Toroidal dipole resonance in an asymmetric double-disk metamaterial

1. Introduction

In 1958, Zel’dovich put forward a new notion, namely toroidal dipole moment, to explain parity violation on the weak interaction in nuclear and particle physics [1]. The toroidal dipole, distinguished from electric or magnetic dipole, is induced by a current flowing on the surface of a torus along its meridian and characterized by a head-to-tail distribution of magnetic dipoles, and thus possesses many interesting properties [2]. In the ensuing decades, people paid more attention to the static toroidal dipole, while few studies focused on the dynamic toroidal dipole usually masked by other electromagnetic multipoles on account of the weak coupling to electromagnetic field. It was not until the appearance of metamaterial that the toroidal dipole response really caused researchers’ attention [3,4]. In 2007, Marinov et al. firstly proposed the toroidal metamaterial that could strengthen the dynamic toroidal dipole moment [5]. Then, the first experimental observation was successfully implemented by N. I. Zheludev in 2010 [6]. Until now, a variety of toroidal metastructures [3,4,7–32] have been designed to obtain toroidal dipole response with different optical functionalities, such as toroidal lasing spaser [3], high-quality-factor Fano resonance [4], all-optical Hall effect [12], invisible nanowires [16], electromagnetically induced transparency [18], perfect absorber [19] and terahertz toroidal meta-modulators [24]. Meanwhile, there have also been some experimental studies on toroidal dipole resonance in the THz and microwave bands. Among the aforementioned toroidal metamaterials, the majority are constructed with split ring resonators (SRRs). Nevertheless, due to structural limit, the SRR-based toroidal metastructures are accompanied by a weak field confinement. By contrast, the double-disk metastructure proposed in our previous work can not only acquire a high-performance toroidal cavity mode, but also present an excellent local field enhancement characteristic. However, the toroidal dipole resonance can only be excited under lateral incidence or certain incident angles, bringing more inconvenience for experimental explorations and potential applications.

In this paper, we proposed an asymmetric metallic double-disk metamaterial to investigate the toroidal dipole resonance by normally incident radiation and carried out an experimental verification in the microwave region. Furthermore, when the offset angle θ of the upper disk with respect to the lower one varies from 0 to 100 degrees, the toroidal dipole resonance always keeps strong resonance performance, providing a more flexible operating space in experiment. Besides, the influence of disk radius asymmetry $\Delta r$ on toroidal dipole resonance has also been explored.

2. Theory and model

Figure 1(a) shows that the toroidal metamaterial consists of asymmetric double disks and an interlayer. In order to obtain the toroidal dipole resonance under normal incident excitation, a radial asymmetry $\Delta r$ is introduced and defined as the difference between the left and right radii of the upper disk, namely $\Delta r = {{r_2} - {r_1}}$ (here, ${r_1} = 10\;mm$ and ${r_2} = 13\;mm$). The upper-disk can be rotated 180 degrees to obtain lower one as displayed. Figure 1(b) presents the view of experiment sample. The bilayer asymmetric disks by copper (conductivity $\sigma = 5.96\times{10^7}\;\textrm{S}/m$) with thickness of 0.035 mm was fabricated on a FR-4 substrate by conventional printed circuit board techniques. The FR-4 substrate is set to 1-mm-thick, with a permittivity of 4.3 and a dielectric loss tangent of 0.025. The periodic arrangement is represented by a 15 mm × 15 mm unit cell. The numerical simulation was performed by a full-wave solver based on the finite-element method (FEM) and the incident excitation propagates along z-axis with x-direction polarization (i.e., normal incidence). An experimental measurement by a vector network analyzer (Keysight N5247A) was carried out to verify the numerical simulation results.

 

Fig. 1. (a) Structural schematic of a unit cell of the designed asymmetric double-disk (note that the three layers of components are displayed separately for eye guidance). (b) The photograph of experimental sample.

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3. Result and discussion

To explore resonant properties of the designed metastructure, first of all, we display the simulated reflection and transmission spectra marked by black line in Fig. 2. As we can see, there are two plasmonic resonances, corresponding to 8.1 GHz and 10 GHz respectively. The low frequency resonance at 8.1 GHz is proved to be a toroidal dipole response by quantitatively calculating the scattered power of electromagnetic multipole as shown in Fig. 3(a). The vortex distribution of magnetic field in Fig. 3(b) also further provides an intuitive proof. In contrast to the SRR-based metamaterials [21,27–29], our asymmetric double-disk metastructure actually possesses much better confinement of magnetic field. Moreover, the asymmetric double-disk metastructure couple strongly with normally incident excitation and cause currents in opposite phases between the upper and lower disks as shown in Fig. 3(c), which accounts for an excitation of toroidal dipole resonance. Meanwhile, the high frequency resonance at 10 GHz is high-order hybrid electromagnetic resonance. Here, we only consider toroidal dipole resonance. The measured data in experiment and the simulated results generally agree well, except an unpleasant heavy loss. As for the somewhat broad linewidth of the measured toroidal dipole resonance, it is mainly caused by the fabrication tolerance, limited metastructure array and imperfect incident microwave beam.

 

Fig. 2. Reflection (a) and transmission (b) spectra of the asymmetric double-disk metastructure from both experiment (red) and simulation (black). The low-frequency resonance at 8.1 GHz corresponds to a toroidal dipole mode. The high-frequency resonance at 10 GHz is a high-order hybrid electromagnetic resonance. Here, we only discuss the toroidal dipole resonance.

