Ultra-Broadband Electromagnetically Induced Transparency in Metamaterial Based on Conductive Coupling

This paper proposes a structure composed of a horizontal metal strip resonator (SR) and four C-shaped ring resonators (CRR) to obtain a broadband electromagnetic induction transparency (EIT)–like effect. The SR and CRRs are divided into bright mode and dark mode according to whether they can be directly excited by the incident electromagnetic wave. The three-level Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda$$\end{document}-type system and electric field are used to explain the mechanism of the EIT-like effect. Meanwhile, by decreasing the distance between the SR and CRRs, a transparency window with a relative bandwidth of 91.93% and a width of 1.4 THz is observed. It is found that when the bright and dark modes are directly contacted, the EIT window increases rapidly through conductive coupling, which can be explained by the surface current. Our work provides a new method for a wide band EIT-like effect, which has a certain value in the fields of slow light, filters, and non-linear optics.


Introduction
Electromagnetic induction transparency (EIT) is a quantum interference phenomenon in a three-level system, which can weaken the absorption at the resonance frequency, resulting in a sharp transparency window [1]. During the generation process of EIT, the atomic medium is usually accompanied by a strong dispersion effect, which significantly reduces the optical speed. Consequently, EIT is used in many important fields, such as slow light propagation [2], optical filters [3], and optical storage [4]. Nevertheless, the EIT effects in an atomic system which require ultra-low temperature and high-intensity lasers cannot be achieved in the actual environment due to the cumbersome conditions. With the development of metamaterials, EIT based on metamaterials has received extensive attention from scholars [5]. In metamaterials, bright-dark mode or bright-bright-mode coupling is commonly used to form EIT without strict environmental conditions [6,7]. With the in-depth study of EIT metamaterials, it has been found that there were still many shortcomings in the EIT-like effect, such as a narrow band, polarization direction sensitivity, etc. [8].
Because EIT is produced by the interference of bright and dark modes, its bandwidth is narrow due to this limitation, which considerably limits the practical application. Therefore, people are committed to studying the broadband EITlike effect. Wu et al. first realized broadband slow light using a 41-layer design based on a double-continuum fano resonance [9]. However, it is difficult to fabricate. Later, Zhang et al. manifested a novel approach of breaking the symmetry to obtain a broadband plasmon-induced transparency [10]. Similarly, a back-to-back self-asymmetric split-ring resonator in Dirac semimetal metamaterials was demonstrated to observe an adjustable broadband transparency window [11]. A scheme of broadband EIT-like effect that was based on Mie resonances in an all-dielectric microstructure was proposed [12]. Nevertheless, its bandwidth is not broad enough. Zang et al. presented a structure with 14 dark modes obtaining a wide transparency window based on a hybridization gap [13]. Unfortunately, this structure is a bit complicated.
Most of the current structures are complex, or the relative bandwidth is not wide enough. In this paper, a structure with a bright mode and four dark modes is proposed to obtain a broadband EIT-like effect at the terahertz region. The structural unit is simple, but realizes a relative bandwidth of 91.93% at 3 dB transmission. The EIT cell consists of a horizontal metal strip resonator (SR) and four C-shaped ring resonators (CRRs). In addition, the transparent window expands gradually with the increasing number of dark modes. Subsequently, the coupling distance between bright and dark modes was adjusted to expand the transparency window. The proposed structure with an obvious slow light effect realizes the wide EIT effect, which is of great significance for the design of metamaterial filters, slow light devices, and broadband polarization devices.
The main contents of this paper are organized as follows: The structure design of the EIT metamaterial and the threelevel Λ-type system are described in "Structure." In the next section, the results and discussion are presented. Finally, the paper is concluded in the last section. Figure 1a illustrates a unit cell of the proposed EIT-like metamaterial, which consists of a horizontal bar and four C-shaped ring resonators made of gold. The dielectric material is quartz with a thickness of 30 µm. Figure 1a shows the schematic diagram of the proposed EIT-like metamaterial. The thickness of gold t 2 is 5 µm. Figure 1b is the front view of the proposed structure. Figure 1c is the perspective view of the proposed structure. The geometric parameters of this design are as follows: P x = 114 µm, P y = 134 µm, L = 80 µm, Fig. 1 a The schematic diagram of the proposed EIT-like metamaterial. b The front view of the proposed structure. c The perspective view of the proposed structure. d Level scheme for EIT in a three-level atom system r = 31 µm, g = 40 µm, w = 4 µm, and d = 10 µm. These geometric dimensions are obtained through extensive simulation and optimization.

