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Abstract
Toroidal dipole moment has attracted much attention in recent years due to their novel electromagnetic response such as non-reciprocal interactions and unusual low-radiating manifestations. However, most of the previously reported toroidal dipole moment are incapable of real-time control of direction and intensity. In this paper, an actively tunable toroidal metamaterials are proposed to achieve programmable toroidal dipole manipulations with electric control. The intensity and direction of toroidal dipole can be sensitively regulated by electrically controlling the loaded diodes. Our proof-of-concept experiments show that the toroidal dipole could be dynamically switched to the electric and magnetic dipole. Meantime, the direction of toroidal dipole also could be controlled. Experimental and numerical results, in good agreement, demonstrate good performance of the proposed toroidal metamaterials, with potential applications in modulators, sensors, and filters.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Electromagnetic metamaterials, which are engineered subwavelength periodic structures, have gained considerable attention in recent years, owing to their unusual electromagnetic properties. Metamaterials have been widely used to arbitrarily manipulate the scattering and propagation electromagnetic waves in unconventional ways, leading to many novel devices and exciting phenomena [1–4]. The traditional electric and magnetic multipoles of metamaterials originating from the individual resonators is due to induced charge and current distribution, respectively.
In 1957, toroidal dipoles first introduced by Y. B. Zeldovich and have already been considered in the context of nuclear and particle physics [5]. In particular, a number of works have led to the theoretical understanding of toroidal electrodynamics in the field of electromagnetism [6,7]. In contrast to the electric and magnetic multipoles, toroidal dipoles can be created by poloidal currents, flowing on a surface of a torus-like shaped structure along its meridians [8–11]. It has attracted considerable interest due to their unusual electromagnetic properties, including non-reciprocal interactions and dynamic non-radiating charge current configurations [12–14]. In the far field zone, far field radiation of toroidal dipole matches with electric dipoles. It can lead to destructive interference effects, vanishing scattering accompanied with strong light confinement and highly nontrivial field distribution [15–17].
Metamaterials is served as a platform for the first observation of toroidal resonances. Metamaterials enable us to access exotic and novel electromagnetic phenomena by controlling the symmetry and character of the response through artificial structuring on a subwavelength scale. In 2007, toroidal metamaterials were first theoretically proposed by K. Marinov et al [18]. In 2010, toroidal metamaterials were first experimentally demonstrated in microwave region by T. Kaelberer et al. [19]. After that, a number of works demonstrate toroidal metamaterials [20–24]. In spite of numerous experimental and theoretical work, most of the methods have been developed and demonstrated only to obtain passive tuning of these toroidal resonance by tailoring the metamaterial design. An active tuning of the toroidal response can enhance the efficiency of the metamaterials in several applications. Therefore, recently active tuning toroidal metamaterials have gained a lot of attention [11,25]. Toroidal metamaterial switch [26], active tuning metamaterials using graphene [27], and reconfigurable terahertz metamaterials based on microelectromechanical systems have been developed and demonstrated to obtain a tunable toroidal response in metamaterials. The majority of tunable toroidal metamaterials realized to date are intensity control. The direction control of toroidal dipole cannot be realized by external biased voltage, conductivity or thermal actuation.
The direction control has far-reaching implications for toroidal dipole. Similar to the coupling of magnetic or electric dipoles tailoring the electromagnetic responses, coupling among toroidal moments also can form exotic electromagnetic states [7,10,28]. The interaction between two toroidal dipoles takes place through the displacement currents and external conduction, the interaction energy for parallel coupling of toroidal dipoles is different from the case of antiparallel coupling of toroidal dipoles. Parallel coupling lowers the entire energy of the toroidal dipoles, whereas antiparallel coupling raises the entire energy [29,30]. Therefore, tailoring the direction of toroidal dipoles has received considerable attention given the potentially significant impact towards controlling radiative fields. Further, achieving a tunable of direction could impact the utility of transfer of electromagnetic energy using toroidal moments, adaptive reflective metasurfaces, and electromagnetic induced transparency.
The concept of digital coding metamaterials was first proposed by T. J. Cui et al in 2014 [31]. The digital metamaterial can manipulate electromagnetic waves and realize different functionalities by programming different coding sequences. It builds up a bridge between the digital world and the physical world. Digital coding metamaterials have experienced a rapid development since they were firstly proposed [32–35]. It has been extended to microwave and terahertz frequencies, as well as to acoustic scenarios [36–38]. Digital coding metamaterials provides a new way to realize the actively tunable toroidal response at microwave frequencies.