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Fig. 3. (a) The calculated scattered power in terms of various multipolar moments to verify that the toroidal dipole resonance plays a predominant role over other multipoles at 8.1 GHz. (b) Magnetic-field distribution at 8.1 GHz. (c) Surface current maps of the upper- and lower-layer at 8.1 GHz (note that the big red arrows represent the direction of current; the instantaneous surface current of the upper-layer disk flows outward away from the center, meanwhile the one of the lower-layer disk flows inward to the center).

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As proposed in Ref. [9], the metallic double-disk can only realize a dominant toroidal dipole response under certain incident angles varying from 0 to 60 degrees, but failing with 90-degree incidence angle (corresponding to the normal incidence in this work). As a matter of fact, the underlying reason for strong toroidal dipole resonance excited by normal incidence in our metastructure is an introduction of the radius asymmetry $\Delta r$ $(\Delta r = {r_2} - {r_1}).$ Therefore, it is necessary to discuss the effect of the radius asymmetry $\Delta r$ on the toroidal dipole excitation as shown in Fig. 4. For the sake of better discussion, ${r_2}$ remains unchanged. When $\Delta r$ gets a small value of 1 mm, the reflection spectrum presents a small dip around 7.0 GHz, indicating a weak toroidal dipole response. As the radius asymmetry $\Delta r$ gradually increases by a division value of 1 mm, a blue shift in the frequency of toroidal dipole resonance occurs along with stronger reflection dips, implying the toroidal dipole is strengthened. Nevertheless, once beyond a certain range, our proposed asymmetric double-disk metastructure will lose its characteristic of toroidal dipole resonance.

 

Fig. 4. The simulated reflection spectra with a variation of the radial asymmetry $\Delta r$.

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For the multi-layer metamaterials, especially ones with an asymmetry or split, the alignment process between different-layer components is of importance in experiment. Accordingly, it is absolutely necessary to investigate the dependent relationship of toroidal dipole excitation on the offset between the upper and lower disks. Consequently, Fig. 5 displays the simulated results about the influence of offset angle θ on the characteristic of toroidal dipole. When the offset angle θ varies from 0 to 100 degrees, the toroidal dipole resonance almost stays a stable excitation frequency, only along with a small fluctuation around 7.9 GHz. Most important of all, it has also been proved that the target resonance always keeps a dominant role over the wide range of offset angle θ, revealing that the toroidal dipole response in our asymmetric double-disk is, to some extent, insensitive to the offset angle. In comparison with the three-dimensional SRR-based metamaterials, our metamaterial lowers requirement of sample fabrication.

 

Fig. 5. (a) The toroidal resonant frequency as a function of offset angle θ. In the inset, the offset angle θ is defined as the rotation angle of upper disk with respect to the lower one. (b) The calculated scattered powers under different offset angles to prove the dominant role of toroidal dipole resonance within a wide range from 0 to 100 degrees.

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In final, an experimental verification was carried on as shown in Fig. 6. The offset angle θ is chosen as 45 degrees. In order to give an intuitive comparison, the black line of Fig. 6(a) is the simulated spectrum where there are two reflection dips. In fact, by calculating the scattered power of multipoles, the two resonant dips respectively correspond to the toroidal dipole and electric quadrupole as displayed in Fig. 6(b). In comparison with the one of offset angle of 0 degrees, the toroidal dipole resonance frequency goes down to 7.95 GHz [yellow shadow region in Fig. 6(a)]. Nevertheless, the electric quadrupole at 8.2 GHz is due to an introduction of an asymmetry induced by the offset angle θ. As for the measured reflection spectrum [red line in Fig. 6(a)], it is in overall consistent with the simulated ones, although there are some deviations. As is known, in contrast to usual low-quality-factor electric multipole resonances, the toroidal dipole resonance is intrinsically a high-quality-factor dark mode, which otherwise cannot be excited under normal incidence, unless the geometric symmetry is significantly broken (i.e., Δr). For the offset angle of 45 degrees, the toroidal mode cannot be excited by the normal incidence as complete as a bright mode can. Therefore, except for influencing factors such as fabrication tolerance, limited metastructure array and imperfect incident microwave beam, it is somewhat overlapped (or screened) in reflection spectrum by the neighboring electric-quadrupole bright mode in Fig. 6(a).

 

Fig. 6. When offset angle θ between the upper and lower disks is 45 degrees: (a) simulated (black) and measured (red) reflection spectra; (b) the calculated scattered power of various multipole moments. The inset in (a) is the photograph of experimental sample of the offset angle of 45 degrees.

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4. Summary

In conclusion, we have numerically and experimentally demonstrated the toroidal dipole response in an asymmetric double-disk metastructure. This work shows that when both the upper and lower disks own a radial symmetry-broken geometry, the metastructure can present an excellent toroidal dipole response under normal incidence, along with a strong vortex distribution of local magnetic field. In the meantime, the dependence of toroidal dipole resonance on the structural parameters (i.e., the radius asymmetry $\Delta r$ and the offset angle θ) is also discussed. Moreover, when the offset angle varies over a large range from 0 to 100 degrees, it always keeps the strong toroidal dipole resonance. In other words, the toroidal dipole resonance is, to some extent, insensitive to the offset angle θ, which maybe provide more convenience for toroidal-resonance-based meta-devices in future.