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A graphene-based metamaterial structure composed of multilayer graphene/dielectric stacking configuration is proposed, which achieves multiple analogue of electromagnetically induced transparencies (EIT) effect at terahertz frequencies. Using the phase-coupling scheme, a theoretical model is established to study the EIT-like effect of the proposed structure, and the theoretical calculations coincide well with the numerical simulated results. By varying the Fermi energy level of the graphene, the EIT-like windows can be dynamically tuned in a wide range of terahertz spectra. Particularly, since the symmetry of the structure, the EIT-like effect is polarization-insensitive and can be performed very well in a large incident angles, nearly 80° for both transverse electric and transverse magnetic waves. The proposed structure has potential applications in tunable terahertz chip-integrated optical devices, especially for dynamic multi-band filters, sensors, modulators and nonlinear devices.
© 2016 Optical Society of America
1. Introduction
Electromagnetically induced transparency (EIT) is a quantum interference effect in laser-driven atomic systems, which induces a narrow transparency window within a broad absorption spectrum [1]. This attractive phenomenon has been widely applied in enhancing nonlinear effect, slow light, and optical data storage since its strong dispersion in the transparency window [2–4]. The implementation of the quantum EIT effect in chip-scale applications [5], which requires stable gas lasers and low temperature environment conditions, is significantly restricted [6]. Recently, mimicking EIT effect in classical systems has attracted tremendous attention, various schemes including metamaterial structures [7–9] and waveguide systems [10, 11] have been engineered to achieve the EIT-like effect. Similar with EIT effect, the EIT-like effect has potential applications in the development of compact elements such as tunable sensors, switchers, and slow light devices. Among these configurations, the metamaterial structures such as cut wires [12], fishnets [13] and split-ring resonators [14], highlight the realization of EIT-like features [6]. However, most of these structures only present a single EIT-like spectral response. Once the devices fabricated, tuning the transparency window is very difficult.
Graphene is a two-dimensional material with unique and fantastic properties in terahertz (THz) regime, such as dynamically tunable, extremely tight field confinement and low losses [15, 16]. With controlling the Fermi energy level of graphene via chemical doping, electrostatic gating or magnetic field, its surface conductivity can be tuned in a broad frequency range [17–19]. These unique properties can be used to design the graphene plasmon based devices which could effectively tune at different frequencies [20]. Thus, the combination of graphene and metamaterial is a promising approach to design dynamically tunable EIT-like devices [21–23]. For most metamaterial structures based on metal and graphene, it is the near-field coupling between bright (radiative) and dark (nonradiative) mode resonators that demonstrates the EIT-like effect [12, 16, 24]. The EIT-like performance of these structures strongly rely on the coupling strength between resonators and entail resonators space separation on the order of tens of nanometers [12, 25]. Such a small spacing, requiring a precise lithographic control, significantly restricts the access to the coupling medium between resonators [3]. Compared with the near-field coupling mechanism, the phase-coupling scheme could be exploited to obviate and substitute the requirement of near-field coupling [3]. However, the EIT-like devices based on phase-coupling scheme are mostly concentrated on planar waveguide structures [4, 26, 27], or working in near-infrared, mid-infrared frequencies [7, 24, 28]. These structures are usually polarization sensitive and invalid for oblique incidence, resulting in attenuation and disappearance of the EIT-like effect. Up to now, there were very few studies about designing the graphene metamaterial structures which could realize EIT-like effect in terahertz region and polarization insensitive.
In this paper, we propose a graphene-based metamaterial structure consisting of multilayer square graphene (SG), spatially separated by a dielectric material. The multiple EIT-like effects are obtained in the THz region by simulation. Exploiting the phase-coupling scheme, the transfer matrix model is used to explain the multiple EIT-like responses. By shifting the Fermi energy level of the SG, the EIT-like windows can be actively tuned in a broad THz region. Particularly, since the symmetry of the structure in x-y plane, the structure has characteristics of polarization-insensitive for incident waves.
2. Structure design and theoretical model
Figure 1 schematically depicts the proposed metamaterial structure composed of graphene/dielectric multilayer stacking unit cell periodically arrayed in x and y direction. Because the permittivity of graphene is negative in THz region, each unit cell structure can be regarded as a hyperbolic medium with large anisotropic permittivity [29, 30]. As previously reported, the hyperbolic medium could act as a subwavelength resonator to realize optical cavity and emission enhancement [31, 32]. When THz wave is incident on the metamaterial structure, a strong electromagnetic resonance can be generated [33].
