1. Introduction

The toroidal moment, firstly proposed by Zel’dovich in 1957 to interpret the parity violation on the weak interaction in nuclear and particle physics [1], fertilizes the area of classical but incomplete electromagnetic theory of multipole decomposition [2–4 ]. Unlike electric and magnetic dipoles resulting from a pair of charges and circulating currents respectively, the intriguing toroidal dipole is caused by a current flowing on the surface of a torus along its meridian [5,6 ]. However, the toroidal dipolar response is much weaker than conventional dipoles and, consequently, difficult to be detected in naturally occurring media [7]. Therefore, to find dominant toroidal dipolar responses by artificial composites is an emerging exploration field. Recently, it was claimed that toroidal metamaterials could be constructed to strengthen the toroidal dipolar response while suppressing other electric and magnetic multipoles [2,5,6,8–13 ]. Though it is well known that a toroidal dipole has a radiation pattern identical to an electric or magnetic dipole [14], an interesting phenomenon associated with the toroidal dipole is the nonradiating characteristic when it interacts with an electric dipole, which can be regarded as a nontrivial anapole [2,5,12,15–18 ]. It has been numerically demonstrated that such a hypothetical nonradiating system can be converted into a directional emission source in dependence of the interface contrast of index of refraction between the dipolar system and surroundings [19]. In addition, it should be interesting to explore the interaction phenomena between a toroidal dipole and other multipoles. For example, it was reported that the toroidal dipolar response could be enhanced by coupling to electric or magnetic dipole, resulting in an analogy of electromagnetically-induced transparency [5,12,15,18 ].

In this paper, a toroidal metastructure by double metallic flat-rings is proposed to numerically investigate the scattering characteristic of its interaction with a dipolar emitter. It is interesting to find that the interaction can be either radiating enhanced or suppressed (i.e., nonradiating), depending on constructive or destructive interference, respectively. Moreover, the nonradiating phenomenon can be sensitively manipulated to be a super-radiating one with both pronounced directionality and linear polarization conversion, once the dipole emitter is shifted away from the axis of toroidal dipole.

2. Numerical model

The passive composite metastructure proposed in this paper is shown in Fig. 1 , comprising double metallic flat-ring elements for the toroidal dipolar response with a silicon-oxide gap layer as spacer (${\epsilon }_{SiO2}=\text{2}\text{.4}$). The flat-ring inner and outer radii are 100 and 300 nm, respectively, and the gap between 40-nm-thick metallic flat rings is 30 nm. This double metallic flat-ring metastructure has been demonstrated to support the dipolar toroidal response by using the multipole decomposition theory in our previous works [20,21 ]. In this work, an active dipole emitter is considered as a light source to excite the toroidal dipolar response. The dipolar emitter, oriented along the y-axis, with a moment $\mu =j{I}_{0}l/\omega$ is numerically modeled by a tiny line ($l=1\text{nm}$) carrying an electric current $I_0=1\text{A}$. The metallic metastructure is considered as silver following the Drude-type dispersion model $\epsilon \left(\omega \right)={\epsilon }_{\infty }\text{-}\frac{{\omega }_p^2}{\left({\omega }^2+i\omega \gamma \right)}$, with a high-frequency permittivity ${\epsilon }_{\infty }=6.0$, plasma frequency ${\omega }_{\text{p}}=1.5\times {10}^{16}{s}^{-1}$, and collision frequency $\gamma =7.73\times {10}^{13}{s}^{-1}$ [20,22,23 ]. To explore the coupling effect between toroidal dipole response and electric-dipole emitter, variable displacements of the emitter with respect to the geometric center of double flat rings are considered [i.e., axial displacement along the y axis ($\Delta y$) and radial displacement along the x axis ($\Delta x$)]. Numerical simulations are performed by a full-wave solver based on finite-difference time-domain (FDTD) method [19,20,24 ]. Far-field emission powers after scattered by the flat-ring metastructure are monitored at three coordinate directions. By excluding negligible components of radiation field, only y-component radiation power of the X-direction emission (${\left|E_y^X\right|}^2$), x-component radiation power of the Y-direction emission (${\left|E_x^Y\right|}^2$), and y-component radiation power of the Z-direction emission (${\left|E_y^Z\right|}^2$) will be presented in this paper, corresponding to far-field probes A, B, and C, respectively (Fig. 1).