In this work, we propose and realize experimentally an actively tunable toroidal metamaterials at microwave frequencies. Our research is distinct from previous studies of tunable toroidal metamaterials, where the loaded diodes were used to realize the intensity and direction control of toroidal dipoles. By sensibly regulating electromagnetic resonance of both resonators in mirrored configuration, we can dynamically switch from open to close of toroidal resonance to a magnetic dipole resonance. More importantly, since each resonator is equipped with two diodes, the proposed metamaterials are able to agilely control the direction of toroidal dipole. Finally, a 5×5 tunable toroidal metamaterials with loaded diodes is fabricated and experimentally tested. The proposed approach promises important advantages in scenarios such as antenna systems and wireless communication.
2. Model structure design
The actively tunable toroidal metamaterials are enabled by electrically controlling the loaded diodes. The schematic of the proposed metamaterials array is illustrated in Fig. 1. In the proposed structure, the unit cell is parametrically tailored for our purpose. In each unit cell, the structure consists of two symmetrical split ring resonators (SRRs). The unit cells of copper are printed on the F4B substrate with a relative permittivity 4.3 × (1 + 0.025i), and the thickness of F4B is 0.8 mm. The detailed geometric parameters of the configuration are as follows: a = 10 mm, b = 0.2 mm, c = 4.3 mm, d = 3 mm, h = 0.8 mm, f = 0.3 mm and e = 0.5 mm. To control the toroidal dipole, four diodes (Macom-14020) and two capacitances (1nF) are loaded in each unit cell. The photograph of the fabricated sample is shown in Fig. 2. Meantime, to apply the direct current bias voltage, the bottom surface is connected with four pieces of separated grounds on the bottom surface through four metallic via holes. Because the surrounded unit cells have same states, the electromagnetic responses of metamaterials were performed with unit cell boundary condition in a commercial software package CST Microwave Studio, which assumes an infinite periodic structure composed of identical unit cells. In our study, the metamaterials are illuminated by x polarized (the electric field E is along the x axis) electromagnetic waves. The diode is “ON” and “OFF” with a biasing voltage of 3.3 V, and 0 V, respectively, the corresponding coding state is 1 and 0. In the simulation, the accurate equivalent circuit model is experimentally obtained, which are C = 0.025 pF, R = 7.8 Ω at the “ON” state, and L = 0.03 pH, C = 0.025 pF at the “OFF” state around 3 GHz.
Fig. 1. Schematic of the proposed actively tunable toroidal metamaterials prototype and geometry of the unit cell.
3. Results and analysis
To explore resonant properties of the proposed structure, the simulated and measured transmission spectral curves for the proposed structure pertaining to the “ON (+x)”, “ON (-x)”and “OFF” state of the toroidal dipole are shown in Fig. 3. In addition, to understand the working mechanism of the toroidal dipolar resonance more clearly, the artistic visuals of the toroidal dipole excitation are also shown in Fig. 3(a) and 3(b). The four diodes are placed in the “1”, “2”, “3”, and “4” position, as shown in Fig. 1. Each diode has one of the two switchable states: “0” and “1”, which can be controlled by applying different biased voltages across the diode. We meticulously designed the “cross power supply method”, which can enable the diodes in position “1” and “4” or “2” and “3” in the same states. In Fig. 3(a) and 3(b), the red and green diode represents the “1” and “0” digits, respectively. When the coding sequence “1-0-0-1” (green-red-green-red as shown in Fig. 3(a)) sequence, the toroidal dipole in + x direction is excited (“ON (+x)”), while as the coding sequence changes to “0-1-1-0” (red-green-red-green as shown in Fig. 3(b)), the toroidal dipole in -x direction is excited (“ON (-x)”). Meantime, when the coding sequence is 0-0-0-0, all diodes are “OFF” state and the SRR cannot be formed, which results in the off of toroidal dipole. From the Fig. 3(c), we can find that the resonance feature at “ON” state in transmission spectrum is observed around 3.15GHz. Contrary to the “ON” state, the resonance feature disappears at “OFF” state. Referring to the related work [39], the modulation depth and quality factor can be easily calculated from Fig. 3(c). The modulation depth is about 14.3 dB, and the quality factor is about 5.5. Moreover, although the structure is highly symmetrical, there are still difference between “ON (+x)” and “ON (-x)” state, which is due to the feeding circuit design. The measured transmission spectral curves are demonstrated in Fig. 3(d). It can be seen that the resonance appears at 3.2 GHz, the modulation depth is about 4.16 dB, and the quality factor is about 4.2. Compared to the simulated resonance peak, there are little deviations in the experimental results. Moreover, for the modulation depth and quality factor, the simulation delivers a higher value than the measurement. This is because in the measurement, a 5×5 array is fabricated, so the received power efficiency is limited. A higher modulation and quality factor can be measured with larger array.
