless power transfer devices made of dielectric cylinders [43].
Trapping electromagnetic excitation inside the object and suppressing the energy leakage leads to a number of nonlinear effects, like the most efficient nonlinear harmonics generation, lower lasing threshold, etc., discussed in the following Sections.
3.2 Excitation of pure anapole states
One of the first observations of anapoles were made under plane wave illumination. Due to reciprocity, it is not the optimal way to excite such nonradiating states. Indeed, plane wave type illumination implies that it is coming from the far-field. Thus, the reciprocity dictates that any excitation inside the scattering object should eventually radiate into the far-field as well. It implies that not only anapole but other states should be excited, which provide open channels for energy radiation. These channels might hinder the anapole observation, although they will be present anyway. There were different attempts to modify scattering object geometry to minimise the contribution of these open channels, like tuning the aspect ratio of the cylindrical particles. However, it is impossible to eliminate them entirely.
It poses several questions - do pure anapoles exist (meaning, can they be measured and actually observed?) Moreover, if ’yes’, how to efficiently excite them? The answer to the first question is positive (see the previous Sections). At the same time, the second question deserves further investigation. Recently, there were some attempts to explore so-called structured light illumination to excite pure anapole states efficiently. In particular, it was shown that radially polarized vortex beams can excite pure electric type anapole states [45], see Fig. 3(a-c), while azimuthally polarized vortex beams are ideal for excitation of magnetic type anapole states [46], see Fig. 3(d). Moreover, other exotic light pulses can be used to selectively excite anapole [48–52] or toroidal modes [44,47,53,54]. It opens a new way to generate arbitrary field profiles, including nonradiating ones, in a finite volume, see Fig. 3(e-g). In general, there are a lot of opportunities to efficiently excite the anapoles.
Fig. 3. (a) Selective excitation of electric multipoles in a dielectric spherical particle by focused radially polarised beam. Reprinted from [44], with the permission of AIP Publishing; (b) Ideal electric anapole excitation by two counter-propagating radially polarized beams and (c) corresponding electric field distributions in the focal region (from left to right: incident field, total field, and scattered filed, respectively). Reprinted with permission from Optica Publishing Group [45] CC BY; (d) similar results for magnetic anapole excitation by azimuthally polarised beams. Reprinted with permission from Beilstein-Institut [46] CC BY; (e) free-space propagation of the doughnut light pulse. Reprinted with permission from Optica Publishing Group [47] CC BY; (f) spherical and (g) Cartesian multipole decomposition of the doughnut light pulse scattering by dielectric disk. Reprinted with permission from [48] © 2021 by the American Physical Society.
One of the challenges here is how to generate such "structured light" excitations. Spatial light modulators can be used to create certain types of nontrivial light beams. The alternative approach, actually, can be based on the usage of metasurfaces [55,56], which will be specifically designed to generate a light beam with the predefined properties. Recent works show some progress in this direction, but the efficiency still should be increased.
There is another way to excite pure anapole states - by using localized sources, like dipole antennas [57,58]. Indeed, it is possible to design dielectric structures which will produce no radiation, which is excited by rod or loop type antennas, see Fig. 4. The careful analysis reveals that they are nothing else as electric and magnetic anapole states [57]. It again brings an analogy with "classical atoms", where the confined electromagnetic excitation does not produce any radiation to the far-field. The whole excitation is confined in the near-field only. Such structures became known as "meta-atoms" - classical electromagnetic analogues of real atoms.
Fig. 4. (a) Experimental setup for the radiation pattern measurements of the nonradiating (anapole meta-atom) electric source prototype based on a dielectric hollow disk ($\epsilon = 235$, $\tan \delta =2.7\times 10^{-3}$, $R_{out}=6.4$ mm, $R_{in}=1.5$ mm, $h=4.24$ mm) that tightly attached to a copper plate (60x60 mm2) and excited by copper monopole antenna (L/2=18 mm, wire diameter=1 mm) placed in the center of the hollow disk. (b) The measured spectra of the transmittance. The left and right insets show the measured and simulated radiation patterns of the nonradiating source at the maximum and minimum of the transmittance, respectively. (c) The measured and simulated vertical component of the near electric field distributions at 1.5 cm above the nonradiating source prototype. The field amplitudes are normalized to their maximum. Reprinted with permission from [57] © 2021 by the American Physical Society.
