Thermally controlled electromagnetically induced transparency metamaterial through the near-field coupling of electric and toroidal resonances

We investigate a thermally controlled electromagnetically induced transparency terahertz metamaterial through the near-field coupling of electric and toroidal resonances. The fundamental unit consists of a composite design incorporating both metal and vanadium dioxide components aimed at inducing toroidal resonance, along with a pair of metal strips generating electric resonance. Simulation results authenticate the coupling mechanism and illustrate that the envisioned EIT phenomenon can be dynamically adjusted by temperature. In a coupled oscillator model analysis, the control over coupling strength primarily emerges from the fluctuating damping rate of the bright-mode oscillator. Moreover, the displacement of the EIT peak is linked to alterations in the inherent resonant frequency of the bright-mode oscillator. This study not only broadens the potential applications for toroidal terahertz metamaterials but also enhances the range of EIT methodologies available, providing practical approaches for the utilization of terahertz slow-light devices, sensors, and switch devices.


Introduction
Electromagnetically induced transparency (EIT), a phenomenon that leads to a narrow optical transparency window within a broad absorption spectrum, was initially observed in atomic systems [1].Leveraging quantum interference effects, this phenomenon effectively reduces the group velocity of incident waves, thereby finding applications in slow light devices [2].Nevertheless, the utilization of EIT in atomic systems has been constrained by demanding experimental conditions [3].Fortunately, the revelation of EIT in metamaterial, as first reported by Zhang et al in 2008, has revitalized the prospects of applying the EIT phenomenon [4].Metamaterials are artificial sub-wavelength structures capable of producing diverse electromagnetic responses that are unattainable in natural materials [5][6][7][8][9][10][11].Currently, a multitude of EIT metamaterials are being proposed and rigorously investigated for diverse materials [12][13][14][15][16], structures [17][18][19], tunable properties [20][21][22][23][24][25][26], and more.Within these EIT metamaterials, the coupling of bright and dark modes resonators emerges as an effective approach.For instance, Shen et al explored a copper-based EIT metamaterial consisting of Reuleaux triangles and Y-type wires in a bright-dark coupling mode.Through analysis of surface current distributions and radiated power of the multipole moments, the coupling responsible for the transparency peak was identified as occurring between the electric dipole and quadrupole [17].Wan et al demonstrated a broadband EIT analogue result arising from near-field interference of electric and magnetic coupling within a metamaterial composed of a ring and a cut wire pair [19].Liu et al conducted numerical and experimental investigations of an EIT metamaterial resulting from the coupling of magnetic dipole modes.Here, a bidirectional EIT phenomenon was induced by the structural symmetry breaking [20].Nonetheless, these EIT metamaterials predominantly focus on electric or magnetic coupling modes, and the utilization of toroidal resonance, a third resonance mode in addition to electric and magnetic resonances, is infrequently incorporated within EIT schemes.Especially, the exploration of an actively tunable EIT effect within toroidal metamaterials remains seldom discussed.
Moreover, the optical properties of phase-change materials can be modified by applying electrical or heatbased triggers, enabling the transition from an amorphous phase to a crystalline phase.This characteristic has highlighted typical phase-change materials such as vanadium dioxide (VO 2 ) [21]and indium antimonide (InSb) [22] as recent key players in actively tunable EIT metamaterial research, offering a novel technical approach for the advancement of multi-functional metamaterials [23,24].Therefore, the utilization of phase-change materials for exploring actively tunable EIT research based on toroidal resonance represents a significant expansion of existing implementation strategies for EIT metamaterials.
Here, we propose an EIT metamaterial stemming from the near-field coupling of electric and toroidal resonance modes.To achieve the actively tunable EIT effect, we employ a composite design incorporating both metal and phase-change materials (vanadium dioxide components) evoke toroidal resonance, serving as bright oscillator.Simultaneously, a pair of metal strips generate electric resonance, operating as dark oscillator.Our simulation results confirm that the origin of the EIT effect lies in the interaction between the bright and dark oscillators, facilitating the transfer of electromagnetic energy from the toroidal resonator to electric resonator through their coupling.Additionally, the suggested EIT-like phenomenon can be manipulated by temperature.
Compared to the current EIT metamaterials based on vanadium dioxide [15,[23][24][25][26], this paper proposes a hybrid structure integrating both metal and vanadium dioxide to induce toroidal resonance, functioning as bright mode resonators.Notably, vanadium dioxide doesn't serve as a dielectric layer in this design.Therefore, in modulating the EIT phenomenon, the focus lies on adjusting the losses of the bright mode resonators.Ultimately, this modulation achieves the transition from the EIT effect to the non-resonant curve.As depicted in figure 1(a), the unit cell consists of two components fabricated on a SiO 2 substrate: two horizontal aluminum strips (HS) and a hybrid split-ring resonator (HSRR).The HSRR is comprised of two VO 2 -based semicircle rings and two vertical aluminum strips and is positioned within the HS to induce toroidal resonance.In figure 1(b), the unit cell has periodic dimensions of P x = 140 μm and P y = 110 μm, with the length of HS denoted as L 1 , measuring 63.6 μm.The length of the two vertical aluminum strips, designated as L 2 , is 21 μm, and the width of the aluminum strips, marked as w, is set at 10 μm.The inner and outer radii of the circular rings are referred to as R 2 and R 1 , respectively, measured as 15 μm and 25 μm.The center-to-center distance, denoted as s, between the unit cell and the HSRR measures 7 μm, while the vertical distance between the center of the unit cell and HS, denoted as d, is fixed at 30 μm.Additionally, the distance g between HS and HSRR is set to 5 μm.Thickness of VO 2 , aluminum, and SiO 2 is 5 μm, 5 μm, and 10 μm, respectively.

