1. Introduction

Toroidal metamaterials, as an intriguing type of metamaterial, merge toroidal moments and metamaterial and become the subject of growing interest for their novel electromagnetic properties [1]. Toroidal moments are created by a unique torus-like-current and characterized by a head-to-tail distribution of induced magnetic dipoles [2]. Although there were extensive research on static toroidal moment in the fields of nuclear, molecules and solid state physics, it is difficult to get access to a dynamic toroidal dipolar moment due to its weak coupling to electromagnetic wave [1,3]. Therefore, in 2010, Kaelberer et al. firstly constructed a three-dimensional (3D) split-ring-based metamaterial and experimentally demonstrated a significant toroidal dipole resonance in the microwave regime, bringing attention to the ignored toroidal multipoles interactions [1]. Up to now, various kinds of toroidal metastructures have been proposed to numerically or experimentally investigate toroidal dipole resonance [3–38], covering microwave [4,8,24,25,29], terahertz [10,15,20,26] and visible bands [6,7]. Simultaneously, a variety of optical phenomena can be realized by toroidal dipolar resonance, such as, all-optical Hall effect [3], resonant transparency [8,12], lasing spaser [9], circular dichroism [17], metamaterial absorber [30], metamaterial rasorber [31]. Especially, toroidal dipole resonance with high quality (high-Q) factor was also exploited to observe photoluminescence (PL) enhancement [36], third harmonic generation (THG) enhancement [23], sensing [37], and optical radiation manipulation [38]. Among these studies, lots of efforts have been devoted to investigate and manipulate toroidal dipole resonance by adjusting structural parameters and implanting special medium, such as a graphene monolayer, a vanadium dioxide (VO₂) layer or optical gain media [9]. Nevertheless, for aforementioned toroidal metamaterials, toroidal dipole mode can only be excited under lateral incidence or certain incident angles. In 2019, Song et al. firstly investigated the two-dimensional (2D) split-ring-based metamaterial where toroidal dipole resonance is insensitive to oblique incidence, but disappearing when polarization angle is large than $45^\circ $. In contrast to 3D metamaterials, the planar 2D toroidal metamaterial is absence of much better confinement of the circulating magnetic field. Actually, as a classic toroidal metamaterial, the double-disk metastucture can concentrate electromagnetic field into a hot-spot at the gap-layer center, possessing the better plasmonic confinement of local electric field than split-ring-based metamaterial [30,31], presenting more potentials in many applications, such as optical force enhancement and optical trapping [16]. Nonetheless, due to the excitation mechanism, the toroidal dipole response can only be achieved with large incident angles. Inspired by its application prospect, it is necessary to explore an insensitive toroidal dipole resonance to oblique incidence, which has not yet been reported.

In this work, we numerically explored an incident-angle-insensitive 3D toroidal metamaterial by an asymmetric double-disk. In comparison with these split-ring-based toroidal metamaterial, this work shows that a significant toroidal dipole mode, accompanied by better confinement capacity for both magnetic and electric field, can be excited and insensitive to the incident angle. Furthermore, under normal incidence, the polarization angle accessible to a dominant toroidal dipole resonance can be expanded to $70^\circ $, broader than literature reports. The local electric field amplitude can also reach a maximum by structural optimization.