Fig. 1 Schematic diagram of the proposed metamaterials structure with TE and TM incidence. (a) 3D view shows the multilayer SG spatially separated by the dielectric material with refractive index 1.5, the THz wave is incident along + z-axis direction. (b) Cross section of one unit cell in x-z plane with N-layer SG. σi is the surface conductivity and Wi is the width of the ith-layer SG. di is the space gap between Wi and Wi+1 SG layers. The period P of the unit cell in x and y direction is 6μm.
The dynamic EIT-like effect of the proposed structure shown in Fig. 1 can be evaluated by employing transfer matrix theory [23, 34]. For simplicity, the propagating and coupling losses are neglected here. The single unit cell structure is composed of N-layer SG as shown in Fig. 1(b). For the ith-layer (i = 1, 2, … N) graphene, the refractive indices above and below it are ni+1 and ni. The incoming and outgoing waves of the Nth-layer structure satisfy the following transfer matrix equation:
where the subscript + and - represent the positive and the negative propagation direction, Sin and Sout represent the input and output waves. Mi and Vi matrices indicate the zero thickness graphene interface and the di thickness dielectric interlayer, which can be defined as:here φi denotes the phase difference between ith and (i + 1)th-layer graphene, expressed as ni+1diω/c. di is the corresponding separation between these two SG layers, ω and c represent the incident wave angular frequency and the light velocity in vacuum. The Fresnel coefficients in Mi matrix are ti,i+1 = 2ni/Δ, ti+1,i = 2ni+1/Δ, ri,i+1 = (ni - ni+1 - Z0σi)/Δ, and ri+1,i = (ni+1 - ni - Z0σi)/Δ for the ith-layer graphene, where Δ = (ni + ni+1 + Z0σi) [25]. Z0 is the vacuum impedance, which is 377 Ω. σi = jfiDiω/{π(ω2 - ω2p,i + iωГp,i)} is the surface conductivity [15]. fi = Wi/P is the filling factor of the periodically arrayed graphene [16]. Di is the Drude weight, denoted as e2EF,i/ћ2. EF,i and ħ represent the Fermi energy level and the reduced Planck’s constant. Гp,i = evF2/(μEF,i) is the plasmon resonance width in the SG layers, and it is usually 10% bigger than the Drude scattering width Гi in the unpatterned graphene [20]. The Fermi velocity vF is about 106m/s and μ is the DC mobility. The plasmon resonance frequency ωp,i in graphene can be calculated by (Di/(ηiεiε0Wi))1/2 [20]. The effective dielectric constant of the medium surrounding the ith-layer graphene can be expressed as (ni2 + ni+12)/2, ε0 is the vacuum permittivity. Meanwhile, the Eq. (1) can be written as:In this work, the incident wave is emitted only from top of the structure and S-,in(N) = 0. Thus, the transmission of the structure can be derived as:Obviously, the Eq. (4) presents a typical form of transmission spectrum of a Fabry-Pérot (FP) resonator [4]. Therefore, the proposed structure can be seen as a multiple FP cavities system, and the FP oscillation can be used to explain EIT-like response. Since the refractive index of the medium surrounding graphene is set as 1.5, the Eq. (4) can be further simplified to T = 1/|H22|2.On the other hand, numerical simulation is also performed by using CST Microwave studio software based on the finite integration method. In the simulation, graphene is modeled as an ultra-thin anisotropic material with thickness tg = 1nm. The in-plane permittivity is ε// = 1 + iσ/ωε0tg, while the out-of-plane permittivity ε⊥ is 2.25. The surface conductivity of graphene is written as σ = ie2EF/πℏ2(ω + iτ−1) [26]. τ = μEF/(evf2) represents the relaxation time. Recently, high-quality suspended graphene with μ > 105cm2V−1s−1 has been achieved experimentally [35]. In this paper, we assume μ of the graphene is 105cm2V−1s−1 in the simulation [26, 36].
