The control of light propagation and concentration due to surface-plasmon resonances has great implication for the fundamentals and applications of nanophotonics [1,2]. In this context, Mie resonances from the optical spectral range [3–6], supported by high-index dielectric nanoparticles and nanostructures, also attract significant attention from the research community [7]. There are light scattering similarities between plasmonic (metallic) particles and their dielectric counterparts. In the both cases the light scattering can be considered as a light reradiation by multipole sources excited in a scatterer by external incident waves. In this case the total scattered fields are imagined to be a superposition of the fields generated by every multipole moment of the scatterer. As a result, the total and differential scattering cross sections can be decomposed on separate contributions related to certain multipole moments. Such multipole approaches can significantly simplify the analysis of the light scattering process and provide important information about material, shape, and size parameters of scattering nanoparticles and nanostructures. However, there are several principal differences between optical response of metal and dielectric nanoparticles. If for metal nanoparticles the optical response is determined by excitation of the free electron conductive current absorbing light energy due to ohmic losses, the optical reaction of all-dielectric nanoparticles is associated with excitation of the displacement currents without the losses of absorption. In the last case, the linear light–matter interaction is solely determined by scattering. Importantly, the strong electromagnetic fields are concentrated in the near-field zone around metal nanoparticles at the resonant conditions [8], whereas the enhancement of the fields for all-dielectric nanoparticles at the resonant conditions is realized in their volumes [5]. The different optical reactions of the metal and dielectric nanoparticles result in differences of their functional properties used in practical applications [9,10].

Recently, the anapole states are observed in many high-refractive-index dielectric particles that support both electric and magnetic resonances and have zero damping loss [11–13]. An ideal anapole state is known as a state with complete scattering cancellation in the far-field and nonzero near-field excitation. From the theoretical point of view, the typical properties of anapole states exhibit zero polarized multipole moments and high near-field enhancement inside the region occupied by the scatterers [11,14–16]. The zero polarized moments of anapole states on any dimensionality could constitute the basis of dark matter in the universe [17]. Based on the nonradiative traits, anapole states are increasingly applied to the fields of nonlinear nanophotonics, dielectric metamaterials, light harvesting, and sensing [15,18–23]. In addition, the total suppressed electric dipole (ED) moment, due to the dipole anapole effect, contributes to the achievement of pure magnetic dipole (MD) scattering [14]. Due to the differences between the optical responses of metallic and dielectric nanoparticles and nanostructures, anapole states have so far been carefully studied, mainly in dielectric nanoparticles with a high refractive index and their structures. In particular, strong ohmic losses of metal nanostructures are the main reason for hindering implementation of the anapole analog in metal nanostructures.

Generally, anapole states, including electric and magnetic anapole states, consist of low- and high-order multipoles [15,24–26]. For example, the excitation of electric and magnetic anapole states in a hybrid metal-dielectric structure is theoretically investigated in our previous work [15]. The key to exciting anapole states relies on the effective generation of toroidal dipole (TD) moments, which can be imagined as high-order dipole terms of the Cartesian multipole decomposition [20,27–32]. In the case of electric anapole states, they are realized due to the totally destructive interference between Cartesian TD and ED moments, resulting in significant suppression of the total ED moment and corresponding dipole scattering [11,29,30]. Importantly, the excitation of the TD moment is accompanied by the circulating magnetic field in scattering systems [33–35]. Therefore, formally the existence of electric anapole states can be expected even in metallic metamolecules that support optical resonances accompanied by the circulating magnetic field in the metamolecule volume [36,37]. Nevertheless, the anapole condition cannot be fully satisfied in the lossy system due to the different sensitivities of different-order multipole moments to ohmic losses.

