Triple-band transparency effect by multiple couplings based on toroidal dipole resonance

We explored multiple couplings properties in composite metastructure. One part is the asymmetric double rings, supporting the narrow toroidal dipole resonance, and the other component is an upright rod that excites the broad electric dipole resonance. When these two resonant modes coincide in the spectrum, dual-band plasmon induced transparency (PIT) behavior can be obtained, which is attributed to in-phase and out-of-phase couplings between the toroidal dipole and electric dipole modes. Meanwhile, the dual-band features will become a single PIT band by varying the rotation offset angle between the upper- and lower-rings. Moreover, by introducing lateral displacement of the rod with respect to the toroidal component, a triple-band PIT effect can be achieved. In particular, under a large lateral displacement, a broadband transparency window appears across a wavelength range greater than 120 nm, where the transmission exceeds 0.9. It is derived from the hybrid coupling between toroidal dipole, electric dipole and induced high-order resonance modes. The toroidal-based PIT metamaterials not only promote the understanding of toroidal dipole moment but also provide a positive reference for toroidal-based meta-devices.


Introduction
Electric and magnetic dipole moments are two fundamental electromagnetic excitations that have been extensively studied.However, as a third independent family of electromagnetic multipole theory, the toroidal dipole moment was first introduced by Zel'dovich in 1958 as an explanation for the parity violation of the weak interaction in nuclear physics [1].The elusive moment is induced by the current flowing on the surface of the torus along its meridians and is accompanied by a vortex field distribution [2][3][4].As a matter of fact, toroidal dipole moment is widely found in nature, ranging from the atomic to the astronomic.Unfortunately, the toroidal dipole response couples weakly with the electromagnetic field and is consequently difficult to detect [4].
The appearance of metamaterials provides an operational platform for an observable toroidal dipole resonance.As is well known, metamaterials [5][6][7], as a thriving research field, have attracted extensive attentions owing to their numerous unique physical properties and applications, such as super-imaging [8], cloaking [9] and negative refraction [6], which are unavailable in nature.Therefore, in 2010, a new type of artificial subwavelength structure, toroidal metamaterial, was first experimentally constructed to demonstrate toroidal dipole resonance [2].Subsequently, more toroidal metamaterials have been theoretically or experimentally proposed to strengthen toroidal dipole resonance and further explore its coupling characteristic with other materials or electromagnetic multiples .For instance, toroidal dipole resonance coupling with an electric dipole or other multipoles not only enhances itself, but also produces an analogy of electromagnetically induced transparency (EIT), called plasmon induced transparency (PIT), in accompany with a transparent window within a broad absorption band [11-16, 24, 28, 29, 39, 44, 45].There are many potential applications based on PIT-based toroidal metamaterials, including the design of optical buffers and highly sensitive sensors, realization of ultrafast optical switches and optical memory.Both single-and dual-band PIT properties have been discussed in reported toroidal metamaterials [10-16, 24, 28, 29, 39, 44, 45].However, triple-band transparency windows induced by multiple coupling mechanisms based on toroidal dipole resonance have not been reported.
In this study, we propose a hybrid metastructure comprising asymmetric double rings (DR) and an upright rod.The asymmetric DR component supports toroidal dipole resonance, characterized by strong field localization, whereas the rod element exhibits an electric dipole resonance property, oscillating at the same resonant wavelength.The results show that when the toroidal dipole resonance couples with electric dipole resonance attached to the rod located at the center of the asymmetric DR, a dual-band PIT can be achieved, which stems from two different hybridization couplings between the toroidal and electric dipole modes.Furthermore, by introducing a lateral displacement of the rod with respect to the asymmetric DR, a high-order resonant mode can be induced, obtaining a triple-band PIT with a broadband transparency window by multiple couplings.This study is expected to provide a positive reference for toroidal-based meta-devices in the future.

Numerical model
To explore the coupling characteristics by toroidal dipole resonance, figure 1 illustrates the geometry of the composite metamaterial design.It consists of an asymmetric DR placed horizontally and an upright rod.For the toroidal DR component, geometric parameters are inner radius r 3 = 95 nm, small outer radius r 2 = 290 nm as well as large outer radius r 1 = 310 nm with a radius defect Δr = r 1r 2 , thickness t = 20 nm.The upper-ring is flipped with respect to the x-axis to get the lower one with a 15 nm thick gap layer.The suspending rod has a length of 200 nm and a diameter of 30 nm.And Δy is defined as the on-axis displacement from the origin 'O' to the center 'C' of the rod.The silver is adopted as the material following the Drude-type dispersion model: with the high-frequency permittivity ε ∞ = 6.0, the plasma frequency ω p = 1.5 × 10 16 s −1 and the collision frequency γ = 7.73 × 10 13 s −1 [5].The incident light propagates along z direction with the polarized electric field in the y-axis.In the numerical simulations, the multimode coupling based on toroidal dipole response is explored by a full-wave solver based on the finite-difference time-domain method [16].