3. Results and discussion
To demonstrate the EIT-like effect of the proposed structure, the transmission characteristics of the two-layer SG system is investigated firstly. In the simulation, we use the open boundary condition in z direction and the periodic boundary condition in x and y direction. By carefully optimizing the geometric parameters (e.g. graphene width, spacing separation between two graphene layers and so on), the simulated transmission spectrum achieves 95% transmittance presented in Fig. 2(a). The transmission spectrum of the two-layer SG system by aforementioned theoretical model is calculated, and is good agreement with the simulation results. From Fig. 2(a), the transmission dips f1 = 3.90 THz, f3 = 4.44 THz, and the transparency window f2 = 4.15 THz. The transparency window of simulated is nearly equal to that derived from the FP model f2 = (f1 + f3)/2 [27]. Besides, the spacing separation d1 = 24 μm in this system is also equal to the FP model value c/(2f2n), n is the refractive index of dielectric material, c is the light velocity in vacuum. This demonstrates that the two-layer SG system satisfies the FP resonant condition and can produce the EIT-like effect. Figures 2(b), 2(c), and 2(d) illustrate the distributions of electric field (|EZ|2) of the two-layer SG system at frequencies f1, f2, and f3, respectively. When located at f1 and f3, the graphene plasmon resonances are excited in the lower and the upper SG layer as depicted in Figs. 2(b) and 2(d). With the excitation of graphene plasmon resonance, the incident wave is partially absorbed and mostly reflected at the resonance frequency, demonstrates resonance dips f1 and f3 originating from the graphene plasmon resonances in the corresponding SG layers [15, 20]. At the transparency window f2, both the upper and the lower SG layers are partially resonant and destructive interference with each other, leading to the FP oscillation as shown in Fig. 2(c). Thus, the incident THz wave can pass through the system and exhibit a transparency window at the resonance frequency.
Fig. 2 (a) EIT-like transmission spectrum of the two-layer SG structure with upper graphene width W1 = 3 μm and lower graphene width W2 = 2.48 μm, the spatial separation d1 is 24 μm and Fermi energy level EF is 0.5 eV for both SG layers. The red solid line and the blue hollow circle denote the simulated and the theoretical calculated results, respectively. (b) - (d) The distributions of electric field (|EZ|2) of one unit cell at transmission dip f1, transmission window f2 and transmission dip f3, respectively. The red dash line represents graphene layers.
Next, the three-layer SG system is constructed by inserting the third SG layer into the aforementioned two-layer SG system as an example to investigate the multiple EIT-like effects. By setting spatial separation d2 between upper and middle graphene at the optimum value 26 μm, two transparency windows can be observed and the transmittance over 90% by simulation, as shown in Fig. 3(a). And the simulated result is also agreed well with the theoretical calculated result. Indicate that the three-layer SG system has two EIT-like responses. Figures 3(b) and 3(c) illustrate the distributions of electric field (|EZ|2) at the two transparency windows f4 and f5 as shown in Fig. 3(a). For the first transparency window at f4, both the bottom and the middle graphene layers are partially resonant, causing the FP oscillation between these two graphene layers. Likewise, the second transparency window f5, which is induced by partially resonant of the middle and the upper graphene layers, causing the FP oscillation. Finally, to further clarify the EIT-like spectral features, the periodic evolution of the transmission spectrum of the two-layer SG system is studied. Figure 3(d) illustrates the theoretically calculation of the transmission spectrum with varying the spacing separation d1. As can be seen that the transmission spectrum is evolved periodically, which arises from the fact that the round-trip phase is controlled by the optical path difference between two mirrors in terms of the FP oscillation [16]. Additionally, the highest transmittance is achieved when d1 = 24.06 μm, which further demonstrates the FP-like spectral features of the proposed structure.
Fig. 3 (a) EIT-like transmission spectrum of three-layer SG system with the third graphene width W3 = 1.96 μm and the spatial separation d2 = 26 μm, the Fermi energy level EF is 0.5eV for all graphene layers. The red solid line and the blue hollow circle denote the simulated and the theoretical calculated results, respectively; (b) and (c) are the distributions of electric field (|EZ|2) of one unit cell corresponding to the frequencies f4 and f5 shown in (a). The red dash line indicates graphene layers. (d) Transmission evolution spectrum with changing the spatial separation d1 of the two-layer SG system.