Fig. 3. Schematic of toroidal dipole excitation at the state of (a) 1-0-0-1 (ON(+x)), and (b) 0-1-1-0 (ON(-x)). (c) Simulated and (d) measured transmission spectral curves for the proposed structure when the diode is at the state of “OFF”, “ON(+x)”, and “ON(-x)”.
The total scattering cross section is related to the induced multipole moments as [10–14]
The results of such calculations are presented in Fig. 4. From the Fig. 4(a). we can find that on the spectral resonance position at ON (+x) state, the toroidal dipole provides the strong contribution at the resonance (the ON (-x) state is ignored because of the highly symmetrical structure). The phases of the electric and the toroidal dipole moments are calculated in Fig. 4(b). It can be seen that the electric and toroidal dipole moments show destructive interference of their radiation due to equivalence of their scattering patterns in the resonance frequency, which results in the low radiation and generates the resonance. Meantime, we can find that the magnetic dipole, electric dipole, and electric quadrupole moments also show the strong contribution at the resonance frequency, which is due to the introduction of diodes and feeding circuit design. The introduction of diodes and feeding circuit design strengthens others dipole at resonance frequency. For example, the diodes provide the resistance in the structure and result in the uneven current distribution. The uneven distribution of current causes the excitation of multipole. Moreover, the introduction of feeding circuit design excites new electromagnetic that we do not expect. While at OFF state (Fig. 4(c)), there has been an apparent downward trend in the multipole moments and the toroidal dipole almost disappears. The toroidal dipole at OFF state exhibit scattering rate that is factors of 109 smaller than that of toroidal dipole at ON (+x) state.
Fig. 4. Contributions of the multipolar excitations to the transmission of the metamaterial array is at the state of (a) ON(+x) and (c) OFF. (b) The phases of the electric and the toroidal dipole moments vs frequency at the state of ON(+x).
To get clear idea about tunable toroidal dipole, the simulated magnetic field and surface current distributions at resonance frequency for the different state are depicted in Fig. 5. The simulated magnetic fields in O-yz plane and O-xy plane are shown in Fig. 5(a) and 5(b), respectively. The density of the displacement currents induced metamolecule are calculated, which are depicted in Fig. 5(c). From the figure, one can see that each SRR forms a circulation of electric currents at ON(+x) state, thus, two SRRs induce magnetic fields toward opposite directions. This is equivalent to the magnetic field distributions of a two-dimensional toroid placed in y-z plane and encircling all two SRRs, which produces a toroidal dipole moment in + x direction. In a similar manner, at ON(-x) state, the directions of the electric current of two SRRs are reversed and produce another toroidal dipole moment in -x direction. However, from the Fig. 5(c). we can find that the current is unevenly distributed (current is significantly suppressed at the diode) and results in the unevenly distribution of magnetic field. The inhomogeneous electromagnetic field further promotes the excitation of electric and magnetic dipole moments. Meantime, from the Fig. 5(b), it can be seen that distinct magnetic fields is excited on both sides of toroidal dipole, which is due to the introduction of feeding circuit design. The strips (feeding circuit design) induce linearly oscillating currents toward opposite directions relative SRRs, which results in the excitation of magnetic dipole. Therefore, magnetic dipole, electric dipole, and electric quadrupole moments (As shown in Fig. 4(a)) are attributed to the introduction of diodes and feeding circuit design. When all diodes are OFF state, the electric current becomes weaker and the circulation of electric currents cannot be excited efficiently, the magnetic field are placed symmetrically along y-direction with respect to x-axis. Therefore, stronger magnetic fields can be excited, but the toroidal dipole cannot be excited.
Fig. 5. Simulated distributions of magnetic field at resonance frequency in (a) O-yz plane and (b) O-xy plane at the state of OFF, ON(+x) and ON(-x). (c) Simulated distributions of surface currents induced on the structure. The black, purple and red arrows illustrate the instantaneous directions of the toroidal, magnetic and electric current densities, respectively.