Structure
The commercial tool CST Microwave Studio is used to simulate this proposed structure. The commercial simulation software CST (Computer Simulation Technology, Germany) Microwave Studio based on the finite difference time-domain (FDTD) is used for numerical simulation in the terahertz range. In our simulation, frequency domain analysis is used. The periodic boundary conditions are set in the x and y directions. The open space is set in the z direction. The incident wave as electric field polarizing x direction propagates along the z-axis, which is perpendicular to the quartz substrate. All simulations use adaptive fine mesh settings.
To understand the mechanism of the EIT effect, a quantum three-level Λ-type system is used [14]. This system is comprised of a ground state �0⟩ and two upper states �1⟩ and �2⟩ as shown in Fig. 1d. When the electromagnetic waves with x-axis polarization propagate, the metal bar is excited strongly, producing a resonance. This is similar to the dipole-allowed transition �0⟩ → �1⟩ . The resonance frequency is 0 , and the damping rate is 1 . The SR acts as a bright mode here and the excitation path is �0⟩ → �1⟩ . The CRRs do not respond directly to incident waves, which is analogous to the dipole-forbidden transition �0⟩ → �2⟩ . The CRRs are dark modes, which are characterized by the transition frequency 0 + Δ and a damping rate 2 . When the SR and CRRs are lumped together, bright-mode SR is excited by the incident waves. Correspondingly, dark-mode CRRs are irritated by the SR, then a new path �0⟩ → �1⟩ → �2⟩ → �1⟩ appears. Destructive interference between two excitation paths produces the EIT effect, which causes the absorption of the transition �0⟩ and �1⟩ to weaken or even reach zero.