With the investigation of two-layer and three-layer SG systems above, it can be deduced that the working mechanism is applicable to the multilayer SG metamaterial structure. For example, the four-layer SG system can be designed to obtain the triple transparency windows by adding the 4th-layer SG, which is shown in Fig. 4(a). Hence, the proposed metamaterial structure has the characteristics of multiple EIT-like responses. In Figs. 4(b)-4(d), we graphed the transmission spectrum of the two-layer, three-layer and four-layer SG systems at different Fermi energy level EF of the graphene. It can be seen that when EF increases from 0.46 to 0.54eV, the transparency windows are blue-shifted obviously. Denote that the proposed structure has the capacity of dynamically tuning the transparency windows. That can realize the multi-band selective filters and switches over a broad THz frequency range. These features have potential applications in tunable sensors, switchers, and slow light devices.
Fig. 4 (a) EIT-like transmission spectrum of the four-layer SG system when the 4th-layer graphene width W4 = 1.45 μm and the spatial separation d3 = 28 μm. (b) - (d) Transmission spectra of the two-layer, three-layer and four-layer SG systems with different Fermi energy level in the graphene, respectively.
Finally, we investigate the dependence of polarization and incident angle on the transmission spectra of the proposed structure. Since the symmetrical design of the square shaped unit cell in x-y plane, the EIT-like effect is polarization insensitive with normal incident THz wave [30]. However, in the case of oblique incidence, the EIT-like effect may be related to the polarization direction. Figures 5(a) and 5(b) show the simulated transmission spectra as a function of frequency and incident angle for TE and TM polarizations. Although only the two-layer SG system presented in Fig. 5, it also can be applicable to the multilayer SG metamaterial structure. It is found that the transmission experiences no variation for both TE and TM polarizations with the incident angle smaller than 60°. For the proposed structure, with surrounded by the same dielectric material, a series of mini photonic bandgaps appears owing to the multiple interferences by the graphene layers [37]. In addition, due to its two-dimensional nature and unique electronic band structure, graphene can support not only TM but also TE plasmons. Inherent to the deep subwavelength confinement of the graphene surface plasmons, the polaritons manifest flat bands above the light line, thus giving rise to the omnidirectional EIT-like effect under different incident angles and keeping the transmittance almost changeless [38]. When the incident angle is larger than 60°, the transmission decreases rapidly for the TE polarization, while decreases slightly for the TM polarization; with the incident angle up to 80°, the transmission at the transparency window can be strongly maintained 73% for the TE polarization and 83% for the TM polarization. Note that the proposed metamaterial structure is able to remain EIT-like effect with incident angle smaller than 80° for both TE and TM polarizations, manifesting that it is insensitive to polarization and incident angle. The proposed structure is very promising to fabricate nearly omnidirectional EIT-like devices.
Fig. 5 The simulated EIT-like transmission spectra of the two-layer SG system with different incident angle for (a) TE and (b) TM polarizations, respectively.
4. Conclusion
In summary, the multiple EIT-like effects have been investigated in THz graphene metamaterial, which consists of multilayer graphene/dielectric stacking unit cell structure, periodically arrayed in x and y direction. With suitable geometric parameters and Fermi energy level in graphene, the multiple EIT-like windows can be realized. Exploiting the phase-coupling scheme, a simplified theoretical model has been established according to the transfer matrix theory, and the proposed structure can be regarded as a series of FP cavities. The simulated results are well consistent with the theoretical calculated results. By shifting the Fermi energy level of graphene, the transparency windows of the EIT-like effect can be dynamically tuned without reconstructing the structure. Importantly, the proposed metamaterial structure is polarization insensitive and able to maintain strong EIT-like effect with incident angle smaller than 80° for both TE and TM waves. The proposed structure can be used in highly THz chip-integrated optical circuits and devices, especially for active multi-band filters, modulators and nonlinear devices.
Acknowledgments
This work was financially supported by the “Hundreds of Talents Programs” of the Chinese Academy of Sciences (Grant No. J08-029), the CAS/SAFEA International Partnership Program for Creative Research Teams, and the Innovative Project of the Chinese Academy of Sciences (Grant No. YYYJ-1123-4).
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