Analogue designs and simulations
To evaluate the coupling effect of the proposed EIT metamaterial, we utilize the full-wave simulation software CST Microwave Studio.In the simulation, a plane wave polarized in the y-direction is incident onto the hybrid metamaterial along the z-direction.The optical characteristic of aluminum is defined by Drude model, with plasma frequency sets at / rad s 2. 24 10 16  ´and damping constant sets at / rad s 1. 22 10 14  ´ [26].The permittivity of SiO 2 is established as 3.9.VO 2 is a material undergoing phase transition, shifting from an insulating state to a metallic one based on temperature modulation.To depict VO 2 's conductivity across various temperatures during this transition, we employ Bruggeman effective medium theory (EMT) for simulating VO 2 layer.The permittivity of VO 2 can be articulated as [27,28]: Where, ε D and ε M represent the dielectric functions of the insulating and metallic domains in the VO 2 layer, respectively, ε D equals 9, while ε M can be defined as follows: e w e = -

=
´signifies the electronic mass, and 0 e denotes the vacuum dielectric constant.The volume fraction f of VO 2 can be expressed using the Boltzmann function: Where, T 0 represents the phase transition temperature.During the heating process, it stands at 341.15 K (∼68 °C), while during the cooling process, it registers at 339.15 K (∼66 °C), thus establishing a transition width of ΔT = 2 °C, f max = 0.95 is the maximum volume fraction of VO 2 at the highest temperature [28].
The correlation between the permittivity and conductivity is: Hence, by integrating equations (1)-( 4), we can derive the conductivity variation of VO 2 layer across different temperatures.Figure 2 illustrates the conductivity of VO 2 across temperature fluctuations during both heating and cooling processes, employing the Bruggeman EMT.