Results and discussion
The simulation results of a single SR, CRR, and complete structure are shown in Fig. 2. When the electromagnetic waves are incident to the structure, the SR resonates at 1.1168 THz, so the SR is the bright mode. The response of CRR to an incident electromagnetic wave is almost a straight line, which proves that CRR is not excited by incident electromagnetic waves, so it works as the dark mode [15]. The blue line in the Fig. 2 represents the relationship between the transmittance and the frequency of the complete structure. There are two dips at 0.8544 THz and 1.4704 THz, and the bandwidth of the transparency window is 0.616 THz. At this time, a wide EIT phenomenon appears at the resonance of 1.1992 THz, which extends from 0.9037 to 1.4087 THz at 3 dB transmission.
In order to obtain a wider EIT, we start by reducing the loss of EIT or enhancing the coupling between the bright and dark modes [16,17]. The loss of EIT includes ohmic loss and radiation loss. On the one hand, we use gold with higher conductivity to reduce the ohmic loss. On the other hand, we increase the number of dark modes to reduce radiation loss [18]. By increasing the number of dark-mode resonators, the loss coefficient is reduced. Correspondingly, the coupling coefficient between the bright and dark modes is increased, achieving the effect of expanding the bandwidth. As shown in Fig. 3, the EIT bandwidth widens as the number of dark-mode resonators increases.
Moreover, the interference phenomena are interpreted by the two-particle model quantitatively [19,20].  Here, particle 1 and 2 represent the bright mode and the dark modes, respectively. x 1 and x 2 represent the transmission amplitude of bright-mode SR and dark-mode CRRs.
1 and 2 represent the damping rates of the two particles. 0 and 0 + Δ represent the resonance frequency of the bright mode and the dark modes. k is the coupling coefficient between the bright mode and the dark modes. q is the coupling strength of the bright modes with the incident waves.
Solving Eq. (1), the effective susceptibility X eff is obtained [12]. Therefore, we obtain the transmission of electromagnetic waves through structures: where d is the thickness of the planar structure and 0 is the wavelength in a vacuum. Through the above calculations, the calculated transmission curve is obtained as shown in Fig. 4. What's more, the calculated transmission spectrum is consistent with the simulated transmission spectrum.
To further illustrate the mechanism of the EIT-like effect, Fig. 5a-c presents the electric field of the single SR, CRRs, and the entire structure, respectively. Figure 5a indicates that the SR is strongly excited by the incident waves. The electric field distributed across the SR at 1.1168 THz is very strong, which is a typical dipolar localized surface plasmon (LSP) resonance phenomenon. However, the electric field of CRRs is very weak as shown in Fig. 5b. Figure 5c depicts the electric field of the entire structure at the resonance frequency; the CRRs are excited by the near-field couplings with SR. The strong excitation of CRRs (dark mode) suppresses the . 4 a The simulated transmission spectrum of the proposed structure. b The calculated transmission spectrum of the proposed structure Fig. 5 a, b, and c are electric fields of a single SR (at 1.1168 THz), CRRs (at 1.1168 THz), and the entire structure (at the resonance of 1.1992 THz), respectively reflection and absorption of SR (bright mode) in a destructive interference way [21]. Consequently, an EIT-like effect is observed. Then, we obtain a greater broadband effect by reducing the coupling distance d. As d gradually decreases, the dip 1 on the left has a slight red shift. Intriguingly, when d changes from 5 to 4 µm, the bandwidth suddenly increases sharply, and a wider EIT is obtained, as shown in Fig. 6a. The full width of the transparency region reaches up to 1.0768 THz. We define the bandwidth Δf = f H − f L , center frequency f 0 = f H + f L ∕2 . After calculation, the relative bandwidth f foc = Δf ∕f 0 at 3 dB is about 91.93%. In order to prove the correctness of the transmission spectrum when the distance d equals 4, Fig. 6b shows the calculated transmission spectrum of the proposed structure. It can be seen from Fig. 6a, b that the calculated curve agrees well with the simulated curves, and the frequencies at the two dips are in good agreement.
The reason for the unique property is that when d is reduced to 4, the distance between the bright and dark modes is close to zero; hence, there is direct contact between them. If there is a metal connection between the resonators, energy is transferred by connecting the metal [22][23][24][25]. There are not only magnetic field coupling and electric field coupling between the bright resonator and dark resonator, but also conductive coupling through the directly connected structure of the metal. These couplings cause the EIT-like effect, and the EIT phenomenon is enhanced due to conductive coupling. As shown in Fig. 6, the increase in bandwidth is due to the left shift of the first dip. To elucidate the mechanism of the conductive coupling, the surface current of the first dip of contactless (d = 10) and contact (d = 4) is shown in Fig. 7.
Comparing Fig. 7a and b, although the frequencies of the two dips of the two structures and the surface currents are different, the characteristics of the current distributions are the same. The currents in the SR and CRRs flow in phase. For Fig. 7b, due to the connection of the bright and dark resonators, the current in the bright mode increases notably and is greater than the current in the bright mode when there is no connection. At this time, the currents of the bright mode and dark modes are connected, and the energy between them is coupled through this current, which is larger than that through capacitance coupling. Therefore, the conductive coupling greatly enhances EIT at the same time. We compare the performance of this broadband electromagnetically induced transparency with other EIT-like phenomena in the terahertz range. As shown in Table 1, our broadband electromagnetically induced transparency exhibits a larger bandwidth.

Conclusions
In this paper, a broadband EIT structure is proposed, which consists of a transverse metal strip and four C-shaped resonators arranged up and down the metal strip. The three-level system and electric field explain the physical mechanism of the proposed EIT-like metamaterial. By increasing the number of dark modes and reducing the coupling distance between the bright and dark modes, a transparent window with a relative bandwidth of 91.93% is observed. It is found that after the bright mode and dark mode are in direct contact, the EIT phenomenon has been significantly broadened, which is explained by the surface current mechanism. Our work has potential applications in metamaterial slow light, broadband filters, and sensors.