 

Fig. 1 Schematic of the metastructure composed of double metallic flat-rings with a silicon-oxide gap spacer. Note that the origin “O” of the coordinate system is located at the center of metastructure.

Download Full Size | PDF

3. Results and discussions

As is well-known, either a toroidal or an electric dipole exhibits a donut-like radiation pattern [14]. Therefore, for the axially symmetric modeling configuration with an on-axis displacement $\Delta y$ for the emitter (Fig. 1), the radiating characteristic can be evaluated by a radially positioned far-field monitor (e.g., probe A or C). When the dipole emitter is put away from the center of the double flat rings (for example, $\Delta y=150\text{nm}$), the probed far-field power has only a slight dip at 1050 nm due to weak coupling to the toroidal response in metallic flat rings. However, when the emitter approaches, the scattering field at the toroidal resonance will exhibit a Fano-type asymmetric line shape, which is a result of strong interference between the narrow toroidal dipolar response and the broad electric dipolar source. If the dipolar emitter is exactly overlapped with the toroidal dipole (i.e., $\Delta y=0\text{nm}$), the radiation field at 1050 nm reaches a resonant peak, implying a pronounced constructive interference between toroidal and electric dipoles. In contrast, destructive interference happens at the adjacent wavelength of 945 nm, leading to a significantly suppressed radiation. The dipolar radiating enhancement peak under the assistance of a toroidal flat-ring structure is beneficial in identification of a single nano-emitter through the radiation intensity. On the other hand, the radiation suppression implies that this composite nano-system can be regarded as a so-called nonradiating source. In literatures, this kind of nonradiating system of interacting toroidal and electric dipoles was known as an anapole [14,16,18 ]. Here, because the realistic metal losses in plasmonic flat-rings results in an incompetent strength of the induced toroidal dipole as compared to the active emitter, we shall not expect a suppression of radiation with a complete destructive interference better than the calculated nonvanishing radiation dip at 945 nm.

To show a straightforward picture of the interference interaction between dipoles, local magnetic fields for the resonant radiation dip and peak are presented in Figs. 2(b) and 2(c) , respectively. In both cases, magnetic-vortex distributions as the unique feature of a dipolar toroidal mode are clearly induced by the dipole emitter, but the underlying interference interactions are not in a same way. For the resonant dip at 945 nm, the toroidal dipole mode shows a 180-degree phase lag [Fig. 2(b)], which results in a destructive interference with the dipolar excitation field and, consequently, the pronounced radiation suppression (nonradiating phenomenon). On the contrary, the toroidal mode at 1050 nm is induced in phase with the dipolar excitation field, so that the constructive interference between them causes a radiating enhancement. Besides, the calculated scattered powers in terms of multipoles also sufficiently justify the excitation of a dominant toroidal dipole around 1050 nm in Fig. 2(d).

 

Fig. 2 (a) Far-field radiating power for different axial displacements $\Delta y$. The nonradiating dips and super-radiative peaks in the Fano-type resonant spectra indicate destructive and instructive interferences between toroidal and electric dipoles, respectively. (b) Destructively induced toroidal mode at the 945-nm wavelength, with out-of-phase magnetic circulating distribution against that of the dipolar emitting source ($\Delta y=0\text{nm}$). (c) Constructively induced toroidal mode at the 1050-nm wavelength, with in-phase magnetic circulating distribution to the dipolar source ($\Delta y=0\text{nm}$). (d) The scattered powers in terms of multipoles when $\Delta y=0\text{nm}$.

