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Abstract
An antisymmetric toroidal dipole array of plasmonic metasurfaces, whose unit cell consisted of a pair of physically connected asymmetric split-ring resonators, is presented in this study. Moreover, a new paradigm was established to control toroidal electric dipole properties. Toroidal electric dipoles and electric and magnetic hybrid pseudo-anapole states are excited owing to imperfect and perfect destructive interference, respectively, which leads to the spatial separation of the electric and magnetic fields and a distinct asymmetric Fano line shape in the transmission spectrum. The imperfect destructive interference was further modified by adjusting the relative position between the even and odd layers of the metasurfaces. The scattered power of the toroidal electric dipole is tuned continuously and linearly, which enables the tailoring of the electromagnetic response. The displacement sensitivity is approximately 0.13 GHz/mm over the range 0–8 mm. The modulation depth of the scattered power of the toroidal electric dipole can reach 740%, realising a toroidal electric-dipole-to-electric-dipole transition. The proposed plasmonic metasurfaces provide a platform to efficiently control near-field enhancement, far-field radiation, and electric-magnetic separation and find potential applications in frequency-selective surfaces, sensors, filters, spectroscopic tests, and many other areas.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Toroidal multipoles are the third family of electromagnetic multipoles with unique characteristics that differ from the widely known electric and magnetic multipoles, which originate from their fantastical field-localisation configuration in a head-to-tail manner [1–3]. However, the scattering of external fields from toroidal multipoles is often relatively weaker because the contributions of other electromagnetic moments mask their contribution [4–6]. In 2007, Marinov et al. reported the first observation of dynamic toroidal dipoles across the microwave regime in novel metamaterials, which exhibited electromagnetic properties which did not occur in natural materials [7]. This allows for the suppression of electric and magnetic moments and, simultaneously, the strengthening of the toroidal moment to a detectable level. Since then, toroidal dipoles of different qualities have been excited in various types of metamaterials, from visible to microwave frequencies [8–10]. Moreover, owing to the fact that toroidal moments enhance device performance in terms of quality factor, circular dichroism, and resonant transparency, the practical use of toroidal metamaterials has received considerable research attention [11–13]. Considering the exotic properties of these spectral features, well-engineered toroidal metamaterials have emerged for designing devices such as nonlinear excitations, lasers, sensors, and active modulation [14–16].
As the unit cell modifies the metamaterials, the properties of toroidal metamaterials make it feasible to tune the toroidal responses by reconfiguring the material, size, and shape of the unit cell. To date, most toroidal metamaterials have been designed to achieve pronounced toroidal dipoles by modifying the toroidal metamolecules. To achieve a pronounced toroidal dipole, split-ring resonators and their variants, metallic double disks, and vertically assembled dumbbell-shaped aperture structures have been designed by mimicking the torus configuration such that the toroidal excitation is amplified [17–19]. Tailoring the near-field properties of the toroidal moment is a significant task for controlling the toroidal moment radiation. Generally, this task is achieved by the selective inclusion of active materials in the hybrid metamolecule design [20,21]. Therefore, to determine whether the excitation or regulation of toroidal moments occurs, researchers pay more attention to the signal toroidal dipole and ignore the interaction between the two toroidal dipoles.
A distinct feature of toroidal metamaterials is their non-radiating nature. The far-field radiation pattern of the toroidal dipole is identical to that of electric dipoles; therefore, destructive interference between the toroidal and electric dipoles yields a non-radiating anapole mode [22–24]. We can precisely tune the strengths of the toroidal and electric moments using geometry to cancel the radiation pattern. Meanwhile, instead of the destructive interference between the toroidal and electric moments in the conventional anapole mode, the pseudo-anapole effect arises when both the toroidal and electric moments tend to a minimum [25,26]. Researchers suppressed the electric-type radiation of metamolecules to the level of other multipoles, leading to the excitation of higher-order multipoles. Therefore, regardless of the anapole or pseudo-anapole modes, metamolecules are still regarded as the main body of the design.