4. Anapole and bound state in the continuum
A similar concept to anapole is dark mode without radiation, which is also referred to as bound state in the continuum (BIC) or embedded state [59–61]. Different from photonic crystal cavity mode localized inside the photonic bandgap, BIC corresponds to a leaky mode with infinite Q-factor or zero radiative decay rate that lies within the continuum background spectrum [59,60]. This intriguing concept was first proposed in the context of quantum mechanics by von Neumann and Wigner through constructing an artificial potential in 1929 [62]. Later, it was demonstrated as a universal phenomenon in waves and has been implemented in water wave [63], acoustic wave [64–66], and electromagnetic waves [67]. Note that ideal BICs only exist in a periodic structure with at least one dimension extending to infinity. To date, four types of BICs, including symmetry-protected BICs [68], Fabry-Perot BICs, Friedrich-Wintgen BICs [64,65,69], and accidental BICs [67], have been demonstrated in theory and experiments. For instance, 1D dielectric metagrating, 2D metasurface and photonic crystal slab support many symmetries protected BICs due to their space symmetry. Figure 5(a) shows the electric field distribution of two represented symmetry protected BICs in 1D circular grating structures [70]. Obviously, both modes show an anti-symmetric profile with respect to the y axis. Such types of BICs were experimentally demonstrated by Lee et al. [68]. As symmetry protected BICs happen at $\Gamma$ point in momentum space, they cannot be excited at normal incidence. They successfully realized a high-Q resonance up to $10^{4}$ in experiments by applying oblique incidence. These BICs also could be converted into quasi-BIC (QBIC) through introducing the in-plane broken symmetry which may be associated with weak bianisotropy [71,72]. A recent theoretical study has found that the Q-factor is inverse proportional to the perturbation square [73]. This suggests a convenient way to engineer the Q-factor of QBICs [72]. A general strategy for the realization of dielectric metasurfaces supporting electric and magnetic QBICs located at the same spectral position has been also recently suggested [74]. Another so-called accidental BICs was theoretically proposed and experimentally demonstrated in a photonic crystal slab [67], as seen in Fig. 5(b-c). In contrast to symmetry protected BIC occurring at $\Gamma$ point, such accidental BICs exist at some isolated points in the first Brillouin zone. Their formation is attributed to the destructive interference between different radiation channels. More interestingly, Zhen et al. [75] reveal that each BIC is linked to the polarization singularity and carries topological charges .
Fig. 5. Anapole and BICs. (a) Electric field distribution $E_z$ for two symmetry-protected BICs $TE_{11}$ and $TE_{21}$. Reprinted with permission from [70] © 2014 by the American Physical Society. (b) Q-factor as a function of $k_x$ along $\Gamma$-X. (c) Eigenfield distribution $E_z$ for the accidental BIC. Reprinted with permission from [Springer Nature] [Nature] [67] © 2013. (d) Real parts of eigenvalues for modes $TE_{34}$ and $TE_{52}$ vs size ratio R. (e) Q-factors of modes $TE_{34}$ and $TE_{52}$ vs size ratio R. Reproduced with permission from SPIE [78] CC BY. (f) Anapole field distribution in a dielectric sphere. (g) Embedded eigenstate field distribution for a layered metallo-dielectric sphere. Reprinted with permission from [81] © 2019 American Chemical Society. (h) Schematic drawing of dielectric cuboid with an air hole in the center. (i) Scattering spectrum and multipole decomposition for the structure shown in (h). (j) Transmittance for metasurface with different artificial loss. Reprinted with permission from Optica Publishing Group [82] CC BY.