Download Full Size | PDF

When the dipole emitter deviates from the central position of toroidal flat rings along radial directions, the interference interaction between toroidal and electric dipoles will lead to a different emission behavior. Without loss of generality, we consider a radial displacement along the x axis, for example, $\Delta x=60\text{nm}$ (Fig. 3 ). It is found that the resonant toroidal response remains in the spectrum as well, indicating that the response of the toroidal dipole is inherent to the flat-ring geometry itself. On the other hand, although the radiating peak at 1050 nm under constructive interference (recorded by probe A/C) is not sensitive to the radial position of a dipole emitter, the emerging peak at 945 nm exhibits a dramatic emission enhancement (recorded by probe B) in contrast to the otherwise nonradiating dip. This can be regarded as a super-radiating effect with an emission power ever 10 times stronger than the 1050-nm enhanced emission.

 

Fig. 3 Far-field radiating power for the radial displacement $\Delta x=60\text{nm}$. The reference data are recorded by probe A/C for $\Delta x=0\text{nm}$.

Download Full Size | PDF

It was claimed that the resonant nanostructure is capable of modulating the emission direction of an adjacent emitter, which is promising for optical nanoantennas for purposes such as steering and/or collecting lights [25–30 ]. Although the dipolar emitter in the center of toroidal flat-ring structure radiates light in a donut-like pattern [Fig. 4(a) ], its emission mediated by the flat rings will evolve towards preferred directions so long as the emitter is shifted along radial directions. Figure 4(b) shows a moderate modification for the donut-like radiation pattern at 1050 nm. However, a pronounced directional radiation pattern can be obtained at the super-radiating wavelength 945 nm for $\Delta x=60\text{nm}$, exhibiting a property as is an optical nanoantenna [Fig. 4(c)]. From corresponding two-dimensional polar plots shown in Figs. 4(d)-4(f), the angular width of main-lobe radiation (angle between half-power radiation directions) can be greatly narrowed to an angle about 70 degree. In literatures, this directivity characteristic was explained by the versatile interference behaviors of the emitter coupling with the resonant mode in terms of multipoles, which will consequently determine the radiation directivity, intensity, as well as polarization [25–30 ].

 

Fig. 4 (a) Donut-like radiation pattern at 1050-nm wavelength for $\Delta x=0\text{nm}$. (b) Modulated radiation pattern at 1050-nm radiative wavelength for $\Delta x=60\text{nm}$. (c) Directional radiation pattern at 945-nm radiative wavelength for $\Delta x=60\text{nm}$. (d)-(f) Two-dimensional radiation in dependence of the polar angle $\theta$ under an azimuthal angle $\varphi =90$ degree, corresponding to (a)-(c), respectively. Note the 1-meter reference distance is used to set the radius of the virtual sphere for calculating field values of radiation pattern in Fig. 4.

Download Full Size | PDF

Figure 5 shows probed emission powers in dependence of radial displacement $\Delta x$. Due to different coupling efficiencies between the dipole emitter and the metastructure, emission powers can be dramatically modulated. When the emitter is in an off-axis position $\Delta x=r_1=100\text{nm}$, the toroidal-dipole-mediated directional emission, ${\left|E_x^Y\right|}^2$, can reach a super-radiating maximum, roughly three orders stronger than radiation power of a bare emitter.

 

Fig. 5 Far-field radiation powers monitored by probe A (a), probe B (b), and probe C (c), under different radial displacements $\Delta x$.

Download Full Size | PDF

4. Summary

In summary, we explore the radiation characteristic of a toroidal-response-mediated dipole emitter under different coupling strategies with axial and radial (off-axis) displacements. Nonradiating dipole-dipole interaction is demonstrated by this coupling system of toroidal flat-rings and dipolar emitter, attributed to the destructive interference between overlapped radiative dipolar fields. Moreover, the otherwise nonradiating system will become a directionally super-radiating one, like an optical nanoantenna, so long as the dipolar emitter has a radial displacement. Our results are promising in manipulating the radiation power and direction of a single emitter, such as fluorescent molecule/atom and quantum dot, by utilizing the intriguing toroidal dipolar response based on the proposed flat-ring metastructure.