With this understanding, this study demonstrates an antisymmetric toroidal dipole array of plasmonic metasurfaces utilising double toroidal electric dipoles with opposite directions and different intensities as the underlying attributes of the unit elements. Previously, a toroidal dipole was considered as a research object. Herein, we propose an innovative reverse toroidal electric dipole structure for controlling toroidal dipoles. The destructive interference between the two toroidal electric dipoles can lead to a spatial separation of the electric and magnetic fields, which simultaneously excites both the toroidal electric dipole and hybrid pseudo-anapole states. In addition, the scattered power of the hybrid pseudo-anapole states is robust against asymmetric structures. We discussed the role of the coupling effect between reverse toroidal electric dipoles. By adjusting the relative positions of the even and odd layers of the metasurfaces, we realised a toroidal electric-dipole-to-electric-dipole transition. The modulation depth of the scattered power of the toroidal electric dipole reaches 740%. In contrast to the aforementioned tuning mechanism, the toroidal electric dipole is modified by adjusting the interaction between two toroidal electric dipoles. These findings open avenues for the active control of the toroidal dipole and anapole mode metasurface design and development.
2. Results and discussions
Our analysis focused on two antiparallel toroidal electric dipoles. The diagrams in Fig. 1 elucidate the key concept of the antisymmetric toroidal dipole array of plasmonic metasurfaces. Each supercell consists of two antiparallel toroidal electric dipoles. When the pairs have the same strength, the two individual toroidal electric dipoles oscillate in antiphase, leading to net suppression of the toroidal electric dipole. Once the strength of the two individual toroidal electric dipoles differed, the enhancement of the toroidal electric dipole was visualised. This highlights the idea that the far-field response can be dynamically controlled by adjusting the relationship between the two toroidal electric dipoles. To highlight the interaction between two toroidal electric dipoles, which is different from the traditional toroidal electric dipole metasurface in which the unit cells are independent of each other, the unit cells of our proposed metasurface are physically connected. Because the interaction between two toroidal dipoles is excited through the external conduction and displacement currents, physical connection aids in further regulating the external conduction and displacement current [27].
Fig. 1. (a) General schematic of the antiparallel toroidal electric dipoles. (b) Schematic of the proposed metasurface. (c) Calculated spectra of the metasurface. (d) Radiating power for induced multipoles of the metasurface. (e) Typical assembly of peaks associated with either electric or magnetic resonances of the metasurface
Figure 1(b) shows a schematic of the proposed toroidal metasurface consisting of a square array of Cu split-rings arranged on top of the substrate. The periods of the square array were x = 16 m and y = 9 mm. To demonstrate the high precision control of toroidal dipole, we use the structure with a = 3.7 mm, b = 6.2 mm, c = 3.4 mm, d = 0.6 mm, e = 0.4 mm, and g = 2.4 mm. The structure was formed by alternately stacking two layers (even and odd). The even and odd layers are staggered by h. The electromagnetic responses and physical mechanisms of the proposed toroidal dipole array are simulated using CST Studio. When h = 0 mm, no dislocations are observed between the even and odd layers of the metamaterial. Figure 1(c) shows the simulated transmission, reflection and absorption spectra. From the figure, we can find that transmission spectrum exhibits a dip at 9.48 GHz and a peak at 9.51 GHz. Specifically, the excitation of the resonance leads to a distinct asymmetric Fano line shape with a characteristic peak/dip pair profile.
To study the contributions to far-field radiation of Fano resonances of Fig. 1(c). We obtained the electromagnetic response by decomposing the scattering power of the multipoles. We estimate cartesian multipoles contributing to the total scattering cross-section using the multipole expansion method. The multipole decomposition of the scattered power is given by [28]
To clarify the coupling mechanism of the Fano resonance, p + T(e)+ T(2e), m + T(m), Q(e) + T(Qe), and Q(m) + T(Qm) can be treated as the total electric dipole, magnetic dipole, electric quadrupole, and magnetic quadrupole moments, respectively. Figure 1(e) shows a typical assembly of peaks associated with the electric or magnetic resonances of the structure. The analysis clearly shows that the resonant peak not only has the minimum total magnetic dipole moment, but also the minimum total electric dipole moment. The scattered power of the total magnetic dipole was much lower than that of the total electric dipole because the magnetic field region had stronger symmetry than the electric field region (Figure S1, Supplement 1), resulting in stronger destructive interference of the radiated fields. The minimum of the total electric and magnetic dipoles helps enhance the near fields. We achieve an attainable nonradiative state without the interference of a particular multipole-like anapole state, which is called the electric and magnetic pseudo-anapole state (hybrid pseudo-anapole states). Therefore, the Fano resonance is formed by the interference between the toroidal dipole resonance and the hybrid anapole states.