In this paper, we demonstrate both analytically and numerically that the plasmonic anapole states (suppression of the electric dipole scattering) can be excited in metallic metamolecules consisting of several nanoparticles by enhancing the destructive interference between Cartesian ED and TD moments of the total system. It is worth mentioning that the excitation of anapole states in this work is independent of high-refractive-index dielectric nanoparticles. The ideal plasmonic anapole state is achieved when the coupling strength between Cartesian ED and TD moments reaches its maximum value. Due to the different sensitivities of ohmic losses between Cartesian ED (low-order electric mode) and TD (higher-order electric mode) moments [33], the coupling strength can be enhanced by the reasonable compensation of ohmic losses of the nanosystem (0.008eV2 for the passive case and 0.014eV2 for the active case). This process is well characterized by the extended coupled oscillator (ECO) model [38]. Compared to the anapole states of high-refractive-index nanoparticle origin, the plasmonic anapole states induce stronger intensity of near fields outside nanoparticles due to the participation of surface plasmons.

The multipole decomposition of the light scattering cross section is calculated in spherical coordinates with using the finite element method (FEM) performed by the commercial COMSOL Multiphysics software. After numerical calculation of the total electric field E(r) inside all disks of the structure, the spherical multipoles are obtained by numerical integration of the following expressions [43]: $$\begin{gathered} a(l,m)=(−i)l−1k2Z0OlmE0[π(2l+1)]1/2×∫e−imφ{[ψl(kr)+ψl′′(kr)]Plm(cosθ)r^·Jsca(r) \\ +ψ′l(kr)kr[ddθPlm(cosθ)θ^·Jsca(r)−imsinθPlm(cosθ)φ^·Jsca(r)]}d3r, \end{gathered}$$(1)b(l,m)=(−i)l+1k2Z0OlmE0[π(2l+1)]1/2∫e−imϕjl(kr)[imsinθPlm(cosθ)θ^·Jsca(r)+ddθPlm(cosθ)ϕ^·Jsca(r)]d3r,(2)where l denotes the order of the multipole components, such as dipole (l=1) and quadrupole (l=2). Jsca(r)=−iω[ε(r)−εh]E(r) (εh is the dielectric constant of the surrounding medium) and Z0=μ0/ε0εh are the scattering current density and impedance, respectively. Olm reads Olm=[(2l+1)(l−m)!/(l+m)!]1/2[4πl(l+1)]−1/2. ψl(kr) is given by the Riccati–Bessel function. Plm(cosθ) are the associated Legendre polynomials. E0 is the electric field amplitude of the incident light. θ and ϕ are the zenith angle and azimuthal angle, respectively. The electric and magnetic multipole scattering cross sections are expressed as CE=πk2∑l=1∞∑m=−ll(2l+1)|a(l,m)|2 and CM=πk2∑l=1∞∑m=−ll(2l+1)|b(l,m)|2, respectively. We consider that all of the electric and magnetic multipoles are located at the center of mass of the Au–SiO2 metamolecules.

The scattering cross sections and corresponding spherical multipole expansions are shown in Fig. 1. The total scattering cross section of the Au heptamer exhibits a dip at the wavelength about 850 nm with nonzero spherical ED scattering [see Fig. 1(a)]. To dope the gain material, the low-refractive-index SiO2 layer is introduced to the Au heptamer [see Figs. 1(b) and 1(c)]. As SiO2 has weak optical response at the near-infrared wavelengths, the SiO2 disks in the nanostructure basically play the role of host medium of the gain materials. The introduction of the SiO2 layer leads to the redshift of the dip. In the passive nanostructure, the total scattering cross section exhibits a dip at the wavelength about 900 nm with nonzero spherical ED scattering [see Fig. 1(b)]. In contrast, when a gain material with the coefficient of 0.28 is introduced to the nanosystem, the ED scattering is completely suppressed at the wavelength of total scattering dip as shown in Fig. 1(c).