2. Numerical model

As reported in [16], the metallic double-disk structure can acquire toroidal dipole resonance by a lateral incident light. Based on such an optical metastructure, our toroidal metamaterial proposed in this paper is composed of asymmetric metallic double-disk, as schematically shown is Fig. 1. To explore toroidal dipole characteristic under normal incidence, it is essential to introduce a radius asymmetry attached to the metallic disk, defined as $\Delta r = {r_2} - {r_1}$ where ${r_2} = 310\textrm{nm}$ and ${r_1} = 270\textrm{nm}$. The upper asymmetric disk can be rotated $180^\circ $ around the z-axis to obtain the lower ones. The 40-nm-thick metal is selected as silver following with Drude-type dispersion model $\varepsilon (\omega )= {\varepsilon _\infty } - \frac{{\omega _p^2}}{{({{\omega^2} + i\omega \gamma } )}}$ (the high-frequency permittivity ${\varepsilon _\infty } = 6.0$, the plasma frequency ${\omega _p} = 1.37 \times {10^{16}}{s^{\textrm{ - 1}}}$, and the collision frequency $\gamma = 8.5 \times {10^{13}}{s^{\textrm{ - 1}}}$) and the gap between the double-disk components is 10-nm-thick. The electromagnetic wave propagates parallel to the z-direction and is y-polarized with electric-field amplitude of 1 V/m. The numerical simulation was performed to explore the characteristic of toroidal dipole response by a full-wave solver based on the finite-difference time-domain method. The E_z-probe is placed in the center of the structure to monitor the E-field magnitude. For future experiments, our metastructure can be fabricated by an electron-beam lithography (EBL) system.

 

Fig. 1. Schematic of the asymmetric double-disk toroidal metamaterial and the polarization configuration of the incident light. Note that the origin of the coordinate system lies in the center of gap layer. The E_z-probe (red arrow) is located at the structural center to monitor the local E-field magnitude.

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3. Results and discussion

In our previous work [29], it has been experimentally demonstrated that the asymmetric double-disk metamaterial can realize the toroidal dipole response under normal incidence within certain microwave frequency band. In this premise, we continue to further explore the dependent relationship of toroidal dipole resonance and incident light. To characterize response excitation of the metastructure, Fig. 2(a) exhibits the local E-field magnitude spectrum, with an obvious resonant response around 1188 nm. According to the H- and E-field maps displayed in the insets of Fig. 2(a), the mode is toroidal dipole resonance. And the probed E-field magnitude reaches a maximum of 30 V/m, indicating an excellent local field confinement characteristic. From an experimental point of view, the silicon oxide (SiO₂) is generally used as the gap-layer medium. The direct result from the gap-filled Ag/SiO2/Ag structure is the resonant wavelength redshifting to 1670 nm, while there is unobvious influence on the probed E-field amplitude. Beyond that, Fig. 2(b) shows the scattering powers of multipoles. For the resonance at 1188 nm, the z-component of the toroidal dipole moment T_z is stronger than other multipolar moments. As discussed in Ref. [29], it is an introduce of structure symmetry break (namely $\Delta \;r$) that lowers the structural symmetry and is responsible for toroidal dipole resonance. Hence, this is taked for granted that it is necessary for exploring the effect of $\Delta \;r$ on local field characteristic for E-field confinement attached to toroidal dipole resonance. As depicted in Fig. 3(a), as the asymmetric factor $\Delta \;r$ gradually increases, the resonant wavelength experiences a blue-shift. And it is found that an optimization structure, within a wide range from 60 nm to 100 nm of ${\Delta }\;r$, will maintains an excellent field concentration capability. Actually, the smaller or larger asymmetric factor (namely $\Delta \;r < 50\textrm{nm}$ or $\Delta \;r > 105\textrm{nm}$) may lead to a weak coupling between the metastructure and electromagnetic wave. Meantime, with the gap increasing, the toroidal dipole resonance exhibits a blue shift, while the local E-field is gradually weakened due to the weak coupling to electromagnetic field in Fig. 3(b). Actually, it is because that an increase of gap distance causes electromagnetic wave radiating massively out.

 

Fig. 2. (a) Simulated local E-field amplitude at (0, 0, 0) coordinate position. (b) The decomposed scattered powers versus wavelength in terms of the multipole scattering theory. The insets of (a) display the H- and E-field maps at resonant wavelength of 1188 nm.

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Fig. 3. Local E-field amplitude monitored by Ez-probe versus wavelength: (a) when the radial asymmetry Δr changing from 10 nm to 110 nm; (b) when the gap changing from 6 nm to 50 nm.