Although single or multiple nanostructures are not able to support ideal BICs, high-Q QBIC mode could be constructed by carefully tuning the structural parameters. Rybin et al. [76] demonstrated that a single high-index dielectric cylinder supports QBIC. They attributed the formation to the strong mode coupling between Fabry-Perot resonance and Mie resonances. Bogdanov et al. [77] presented an in-depth study on the fundamental physics behind such QBICs’ formation. They found that QBICs are induced by the Friderich-Wintgen destructive interference. A multipole decomposition analysis indicates that QBIC formation is always accompanied by a remarkable change in the far-field pattern. They also established a direct link between the formation of high-Q resonance and the Fano lineshape in the scattering spectrum. Later, Huang et al. [78] unveil the universal recipe of finding out a series of high-Q QBICs in a single dielectric nanocavity, including rectangular wire, disk, and cuboid. In general, such QBIC could be easily explored by constructing a pair of high-Q and low-Q leaky modes that shows crossing or avoid-crossing feature. Figure 5(d-e) shows an example of building such a BIC in a single rectangular nanowire for the transverse electric case. The refractive index of a nanowire is set as $n=4$. It can be observed that the Q-factors for modes $TE_{34}$ and $TE_{52}$ are boosted and suppressed to maximum value 1309 and minimum value 33, respectively at critical size ratio, around which the avoid crossing has been found in the real part of eigenvalues. QBICs are correlated with the suppressed radiation in the corresponding dominant channels. An alternative explanation is that the electric field is quenched to the minimum at momentum space by performing Fourier transformation on the eigen-field. Experimental results show that even a single subwavelength Si nanowire supports high-Q resonance up to 380. The high-Q nature of QBIC ensures the extreme field confinement inside the structures and thus hold great promise in enhanced light-matter interaction, such as lasing and enhanced nonlinear harmonic generation [79,80].
Although anapole and BIC share some similar features, it is necessary to point out that they are fundamentally different in the physical nature [83]. Anapole represents a nonradiating state where far-field scattering is suppressed but strongly localized near the object field. In the scattering spectrum, anapole corresponds to the scattering dip. It is not necessary to be a mode. Usually, the resonant frequency of a real mode happens between the scattering maximum and minimum. Ideally, anapole corresponds to a scattering zero state if the minimum value is reduced to zero. However, it is well established that eigenmodes correspond to a pole of the scattering matrix. Anapole can become a real mode only when the scattering spectrum displays the Lorentz profile. Anapole state could be accessed by either the plane wave or structure beam excitation. On the contrary, an ideal BIC could not be excited by any external incidence as it is perfectly decoupled from the background. It could be detected only after BIC is transformed into QBIC with a finite-Q factor. Monticone et al. perform a systematic study on distinguishing the anapole, and BICs [81]. Because anapole is not an eigenmode of an open cavity, the anapole field distribution cannot be sustained if no external incidence is applied. However, BIC or embedded state is a special leaky mode that can sustain even without external excitation. This essential difference is perfectly reflected in Fig. 5(f-g). Also, they apply the Lorentz reciprocity theorem to explain why the BIC cannot be excited by the external excitation. Note that anapole and BICs are not completely independent of each other. In the pioneering work of anapole, Miroshnichenko et al. has demonstrated that a single nanodisk with suitable aspect ratio supports anapoles [23]. Later, Yang et al. [84] found that anapole can enhance electric field 3 orders of magnitude through introducing a rectangular air slit inside the nanodisk. Two groups independently demonstrated that radiationless anapole state could be transformed into QBIC or even BIC [82,85]. For example, Algorri et al. [82] first demonstrated that anapole still sustains in a cuboid structure with a square hole in the center, as schematically shown in Fig. 5(h) and demonstrated in Fig. 5(i). By arranging such a structure into array and carefully tuning the structure parameters, the Q-factor of such modes could be significantly improved to several million and eventually become BICs, as confirmed in Fig. 5(j). However, we would like to emphasize that such a BIC is no longer an anapole but dominated by the toroidal dipole. Lattice invisible effect (lattice anapole state), when the light goes through the metasurface without amplitude and phase perturbations with the excitation of nanoparticles’ multipole moments, has been recently theoretically demonstrated for metasurfaces composed of cubic [86] and spherical [87,88] silicon nanoparticles. It has been found that this effect is realized due to destructive interference between the fields generated by the basic multipole moments of nanoparticles in the backward and forward directions.
5. Applications
All-dielectric metamaterials can reproduce many unique phenomena that are observed in plasmonic materials due to the resonant behavior of electromagnetic waves in dielectric nanoantennas. Furthermore, as opposed to plasmonic-driven phenomena predominantly resulting from the electric field localization, magnetic resonances in dielectric nanoparticles provide an additional degree of freedom for manipulating light-matter interactions. Therefore, all-dielectric resonant photonics offer unexplored new effects and functionalities. This section discusses recent progress in the applications of a fast-growing direction of all-dielectric photonics, i.e. the excitation of dynamic anapole states and their exploitation for exploring new functionalities.