Results and discussions
The proposed hybrid metastructure exhibits distinct optical properties, and the calculated transmission spectra in dependence on the axial displacement Δy are characterized in figure 2. When the rod is suspended 150 nm above the asymmetric DR, both are not yet coupled with each other, contributing to a solitary transmission dip, as shown in figure 2(a).In fact, from the perspective of electromagnetic excitation, apart from supporting a broad electric dipole resonance mode attached to the rod, our metastructure also supports a narrow toroidal dipole mode by asymmetric DR.Therefore, there is an overlap in the transmission spectrum.In addition, by calculating the multipole scattered powers in terms of various multipolar moments in figure 3(a), one can see that the toroidal dipole (T) and electric dipole (P) dominate this mode around the resonance wavelength.Although the scattered power of magnetic dipole moment is comparable to that of electric dipole moment, there is no corresponding resonant characteristic in the spectrum.Therefore, it is reasonable here that the electric   To fully visualize the coupling and reveal the physical mechanism, we label the two resonant peaks separately as R1 and R2 and present 2D distributions of the corresponding magnetic field, as illustrated in figure 4. For the short-wavelength mode R1 in figure 4(a), the magnetic field on the metal rod is anti-clockwise, while those on the asymmetric DR oscillates clockwise, with out-of-phase magnetic vortex distribution, indicating destructive interference between the toroidal and electric dipole resonances.Conversely, in the case of the long-wavelength mode R2 in figure 4(b), the magnetic field distribution of the asymmetric DR is just the reverse of that at R1 resonance, with an in-phase magnetic vortex distribution relative to the rod, suggesting a constructive interference mechanism between these two different modes.
To further probe the excitation mechanism for these two coupling modes, the symmetric DR metastructure without a radial defect (Δr = 0 nm) is investigated for contrast analysis.It should be emphasized that other parameters (including those of the rod) remain invariable.In the absence of radial defect Δr, symmetric DR under the same incident radiation can still excite a strong toroidal dipole mode.When the toroidal dipole mode couples with the electric dipole resonance by the rod, there is a single-band PIT effect as indicated in figure 5(a).According to the 2D magnetic field distribution in the right black box, it results from an out-of-phase coupling between these two resonance modes.This result is similar to that of resonance R1, as discussed above.However, when the radial asymmetry Δr is 10 nm, there is a tiny split in the transmission spectrum in figure 5(b).As Δr continues rising to 20 nm, a prominent split can be observed in figure 5(d), producing a dual-band PIT effect.In other words, the radial asymmetry Δr causes another coupling mechanisms (i.e., coupling).Therefore, based on the comparative analysis, it is easy to understand that it is an introduction of radial defect Δr that makes the dual coupling mechanisms accessible in our designed constituent metastructure.Furthermore, we discuss the effect of the rotation offset angle on the transmission spectrum and the coupling interference between the constituent elements.Keeping the lower ring fixed, the angle at which the upper ring rotates around y-axis (following the white arrows) is considered as the rotation offset angle, varying from 0°to 180°as shown in figures 6(b) and (c).As can be seen from figure 6(a), the dual-band transparency window is gradually transformed into a single-band window with a transmission peak marked by R4 as the rotation offset angle increases, which can be interpreted as the removal of the defect factor.Actually, when the offset angle is 180°, the structure shown in figure 6(c) is equivalent to the symmetric DR metastructure presented in figure 5(a), possessing almost the same transmission performance.However, it is worthwhile to mention that the central resonance wavelength of peak R4 is smaller than that of R3.This is because the effective radius r 2 of the structure in figure 6(c) is smaller than the radius of the symmetric DR, as shown in figure 5(a).
From the above discussion, it can be seen that the position of the upright rod relative to the asymmetric DR along y-axis is crucial for the coupling between them.Here, we define Δx as the lateral displacement of the rod along x-direction with respect to the asymmetric DR component, and further investigate the influence of Δx on the transmission spectrum and hybrid coupling in figure 7. Intuitively, Δx induces a new resonance mode at a short wavelength of 910 nm, and the transmission intensity increases with increasing Δx, producing a tripleband PIT phenomenon, as shown in figure 7(a), which is rarely reported in toroidal metamaterials.To make the discussion easier to follow, we provide the magnetic field distribution at resonance R5 labelled for the case of Δx = 60 nm.It is apparent from the simulation results that the movement of the rod along the +x direction enhances coupling with the right half of the asymmetric DR, forming a high-order hybrid resonance mode as shown in figure 7(b).By continuing to increase the value of Δx, in principle, a broadband transparency window across a wavelength range greater than 120 nm can be achieved, where the transmission exceeds 0.9.The underlying reason for this is the hybrid coupling between the toroidal dipole, electric dipole and induced highorder resonance modes.This result is superior to those in reported toroidal-based PIT metamaterials, which are usually characterized by a solitary and narrow transparency window [16,29,44].

Summary
In summary, multiple couplings based on toroidal dipole response are explored in a composite metamaterial composed of an asymmetric DR and an upright rod.These two constituent elements support the toroidal and electric dipole resonances, respectively, excited at the same resonant wavelength and characterized by a solitary transmission dip.When the rod element is moved to the center of the asymmetric DR, a double-band PIT effect occurs owing to the constructive and destructive interferences between these two resonance modes.Meanwhile, the dual-band transparency windows can be tuned to a single-band window by introducing a rotating offset angle.Moreover, under the impetus of a lateral displacement, our coupling system will achieve a triple-band PIT along with a high transmission intensity within a broadband transparency window.These results promote an understanding of the toroidal dipole moment and provide a positive reference for optoelectronic devices applications.

Figure 1 .
Figure 1.Schematic drawing of the asymmetric DR with a rod suspended above it and the incident light polarization configuration.The origin 'O' of the coordinate system is located at the center of the asymmetric DB component.

Figure 4 .
Figure 4.The 2D magnetic field distributions.(a) For the short-wavelength mode R1.(b) For the long-wavelength mode R2.

Figure 5 .
Figure 5.The transmission spectra of the hybrid metastructure for the different radial asymmetry Δr.(a) Δr = 0 nm, (b) Δr = 10 nm, (c) Δr = 15 nm, (d) Δr = 20 nm.The inset in the right black box are the symmetric DR metastructure without a radial defect (for the case of Δr = 0 nm) and the 2D magnetic field distribution of resonance R3 marked by pink triangle in (a).