For explanation of the ED scattering suppression we use the multipole decomposition in the framework of the Cartesian multipoles determined in the long wavelength approximation (LWA) [29]. In this case the full Cartesian ED moment can be presented as a sum of infinite series of dipole terms [44,45]. For a relatively small scatterer, the series can be restricted to only the first several terms. If we take only the first two terms, the full electric dipole moment can be approximated by the following expression: D=p+ikT [29]. Then the electric dipole part of the scattering cross section is given by σscaED=c2k4Z06πI0|p+ikT|2,(3)where I0 and c are the illumination intensity and the speed of light in a vacuum, respectively. The Cartesian electric dipole moment p reads p=−∫d3rJsca(r)/iω, and the electric toroidal dipole moment T reads T=∫d3r{[r·Jsca(r)]r−2rsca2J(r)}/10c. When achieving the anapole condition, that is, p=−ikT, the destructive superposition of the Cartesian electric and toroidal dipoles leads to the cancellation of the full electric dipole moment of the nanosystem and its contribution into the scattering cross section.

In order to better understand the achievement of anapole condition via loss-compensation mechanism, we adopt an ECO model to analyze the process of the compensation of ohmic losses. In reality, the anapole mode belongs to the category of Fano resonance, which can be well characterized by the ECO model. Herein, the two-coupled oscillators model can be expressed as [38] x¨1+γ1x˙1+ω12x1−υ12x2=0.5P⃛tot+α1Eext,(4)x¨2+γ2x˙2+ω22x2−υ21x1=0.5P⃛tot+α2Eext,(5)where x1 and x2 are the displacements from the equilibrium position of the first and second oscillators with the harmonic solution form of x1(t)=c1eiωt and x2(t)=c2eiωt [48]. γ1 and γ2 are the damping coefficients of the first and second oscillators accounting for intrinsic losses, respectively. ω1 and ω2 are the natural frequencies (eigenmodes) of the two oscillators. υ12 and υ21 describe the coupling strength between the two oscillators (υ12=υ21). Ptot=α1x(t)+α2x(t) is the total dipole moment of the system describing the radiative damping, where α1 and α2 are the polarizabilities corresponding to the plasmon amplitude to their induced dipole moments. The oscillators are driven by the harmonic forces α1Eext and α2Eext, where Eext=E0eiωt. Note that the first and second oscillators indicate the Cartesian ED and TD moments, respectively. The total scattering cross section is |c1+c2|2 [38], where c1 and c2 are obtained by solving Eqs. (4) and (5).

In summary, it has been theoretically shown that low- and high-order anapole states can be excited in metallic metamolecules. As both the electric and magnetic modes are supported by the metallic metamolecules, it offers the necessary conditions of the origin of the anapole states. The excitation of the ideal plasmonic anapole state depends on the effective excitation of the TD moment and fine-tuning it to ensure the completely destructive interference with the Cartesian ED moment. The totally destructive interference is achieved by enhancing the coupling strength. We have overcome a major limitation of plasmonic (metal) systems in realization of anapole states, related with strong ohmic absorption of light, by doping of the system by a gain material. Note that the spectral position and width of the gain effects depend on the materials used, and the environment and can vary widely. In this case, from the practical point of view, the implementation of the anapole state can be achieved in a required spectral gain range by adjusting the dimensions and material parameters of the total structure. The developed coupled oscillator model with active terms has been used for clarification of the physical process in the anapole state formation. Compared to the high-refractive-index nanoparticles, the plasmonic anapole modes can be excited in the metallic metamolecules with greater enhancement of near fields, which indicate the remarkable energy concentration performance and have more important implications for enhancing Raman scattering, fluorescence, and nonlinear effects. The theoretical results in this paper can be realized with the current nanofabrication technology, which not only open a route to study the anapole modes but also provide a new way of thinking to achieve the total suppression of noise modes and increase the signal-to-noise ratio of the target modes. Application of the considered structures, as buildings blocks for material developments, can extend varied physical approaches [50–52] to the creation of new materials and metamaterials with special functional properties.

Acknowledgment. We thank Hai-Long Wang, Hai-Tao Liu, and Dong Xiang for their valuable suggestions.