5.1 Enhanced light emission and nonlinear harmonic generation
Anapole-driven cancellation of far-field radiation can be used for the near-field enhancement and the energy concentration at the subwavelength scale that can be utilized for boosting nonlinear optical processes [89]. It has been shown that the near field enhancement at anapole state can be further boosted by geometry optimization. As an example, introducing an air slot in an individual Si nanodisk at its anapole state has led to a strong electric field within the slot volume with intensity enhancement of $|E/E_0 |^{2}$ $\approx$ $10^{3}$ that is two orders of magnitude higher than the intensity enhancement at anapole state in a conventional Si nanodisk [84]. The advantage of such an air hole is that adjacent emitters such as molecules have access to the electric hotspot that would be useful for nanophotonic applications like single-particle enhanced spectroscopies. A similar strategy has been used for a $\Phi$-shaped Si nanoparticle at the near-infrared spectrum [34].
Totero Gongora and collaborators [90] have proposed that the anapole-driven field localization can be applied for designing a novel type of nanolasers. In their study, an InGaAs nanodisk optimized to support anapole state was optically pumped to emit at the wavelength of 948 nm. Unlike typical nanoparticle lasers, the emission of this type of laser is confined in the near field and can be efficiently coupled to a nearby waveguide. More recently, room-temperature lasing has been shown in the anapole metasurface of split-nanodisk InGaAsP resonators [91]. The recent advances in the field of lasers based on radiationless states, including anapole and BIC, have been reviewed in Ref. [83] and more recent Ref. [42].
The first efficient harmonic conversion based on anapole excitation was exhibited by the efficient third-harmonic generation (THG) in individual Ge nanodisks in 2016 [92] as shown in Fig. 6(a). TH conversion at the wavelength of 550 nm with the efficiency of 0.0001$\%$ has been obtained from fundamental pumping (1650 nm). This TH conversion was four orders of magnitude larger than that of a Ge reference thin film. Soon after, it was demonstrated that a supporting metal structure can further boost the conversion efficiency at anapole state. Shibanuma et al. [93] showed that the anapole-driven field enhancement in a Si nanodisk supported by a gold nanoring could increase third harmonic conversion efficiency to 0.007$\%$. In another study, a virtual image of an individual Si nanodisk at anapole state generated by a supporting gold film beneath the Si nanodisk was used to produce a substantial near-field enhancement to facilitate the nonlinear processes [94]. Figure 6(b) shows THG for the Si resonator on a gold mirror (ROM) and insulator (ROI) as a function of total pump power. For the same pump configuration, the TH emission of the anapole state was enhanced up to two orders of magnitude by the gold mirror. This enhancement resulted in an unprecedented total TH conversion efficiency value, 0.01$\%$. More recently, mirror-induced anapole has been developed using two metal mirrors sandwiching a Si strip to boost both electric anapole and magnetic anapole at the near-infrared [95]. There are also some studies on anapole-assisted second harmonic generation (SHG), including the enhancement of SHG by free-standing III–V nanodisks [96] and AlGaAs nanodimers [97].
Fig. 6. (a) THG in individual Ge nanodisks. Reprinted with permission from [92] © 2016 American Chemical Society. (b) Boosting THG of Si nanodisk by gold mirror. Reproduced with permission from Springer Nature [94] CC BY. (c) SHG in LiNbO$_3$ nanodisk placed on the epsilon-near-zero substrate of alternating silver and SiO$_2$ layers. Reproduced with permission from De Gruyter [50] CC BY. (d) The photothermal nonlinearity is investigated via analyzing the scattering of CW laser light by Si nanodisks. Reproduced with permission from Springer Nature [52] CC BY.