To further verify the near-field localisation characteristics, the field and current distributions in the metamolecules are shown in Fig. 2. From the figure, we can find that whether at 9.48 GHz or 9.51 Ghz, the electric field is localized in the gap, and the magnetic field corresponds to the closed vortex circulating around the central axis of the metamolecule. We found that the magnetic field exhibited stronger antisymmetry than the electric field. At 9.48 GHz, opposite circular displacement currents on the right and left sides of the structure generate a circular magnetic moment distribution (Fig. 2(c)) perpendicular to the structure surface. This provided a toroidal electric moment (Fig. 2(g)) oriented parallel to the structural surface. The resonant electric field enhancement |E|/|E0| and the magnetic field enhancement |H|/|H0| were approximately 523 and 22, respectively. The electric field enhancement of the structure is approximately 24 times higher than the magnetic field enhancements, which leads to the separation of the electric and magnetic fields. Strong electric fields exhibit high asymmetry, resulting in imperfect destructive interference and a strong toroidal dipole. In contrast, at 9.51 GHz, the resonant electric field enhancement |E|/|E0| (Fig. 2(b)) and magnetic field enhancement |H|/|H0| (Fig. 2(d)) were approximately 93 and 467, respectively. The magnetic field enhancement of the structure was approximately five times higher than the electric field enhancement. A strong magnetic field caused the structure to exhibit an asymmetric mode. Although such a strong current configuration gives rise to magnetic moments, the latter cancel one another, leading to a negligible total magnetic moment and resulting in the magnetic pseudo-anapole mode.
Fig. 2. Distribution of induced electric field at (a) 9.48 GHz and (b) 9.51 GHz in the xy-plane, respectively. Distribution of induced magnetic field at (c) 9.48 GHz and (d) 9.51 GHz in the xy-plane, respectively. Surface current distribution at (e) 9.48 GHz and (f) 9.51 GHz in the xy-plane, respectively. Distribution of vectorial magnetic field at (g) 9.48 GHz and (h) 9.51 GHz in the yz-plane, respectively.
Meanwhile, owing to the asymmetry of the electric field at 9.48 GHz, the current in the middle of the metamolecule dominates its left and right sides, oscillating in the antiphase, which results in a dipole-like resonant decrease in the metamaterial transmission. However, at 9.51 GHz, both the magnetic and electric dipole radiation of this current configuration are cancelled, thus considerably reducing radiation losses and increasing metamaterial transmission.
We now focus on the influence of the structural parameters on the electromagnetic characteristics. The resonant characteristics for different asymmetric parameters, L = c/(2 g), were investigated. Asymmetry parameters were varied by varying the width of the vertical arm (g). Figure 3(a) shows the transmission spectra for different values of L. As L decreases (the asymmetry profiles of the structure are highlighted), the Fano line shape is continuously modified from disappearing to a characteristic asymmetric resonance. Our numerical analysis showed that as L decreases from 1 to 0.7, the resonant frequency has a redshift, whereas it has a slight blue shift when L further decreases from 0.7 to 0.6. The magnitude of the resonance depends on asymmetry. As L increases, the symmetry of the structure improves, which reduces the radiation losses and increases the metamaterial transmission. The resonance weakened until it disappeared in the fully symmetric structure. Therefore, the excitation of the toroidal dipole and hybrid pseudo-anapole states originates from the structural asymmetry.
Fig. 3. (a) Transmission spectra under different asymmetry parameters L. Numerically calculated contribution of (b)total electric dipole, (c) electric dipole, and (d) toroidal electric dipole at different values of L.