Results and discussions
To confirm the near-field coupling effect in the toroidal resonance that underlies the proposed EIT phenomenon, we first simulate the individual HSRR. Figure 3 displays the transmission curves of the individual HSRR, where the initial temperature is set as 339.1 K (in cooling process).As illustrated in figure 3, the individual HSRR demonstrates resonance at 1.82 THz when subjected to a y-polarized plane wave (indicated by the red line).Figure 4 portrays the distributions of fields and currents within the HSRR at 1.82 THz.Observing figures 4(a)-(c), we note the emergence of two counter-rotating circulating currents originating from the electric field distributions.Subsequently, these currents give rise to magnetic dipoles arranged in a head-to-tail configuration, consistent with the hallmarks of toroidal resonance.Upon combining the HSRR with the HS, a narrow transparency peak emerges at 1.78 THz between the two resonance dips: dip I (1.73 THz) and dip II (1.86 THz), as depicted in figure 3 (denoted by the black line).As the individual HS remain unresponsive to the incident wave under the current polarization mode (as evident in the blue line of the transmission spectra in figure 3), the proposed transparency peak is a result of the near-field coupling between the bright mode HSRR and the dark mode HS.
To further elucidate the coupling mechanism, figure 5 illustrates the field distributions before and after nearfield coupling through bright and dark resonators.In figure 5(a), the individual HS remain non-interacting with the incident wave, thus acting as dark resonators.However, with incorporation of HSRR into the unit cell, the  In order to facilitate the extension of EIT metamaterial applications to other frequencies, this study investigated the influence of changes in structural parameters on the transmission spectrum.Figure 6 shows the transmission spectrum of the EIT structure when adjusting the length L 1 of the HS.From figure 6, it can be observed that compared to L 1 = 32 μm, reducing L 1 leads to a blue shift in the frequency of the transparency window, and the amplitude of transmission dip II increases.Conversely, increasing L 1 leads to a frequency red shift, and the amplitude of transmission dip I increases.
Figure 7 depicts the transmission spectrum of the EIT structure when adjusting the length L 2 of the two vertical aluminum strips.Compared to L 2 = 21 μm, reducing L 2 causes a blue shift in the frequency of the transparency window, and the amplitude of transmission dip II significantly decreases.Conversely, increasing L 2 results in a frequency red shift, and the amplitude of transmission dip II significantly increases.Figure 8 illustrates the transmission spectrum of the EIT structure when adjusting the center-to-center distance s.From figure 8, it can be seen that reducing s leads to a decrease in EIT coupling strength, with a smaller amplitude of the transparent peak.When s = 0 um, the EIT phenomenon disappears, and the transmission spectrum becomes a resonance curve.Figure 9 displays the electric field distribution of the structure at transmission dip when s = 0, indicating that electromagnetic energy is concentrated on the HSRR and does not propagate to the HS structure.This absence of EIT phenomenon is primarily due to the symmetrical action of HSRR on both ends of HS when s = 0. Once this symmetry is broken, electromagnetic energy will undergo a transition from bright to dark modes, leading to the emergence of the EIT phenomenon.As the numerical value of s increases, the EIT coupling strength also increases, as reported in [29,30].
As previously mentioned, the suggested EIT phenomenon undergoes dynamic alterations through temperature adjustments.Figure 10 illustrates suggested EIT phenomenon as the temperature cools from 339.1 K to 322.7 K.The transparency peak, as shown in figure 10, undergoes a slight shift towards lower frequencies, alongside a decrease in coupling strength, as the temperature cools.Eventually, the transparency window diminishes, and the transmission spectra demonstrate inactivity with the incident wave at T = 322.7 K.The displacement of the transparency peak can be linked to the reduction in the HSRR's resonance frequency.In accordance with the resonance frequency formula of HSRR: ( ) = p [31], a reduction in VO 2 conductivity corresponds to an elevation in the effective R, resulting in a decline in in f.Furthermore, the defining attribute of the EIT phenomenon is its slow light effect.As a result, the incident wave's group delay, a key parameter describing the slow light effect [32], is computed as the temperature cools.Displayed in figure 11, at a temperature of 339.1 K (in cooling process), the transparency peak exhibits a group delay of 13.4 ps.Compared to the previously reported EIT metamaterial composed of two electric dipole resonances forming bright-bright modes [22], and the EIT metamaterial coupling electric and magnetic dipole resonances forming bright-dark modes [26], the group delay of this design at the transparent peak is significantly higher than the previous values of 3.19 ps [22] and 5.72 ps [26].More importantly, a distinctive feature is  observed in this design where the group delay reaches its maximum at the transparent peak within the transparent window, a characteristic not present in the previous two designs.This can be primarily attributed to the excitation of electric quadrupole resonance in the HS described in this paper.This delay gradually diminishes as the temperature cools further, attributed to the reduction in coupling strength.
To further dissect the regulatory mechanism underlying the tunable EIT effect, we employ the two-coupled oscillator model for theoretical analysis to align with the simulated transmission curves.Within this model, the HSRR and HS are treated as bright harmonic oscillator and dark harmonic oscillator, respectively.The coupled differential equation governing the amplitudes x and resonance frequencies ω of the two harmonic oscillators is as follows [33,34]:   Here, the HSRR is oscillator 1, while HS is treated as oscillator 2. The parameters m, g and g correspond to the effective mass, damping rate and geometric attributes under the incident wave E E e , i t 0 = w respectively, and κ denotes the coupling coefficient.As HS functions as a dark mode oscillator, g 2 is set at 0. Upon solving the differential equation, the metamaterial's susceptibility eff c is characterized by the following: Where, A g g , = and K represent scale factor.The transmission curves are obtained from 1effi c and reflection, in which effi c denotes the imaginary part of susceptibility and exhibits the absorption  in metamaterial.Figure 12 presents the fitted transmission curves of the proposed EIT metamaterial as the temperature cools.As illustrated in figure 12, the transmission curves derived from fitting exhibit a strong correspondence with the simulated outcomes.Figure 13 displays the fitted values of the damping rates , 1 g , 2 g and the coupling coefficient κ corresponding to different temperature.The observations from figure 13 reveal a notable increase in the damping rate , 1 g accompanied by a substantial decrease in the coupling coefficient κ, as the temperature cools.Meanwhile, the damping rates 2 g remain constant.The heightened damping rate of the bright-mode resonator (HSRR, oscillator 1) leads to a reduction in resonance strength, ultimately resulting in a diminished coupling strength of the EIT effect.This cascade of effects leads to the vanishing of the transparency window, with the transmission spectrum closely resembling a non-resonant curve.
Next, we explore a scenario wherein, during the heating process, the transmission curves of EIT metamaterial at identical temperatures are compared.Figure 14 depicts this comparison between transmission curves in both heating and cooling processes.As depicted in figure 14, compared to cooling process, the EIT phenomenon significantly diminishes during the heating process at the same temperature.Moreover, a notable red shift in the transparency peak is observable, manifesting in frequency shifts of 0.03 THz at 339.1 K, 0.04 THz at 332.1 K, and 0.04 THz at 329.2 K, for instance.This shift is attributed to the differing conductivity of VO 2 at the same temperature during both coupling and heating processes.Remarkably, at 332.7 K in both heating and cooling processes, the EIT phenomenon completely disappears.