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Although some research on polarization-insensitive toroidal metamaterials have been reported, these structures only work under the certain incident angle [30,31,34,35]. In our work, the resonant characteristic of toroidal dipole mode under different incident angles is also discussed as shown in Fig. 4. Firstly, as far as the resonant wavelength is concerned; toroidal dipole mode is independent of the incident angle both for TM and TE oblique incidences. Secondly, for the case of TM oblique incidence, when the incident angle varies from $0^\circ $ to $90^\circ $, the E-field magnitude attached to toroidal dipole mode is getting stronger, implying that the coupling with our metastructure is also enhanced as shown in Fig. 4(a). To some extent, the local field confinement characteristic holds the stable level, possessing persistent capturing potential. As a matter of fact, for the lateral incidence case (i.e., $90^\circ $-incident-angle), our metastructure can still obtain a significant toroidal dipole resonance, even though without an introduce of the asymmetry factor (i.e., $\Delta r = 0$), which has been reported in the Ref. [3]. It is mainly because that the incident polarized electric field is parallel to toroidal dipole moment, according with the classical current-carrying toroidal solenoid configuration. Nevertheless, under the normal incidence (i.e., $0^\circ $-incident-angle), the asymmetry factor $\Delta r$ is the root cause for realizing toroidal dipole resonance. This seems to explain why resonant intensity of toroidal dipole mode is getting stronger when the incident angle varies from $0^\circ $ to $90^\circ $. For the case of TE oblique incidence in Fig. 4(b), with the incident angle larger than $60^\circ $, the E-field magnitude monitored in the center of the interlayer suffers an obvious loss. Despite this fact, our metastructure still exhibits a perfect magnetic field vortex distribution for $90^\circ $-incident-angle situation in the inset of Fig. 4(b), retaining obvious toroidal dipole resonance characteristic.

 

Fig. 4. Excitation of toroidal dipole resonance under different incident angles: (a) TM oblique incidence and (b) TE oblique incidence. The inset of (b) displays magnetic field vortex distribution when the incident angle for TE case is $90^\circ $, indicating a good realization of toroidal dipole response in spite of weakened electric field amplitude.

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Furthermore, we also discuss the effect of different polarization angles on toroidal dipole mode under normal incidence as shown in Fig. 5. For the SRR-based metastructure mentioned above, the toroidal dipole resonance almost vanished when polarization angle is larger than $45^\circ $ due to structure anisotropy. Here, the polarization angle is defined as the angle between polarization electric field of incident light and y-axis. It can be seen that resonant wavelength of toroidal dipole mode is independent of polarization angle, but, on the flip side, E-field magnitude experiences a diminishment when the polarization angle exceed $45^\circ $, even to a “disappearance” due to weak coupling with electromagnetic wave with the polarization angle continuing to increase to $80^\circ $. However, even in a weak-coupling scheme, there is still a good magnetic field vortex distribution attach to toroidal dipole resonance when the polarization angle is $70^\circ $ as illustrated in the inset of Fig. 5, which stems from the particularity of our metastructure.

 

Fig. 5. Excitation of toroidal dipole resonance for different polarization angles under normal incidence. The inset gives magnetic field vortex distribution when the polarization angle under normal incidence is $70^\circ $.

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4. Summary

In conclusion, we have presented an interesting asymmetric double-disk metamaterial to investigate toroidal dipole resonance characteristic in the near infrared band. When incident angle varies from $0^\circ $ to $90^\circ $, the proposed metastructure system both realizes a strong toroidal dipole resonance over other electromagnetic multipoles and keeps an excellent field confinement characteristic. Under normal incidence, the polarization angle can be extended to $70^\circ $ where toroidal dipole mode can keep the stable vortex distribution of magnetic field. The influences of structural parameters on toroidal dipolar resonance in terms of the resonant wavelength and plasmonic confinement capability of local electric field are also investigated. With the radial asymmetry Δr gradually increasing, there appears a blue-shift for toroidal dipole resonance, while the local electric field amplitude can also maintains a strong and stable level from 60 nm to 100 nm. The work will offer more potentials in optical devices based on toroidal moment.