To obtain a clear idea of this effect, Fig. 3(b) shows the scattered power of the total electric dipole for different L values. The total electric dipole is highly sensitive to asymmetry parameter L. The total electric dipole changes from 0.214 to 1.344 as L decrease from 1 to 0.6. The asymmetric and antisymmetric modes were compared. It can be seen that the change of scattered power mainly comes from the asymmetric mode. For further analysis, Figs. 3(c) and 3(d) illustrate the scattered powers of p and T(e) for different values of L. From the figure, we can see that regardless of the electric or toroidal electric dipole, the scattered power is more easily influenced by L in the asymmetric mode. However, the scattered power hardly changes in the antisymmetric mode, particularly in the toroidal electric dipole. Interestingly, the scattered power of the toroidal electric dipole is robust against the asymmetric structure in the hybrid pseudo-anapole states. There was no significant change in the scattered power of the toroidal electric dipole when L decreases from 0.89 to 0.6.
The effect of the symmetry of the structure on the scattered power was confirmed by calculating the corresponding electromagnetic near-field distribution (Figure S2, Supplement 1). To obtain a clearer picture, the evolution of the field enhancement with increasing L is shown in Fig. 4. A change of one order of magnitude in the electric-magnetic separation is obtained when L increases. At the resonant dip (asymmetric mode), the electric and magnetic field enhancement increased with increasing L. At the resonant peak, as L increases, the electric field enhancement increases, whereas the magnetic field enhancement decreases. The rate of change of the electric-magnetic separation is much greater in the asymmetric mode than in the antisymmetric mode. Therefore, the change in the scattered power mainly comes from the asymmetric mode, as shown in Fig. 3.
Fig. 4. Electric and magnetic field enhancement, and electric-magnetic separation factor under different asymmetry parameters L at resonance (a) dip and (b) peak, respectively.
Based on the simulation, we determined that the electric toroidal dipole can be controlled by tailoring the asymmetry of the structure. Subsequently, we explored the effect of h on the resonance frequency. Figure 5(a) shows the simulated and experimental results, which depict the transmitted intensity response with increasing h. In the experiment, a metamaterial sample was fabricated by integrating metallic structures on a Rogers RO3003 dielectric substrate using a standard printed circuit board technique, as shown in Fig. 5(a). The sample was characterised using free-space transmission measurements. A vector network analyser (Agilent E5071C) and two linearly polarised horn antennas were used to transmit the EM waves onto the sample and receive the transmitted signal. We first focus our attention on the spectral position of the metamaterial transmission dips, and it is visualised that there is a resonance frequency shift from 8.49 GHz to 9.48 GHz for the resonance in simulations. The results show a linear relationship between resonant frequency and displacement. The sensitivity Δf/Δmm defined as the spectral shift per displacement, is studied in Fig. 6. By employing linear fitting, the sensitivity was approximately 0.13 GHz/mm over the range 0–8 mm. The measured frequency shift follows the trend observed in the simulations. The spectral shift originated from the strong interactions between the unit cells (between the even and odd layers). We analysed the spectral shift based on the Lagrangian of the coupled system [29,30]. The even and odd layers are regarded as LC circuits. We assume that (1) the even and odd layers have the same oscillatory charge and (2) when driven near the resonant frequency, the response of each layer is approximated by a simple harmonic oscillator with resonant angular frequency $\mathrm{\omega}$0. Thus, the Lagrangian of the coupled system can be written as
Fig. 5. (a) Simulated and measured transmission spectra under different h. (b) Radiating power for induced multipoles of the metasurface under different h. (c) Surface current distribution in the xy-plane under different h.