Although anapole states are typically excited in high-index dielectric materials such as Si and Ge, these materials have a narrow transparent spectral region where cavity losses (extinction coefficient) are low. Lithium niobite (LiNbO$_3$) is a material with large nonlinearity and a wide transparent spectral region, but it is not proper for anapole excitation due to its low refractive index. However, it has been shown that low-index materials such as LiNbO$_3$ can be used for anapole excitation if supported by a metallic substrate to produce a large contrast between the index of the substrate and that of nanoantenna [98]. Recently, Li et al. [50] theoretically investigated the second harmonic generation (SHG) in LiNbO$_3$ nanodisk placed on the epsilon-near-zero substrate to increase the index contrast between nanodisk and substrate to excite anapole efficiently. Figure 6(c) illustrates the proposed structure that composed of a LiNbO$_3$ nanodisk on a substrate of alternating silver and SiO$_2$ layers. A strong electric field intensity enhancement of about 35 was obtained at the fundamental wavelength due to the anapole excitation that induced an enhanced SHG with the conversion efficiency of 0.01$\%$. Anapole states’ largely unexplored research direction is the nonlinear responses of dielectric nanoantenna under ultrashort pulse illumination. Pulsed laser illuminations can excite some nonthermal processes [99] that may influence the nonlinear responses of dielectric nanoparticles. Shi et al. [100] experimentally investigated the anapole-enhanced third-harmonic generation by few-cycle laser radiation at the deep-ultraviolet spectral range. The study demonstrated that a few-cycle laser illumination improves the THG emission of amorphous Si disks at anapole state and makes the spectral bandwidth broader from 30 to 34 nm. More recently, sum-frequency generation (SFG) by two pulsed laser beams at a telecommunication frequency of $\omega$, and its duplicate frequency, 2$\omega$, has been reported in individual AlGaAs nanocylinders [101].
Anapole excitation can enhance other nonlinear optical properties of dielectric nanoparticles, for instance the Raman scattering of Si nanoparticles [102]. Furthermore, the influence of anapole state on the nonlinear Kerr effect was studied in GaP metasurfaces [103]. The authors investigated metasurfaces composed of random and ordered GaP nanodisks. Theoretically, they demonstrated that the excitation of anapole state increases the second-order refractive index by order of magnitude. Anapole-assisted field enhancement has been recently applied to boost photothermal nonlinearity [52]. The photothermal nonlinearity was investigated by analysing the CW laser light scattering by Si nanodisks. The nonlinear dependency of the scattering by individual Si nanodisk on irradiance intensities is shown in Fig. 6(d). The saturation scattering (SS) and the scattering reduction shown in the red shaded area are associated with the anapole excitation that led to a deviation from the linear scattering (red dashed line). The enhanced near field at the anapole state directly increased the light absorption and thus the temperature of Si nanodisk. The temperature rise of Si nanodisk induces a refractive index variation called thermo-optic effect [104,105]. For the intensities higher than $1.3 \times 10^{6} W/cm^{2}$ (the yellow shaded area), the scattering increased due to the transition from anapole state to electric dipole (ED) mode denoted as reverse saturation scatterings (RSS). The lower pictures in Fig. 6(d) shows the dominant modes inside the nanodisk in the three regimes of linear, saturation, and reverse saturation scatterings. The results showed an effective nonlinear refractive index of 0.4 $cm^{2}/MW$, indicating a photothermal nonlinearity that is three orders of magnitude stronger than that of bulk silicon.
5.2 Other applications
Field confinement by anapole excitation is typically conducted in the transparent spectral regions of materials with a high real part of complex refractive index n but a zero-imaginary part k (extinction coefficient) to avoid cavity losses. To apply the field confinement for enhancing light absorption and generating charge carriers, however, cavity losses are needed [72,106] . It has been recently shown that the constructive interference of toroidal dipoles in a hexagonal array of Si nanodisks near the anapole state condition leads to coupling of toroidal dipoles called coupled toroidal dipole modes [107]. The coupled-mode realized strong toroidal dipole (TD) resonances in the nanodisks near the anapole condition (panel (a) of Fig. 7). Enhanced optical field confinement at the coupled toroidal dipole resulted in an optical absorption 30 times larger than an unstructured Si film of the same thickness as seen from the absorptance curves.
Fig. 7. (a) Constructive interference of toroidal dipoles in a hexagonal array of Si nanodisks near anapole state leads to coupling of toroidal dipoles that in turn results in absorption enhancement. Reprinted with permission from [107] © 2020 John Wiley and Sons Inc. (b) The Photocatalysis experiment shows the increased catalytic activity at anapole state. The largest amount of Ag is deposited on the nanodisk of 290 nm that is at anapole state. Reprinted with permission from [108] © 2020 American Chemical Society.