To further demonstrate the features of the structure, we simplify the structure as shown in Fig. 6. As it shown that the feeding circuit design has been removed, the diode that in state “1” and the capacitances are visualized as path. From the figure, we can find that when the state is changed, the position of the split of SRR are changed. Therefore, the direction control of toroidal dipole is attributed to the change of the position of the split of SRR. When the excitation perpendicular to the metamaterials is adopted, the induced current flows on the strip metal, and the gap between strip metals will form capacitance, and the structure will produce electric dipole. When the position of the split of SRR is changed, the electric dipole moment presents opposite sign due to the 180 degrees shift in the split position, and results in the reverse-current path [40]. The reverse-current excites the reverse-magnetic field, eventually the direction of toroidal dipole is changed.
In order to further verify the exactness of the simulation results, an equivalent LC-circuit model is used to model the transmission characteristics of the proposed. Taking ON(+x) state as an example, the inductances, the resistance and the ring capacitance may be represented in terms of distributed elements giving rise to the equivalent circuit shown in Fig. 7. The split gaps and diodes provide the capacitance (C1∼C2), the copper arms provide the inductances (L1∼L3), the dissipations in them provide the resistance (R). The coupling between two adjacent SRRs provide the capacitance (C3), as for the coupling inductance, it is 5 orders of magnitude smaller than L1∼L3. Therefore, the influence of coupling inductance in not considered in our case. C1∼C3, L1∼L3 in the resonant circuit could be given by the following formula [41]:
where t is the thickness of the dielectric, l is the length of the copper arm, w is the width of the copper arm, h is the thickness of the copper, µ0 is the permeability of vacuum, ɛ is the dielectric constant, G(x) is a function that for ${}_{\to}x\to 0$ behaves as -log(x), C0 is the numerical factor in the range 0.2 ≤ C0 ≤ 0.3. According to Eqs. (7–9), the values of C1∼C3 and L1∼L3 are calculated and shown in Table 1. The resonance frequency can be written as [41]:To demonstrate the features of LC-circuit, we have established equivalent circuit model in a commercial software ADS (Advanced Design System) based on the calculated results of Table 1. Both the simulated and measured transmission curves at ON(+x) state are shown in Fig. 8 for better comparison, with the simulated resonance frequency at 4.52 GHz, and the calculated resonance frequency at 4.612 GHz. We can find that there were several discrepancies in the transmission spectral curves, which is due to the coupling capacitance error. Furthermore, compared to the Fig. 3(c), we can find that in the non-simplified structure, the resonance frequency shows a redshift, and the intensity is greatly diminished, which is due to the introduction of the feeding circuit design. The feeding circuit design provides the inductance and resistance. According to Eq. (10), the resonance frequency shows a decrease with the increase of the inductance, and results in the red shift of resonance frequency. Meantime, resistance bring more losses and results in the decrease of strength of resonance.
Fig. 8. (a) Simulated and (b) calculated transmission spectral curves for the proposed structure at ON(+x) state.
4. Conclusion
In summary, an actively tunable toroidal metamaterials at microwave frequencies are experimentally and theoretically investigated according to the analysis of the multipole scattering intensity, and the distribution of magnetic field and surface currents. Each resonator is equipped with two diodes, the intensity and direction of toroidal dipole can be sensitively regulated by electrically controlling the loaded diodes. We have experimentally demonstrated that when the coding sequence is 0-0-0-0, toroidal dipole is “OFF” state. For the “1-0-0-1” sequence, ON (+x) state is excited, while as the coding sequence changes to “0-1-1-0”, ON (-x) state is excited. A simple equivalent LC-circuit approach is used to demonstrate the features of the structure, that further prove the feasibility of the proposed structure. The as-proposed design method based on the digital coding metamaterials extends the brand-new application field of the toroidal dipole. Moreover, when each unit cell can be independently programmed to realized “ON(+x)” or “ON(-x)” state by controlling the applied voltage bias of the diode, more exotic and novel electromagnetic phenomena can be discovered.
Funding
National Natural Science Foundation of China (11774054, 12075036, 62005025); Science and Technology Research Project of Education Department of Hubei Province (D20201303); Natural Science Foundation of Jiangsu Province (BK20161410).
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.
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