Substituting Eq. (3) to the Euler-Lagrangian equations $\frac{d}{{dt}}\left( {\frac{\partial\mathcal{L}}{\partial\dot{Q}}} \right) - \left( {\frac{\partial\mathcal{L}}{\partial Q}} \right) = 0$, where the Ohmic loss is neglected for a phenomenological explanation, we get the equations:
Adopting the root form of $Q = Aexp({i\omega t} )$ and the normalized coupling coefficients are ${K_e} = \frac{{{M_e}}}{L}$, ${K_m} = \frac{{{M_m}}}{L}$, and $\eta = \frac{{L^{\prime}}}{L}$, the full express for the resonant frequency of the metasurface thus becomes:
The response of each layer $\mathrm{\omega }$0 is 8.96 GHZ. As we know that the interaction between two electric or magnetic dipoles has the form of $E = \frac{{3({{d_1}.r} )({{d_2}.r} )- {d_1}{d_2}}}{{4\pi {\varepsilon _0}r_{}^3}}$. where ${d_i}$ is either an electric or magnetic dipole. r represents the vector connecting them. Comparing the effective dipole configurations and equivalent coupled LC circuits, we can conclude that ${K_m}$ is positive and the ${K_e}$ is negative. Meanwhile, as shown in Fig. 5(c), with an increase in h, the circular currents excited in the even and odd layers change from in-phase to out-of-phase, and the relative orientations of the electric dipoles decrease, resulting in an increase in the interaction of the electric dipole. Therefore, the absolute value of ${K_e}$ increased with an increase in h.
Figure 7 shows the effect of h, ${K_e}$, ${K_m}$, and $\eta $ on resonant frequency. From Fig. 7(a), we can see that the resonant frequency decreases with an increase in h. Meanwhile, the resonant frequency increases with an increase in the absolute value of ${K_e}$. But, the impact of ${K_m}$ and $\eta $ is much lower than that of ${K_e}$ on resonant frequency, except ${K_m}$ and $\eta $ is much greater than 0.5. Among the interaction terms, the electric dipole is the strongest, whereas the magnetic dipole is much weaker. We assume that the ${K_m}$ and $\eta $ remain unchanged, while ${K_e}$ changes during the process of change of h. ${K_m} = 0.1$, $\eta = 0.01$, ${K_e}$ changes from -0.25 to -0.19. In Fig. 7(d), this phenomenon is numerically demonstrated by changing h and varying the resonant frequency. The calculated frequency shift follows the trend observed in the simulations.
Fig. 7. (a) Relation between resonant frequency and (a) ${K_e}$, (b) ${K_m}$, or (c) $\eta $, respectively. (d) Calculated resonant frequency under different h.
The effect of h on the intensity of individual scattered multipoles was also considered. The relationship between h and the dipole strength in the metamaterial was further investigated in Fig. 5(b). With a gradual increase in h, the amplitudes of the electric and magnetic dipoles first increased and then decreased, owing to the destruction of the symmetry of the structure (Figure S4, Supplement 1). When h = 4, the structure reaches the maximum degree of fragmentation, resulting in the maximum scattered power of the electric and magnetic dipoles. However, the amplitude of the toroidal electric dipole gradually decreases. With an increase in h, the electric toroidal dipole between the even and odd layers changed from a symmetric to an antisymmetric arrangement, which further strengthened the net suppression of the toroidal electric dipole, resulting in a decrease in the toroidal electric dipole. Clearly, h leads to a larger modulation depth for the toroidal electric dipole. The modulation depth is defined as (%) = $\frac{{{P_{h = 0}} - {P_{h = 8}}}}{{{P_{h = 8}}}} \times 100$. We obtained an extremely large modulation depth of 740% and realised the active switching of the nonradiating toroidal configuration into a highly radiating electric dipole.
3. Conclusion
In summary, we proposed and experimentally demonstrated an antisymmetric toroidal dipole array of plasmonic metasurfaces. A rigorous multipole analysis clearly revealed that the scattered power of the toroidal electric dipole can be controlled by tuning the asymmetry of the structure. The modulation depth of the scattered power of the toroidal electric dipole reached 740%, suggesting the high practicality of the device for modulation applications. In the process of dynamic tuning, the metasurface not only realises the toroidal electric-dipole-to-electric-dipole transition, but also achieves toroidal electric-dipole switching in an electric quadrupole metasurface. Meantime, the metasurface exhibits the linear relationship between the resonant frequency and displacement, the sensitivity is approximately 0.13 GHz/mm over the range of 0-8 mm. Our results offer a new scheme for extensive tuning of radiative scattering and sensors, leading to promising practical and fundamental advances and applications.
Funding
Science and Technology Research Project of Education Department of Hubei Province (D20201303); National Natural Science Foundation of China (62005025).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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