Anapole-assisted absorption enhancement has been recently used for the improvement of photocatalytic activity in dielectric nanoparticles, including oxygen-vacancy-rich TiO$_{2-x}$ nanodisk in the visible spectrum where it naturally has a moderate refractive index (n < 3) and exhibits weak absorption due to small value of k [108]. Figure 7(b) shows the single-particle photocatalysis experiment in which the sample was immersed in AgNO$_3$ solution and then excited by a laser with the wavelength of 532 nm where the nanodisk of 290 nm height supports the anapole state. Dark-filed images were then taken in the air for the sample before and after the excitation that showed a bright spot for the disk of 290 nm height, demonstrating the reduction of Ag$^{+}$ ions. SEM image and EDX map of the sample also showed that the largest amount of Ag was deposited on the nanodisk of 290 nm, confirming an increased catalytic activity at the anapole state. More recently, Huettenhofer and colleagues [109] investigated the effect of anapole-anapole interactions on the anapole-assisted absorption in GaP nanodisks that is a typical semiconductor photocatalyst. The results showed that anapole-assisted absorption of an individual amorphous GaP nanodisk increases up to three times due to Rayleigh anomalies [110,111] when embedded in an optimum 2D array of the nanodisks supporting anapole excitations.
Anapole’s unique characteristics, such as enhanced near-fields, are enabling the design of optical sensors and the development of advanced sensing techniques [22,112]. Recently, an anapole-based gold metamaterial incorporated into graphene has been proposed for sensing in THz region [113]. The proposed metamaterial shows a high-Q resonance in THz that can be tuned by the Fermi energy of graphene. Optical sensing techniques based on all-dielectric metaphotonics are also an emerging direction in biomedical monitoring applications [114]. Sabri et al. [115] proposed that anapole excitation in dielectric nanoantennas can be used for biosensing applications due to its interesting properties such as the confined electromagnetic field inside nanoantenna with no strong extension into the surrounding environment.
The transfer of localized electromagnetic energy by the propagation of anapole states along a chain of nanoparticles [35] is expected to develop the currently integrated waveguides and photonic communications. More recently, Huang et al. [116] reported the first experimental evidence of anapole-based ultracompact energy transfer in the long-range using a metachain of subwavelength Si disks at the mid-infrared. The anapole metachain is capable of confining electromagnetic energy to 1/13 of incident light wavelength. It also showed a highly efficient energy transfer at the telecommunication C-band (1550 nm) when introducing a $90^{\rm o}$ bend.
Nonradiating anapole state with enhanced near filed using metamaterials also provides a great platform for other applications such as cloaking [117,118], optical modulation [119], and tunable optical devices [120]. Moreover, anapole state realized in small particle clusters with special symmetry may lead to the development of metasurfaces with new functional properties. For example, it has been demonstrated that the anapole excitation in trimers of disk-shaped particles results in the polarization-independent suppression of reflection with the resonant enhancement of local electromagnetic fields in the metasurface composed of such trimers [121].
6. Conclusion and outlook
In summary, in this review, we presented recent progress on the existence and application of anapole states in nanophotonics. During the last decade, there was a rapid growth of understanding of the physical origin of anapoles based on a proper mathematical description, providing deeper insight into such unusual states. One of the main peculiarities of anapole states associated with the excitation of a nontrivial electromagnetic field configuration is finite volumes producing zero scattering to the far-field. Thus, they can be related to a class of nonradiating sources in electrodynamics in a wide frequency range - from visible to microwave parts of the spectrum. They open new opportunities to enhance the number of linear and nonlinear systems responses.
After providing a consistent description of anapole states, allowing us to properly analyse their unique properties, we focused on their new developments in nanophotonics. In particular, how anapoles can be efficiently excited, their relation to high-Q resonant states, and significantly enhanced nonlinear properties. It certainly does not exhaust all their properties and applications but provide a glimpse on the rapid development of such an exciting topic. It shows how the trapped electromagnetic excitation can improve almost any aspect of various photonic structures. Although, in general, anapole states are not necessary resonant ones, recent publications demonstrate that they can be tuned to become high-Q resonant modes, revealing their full potential in light manipulation in the near-field nanophotonics. Despite being peculiar and controversial states, anapoles find their application in wireless power transfer, nonlinear harmonic generations, and compact light sources for near-field communication and light routing. It is expected that more exciting applications will emerge in the near future, as our understanding and appreciation of anapoles will grow.
Funding
Deutsche Forschungsgemeinschaft (390833453); Australian Research Council (DP200101353).
Disclosures
The authors declare no conflicts of interest.
Data availability
The dataset underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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