Conclusions
In summary, we have introduced a novel terahertz EIT analogue centered on the near-field coupling effect from toroidal resonance to electric resonance, which has been both proposed and simulated.This approach involves the utilization of a HSRR to initiate toroidal resonance, serving as bright oscillator, while a set of horizontal metal strips generates electric resonance, functioning as dark oscillator.To achieve an actively tunable EIT effect, we have designed the HSRR as a combination of two VO 2 semicircle rings and two vertical aluminum strips.Our simulation results provide compelling evidence that the suggested EIT phenomenon undergoes dynamic alterations through temperature adjustments.To elucidate the underlying mechanism of this tunable characteristic, we have employed a two-harmonic oscillator model for theoretical analysis of the effective parameters.The theoretical findings indicate that the decrease in coupling strength is attributed to the diminished resonance in the bright-mode oscillator.Furthermore, the red shift of the EIT peak during cooling processes can be traced back to the reduction in intrinsic resonant frequency of the bright-mode oscillator.Compared to the cooling process, the EIT phenomenon significantly diminishes during the heating process at the same temperature, and a notable red shift in the transparency peak is evident.In essence, our study paves a new way for harnessing near-field coupling effects to achieve tunable EIT phenomena, paving the way for advancements in toroidal resonance development.It provides technical solutions for applying EIT metamaterials in terahertz slow-light devices, sensors, and switch devices.

Figure 1
Figure1schematically depicts the structure of the proposed EIT metamaterial in a stereoscopic representation.As depicted in figure 1(a), the unit cell consists of two components fabricated on a SiO 2 substrate: two horizontal aluminum strips (HS) and a hybrid split-ring resonator (HSRR).The HSRR is comprised of two VO 2 -based semicircle rings and two vertical aluminum strips and is positioned within the HS to induce toroidal resonance.In figure 1(b), the unit cell has periodic dimensions of P x = 140 μm and P y = 110 μm, with the length of HS denoted as L 1 , measuring 63.6 μm.The length of the two vertical aluminum strips, designated as L 2 , is 21 μm, and the width of the aluminum strips, marked as w, is set at 10 μm.The inner and outer radii of the circular rings are referred to as R 2 and R 1 , respectively, measured as 15 μm and 25 μm.The center-to-center distance, denoted as s, between the unit cell and the HSRR measures 7 μm, while the vertical distance between the center of the unit cell and HS, denoted as d, is fixed at 30 μm.Additionally, the distance g between HS and HSRR is set to 5 μm.Thickness of VO 2 , aluminum, and SiO 2 is 5 μm, 5 μm, and 10 μm, respectively.To evaluate the coupling effect of the proposed EIT metamaterial, we utilize the full-wave simulation software CST Microwave Studio.In the simulation, a plane wave polarized in the y-direction is incident onto the hybrid metamaterial along the z-direction.The optical characteristic of aluminum is defined by Drude model, with plasma frequency sets at / rad s 2.24 10 16  ´and damping constant sets at / rad s 1.22 10 14  ´[26].The

Figure 1 .
Figure 1.Diagram layout: (a) 3D schematic perspective and (b) overhead depiction of a unit cell.
is the plasma frequency.Here, N 1.3 10 cm 22 3 = ´stands for the carrier concentration within the medium, m m 2 e = * represents the effective mass, m 9.1094 10 kg e 31

Figure 2 .
Figure 2. The conductivity of VO2 across temperature fluctuations.

Figure 3 .
Figure 3. Transmission curves of individual HSRR, individual HS and EIT scheme when temperature is 339.1 K (in cooling process).

Figure 4 .
Figure 4.The corresponding field distributions of individual HSRR at 1.82 THz: (a) electric field, (b) magnetic field and (c) surface current.

Figure 5 .
Figure 5.The field distributions for various configurations: (a) individual HS and (b) amalgamation of HS and HSRR at 1.78 THz.

Figure 6 .
Figure 6.Transmission curves of EIT metamaterial as L 1 from 28 um to 36 um.

Figure 7 .
Figure 7. Transmission curves of EIT metamaterial as L 2 from 19 um to 23 um.

Figure 8 .
Figure 8. Transmission curves of EIT metamaterial as s from 7 um to 0 um.

Figure 9 .
Figure 9.The electric field distributions of EIT metamaterial at transmission dip when s = 0 um.

Figure 10 .
Figure 10.Transmission curves of EIT metamaterial as the temperature cools from 339.1 K to 322.7 K.

Figure 11 .
Figure 11.Incident wave's group delay as the temperature cools from 339.1 K to 322.7 K.

Figure 12 .
Figure 12.Fitting transmission curves of metamaterial as the temperature cools from 339.1 K to 322.7 K.

Figure 13 . 1 g 2 g
Figure 13.Fitting values of , 1 g 2 g and k as the temperature cools.

Figure 14 .
Figure 14.The comparison of transmission curves for EIT metamaterial during heating and cooling processes: T is (a) 322.7 K; (b) 329.2 K; (c) 332.1 K; and (d) 339.1 K.