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We presented a dynamically tunable electromagnetically induced reflection (EIR) based on the complementary graphene metamaterials composed of the wire-slot and split-ring resonators slot (SRRs-slot) array structures for the terahertz region. In this structure, the dark mode excited by the near field coupling between wire-slot and SRRs-slot structures, induces a reflection window. Moreover, the reflection window can be actively controlled by varying the lateral displacement between two slot-type resonant structures or Fermi energy of graphene without reoptimizing and re-fabricating structure. In addition, the large positive group delay obtained within the reflection peak can be also tuned over a broad terahertz region by changing the Fermi energy of graphene. Therefore, the work opens up the possibility for the development of compact elements such as modulators, tunable sensor, switches and slow light devices.
© 2015 Optical Society of America
1. Introduction
Electromagnetic metamaterials, composed of the subwavelength structure cells arranged in periodic arrays, are a kind of artificial structural material with extraordinary electromagnetic properties unavailable in nature materials, such as negative refractive [1], invisible cloaking [2], and superlens [3]. As a new branch of applications, recently, implementations of EIT in metamaterials, referred to as EIT-like effect, have attracted a lot of attention due to the potential applications in developing novel devices, such as slow light components and highly sensitive sensors [4,5]. Compared with EIT in atomic system, moreover, the EIT-like effect can avert the scathing experimental requirements, such as macroscopic apparatus, stable gas lasers, and low temperature. So far, various metamaterial-based EIT structures have been proposed and demonstrated from microwave to optical frequencies [6,7]. Unfortunately, the transparency windows obtained in metamaterial structures can merely work at a fixed wavelength, which hinders their practical applications.
To achieve tunable transparency window, currently, various approaches have been employed and demonstrated by integrating metamaterial with the different active materials. For example, Gu et al. experimentally demonstrated an optically reconfigurable EIT effect in planar metamaterials by integrating photoactive silicon islands into functional unit cells [8]. Cao et al. experimentally demonstrated a thermally tunable hybrid metal superconductor EIT metamatrial by cooling below the high-temperature superconducting phase transition temperature [9]. Duan et al. numerically analyzed a dynamically tunable plasmonically induced transparency (PIT) planar hybrid metamaterial, where VO2 stripes are filled in the cut-out slots as components of a plasmonic system. Results show that the PIT effect of the hybrid metamaterial can be agilely tuned by adjusting the temperature of VO2 stripes without modifying the structure accurately [10]. However, those tunable methods depend highly on the nonlinear properties of the active materials, which inevitably results in a low modulation depth and range. In addition, the possibility for massive fabrication is still limited by complex structures and processes, so that they are currently available only for laboratory experiments.
Since discovered in 2004, graphene has attracted considerable attention due to its unique electric, mechanical, and thermal properties [11–13]. Particularly, it is found that graphene conductivity can be dynamically controlled by shifting the Fermi energy EF, which is difficult to attain in conventional metal structures [14] By patterning graphene membrane, recently, different tunable devices have been realized by chemical doping or electrostatic gating [15,16]. These reported structures mainly focus on the graphene metamaterials, which realize electrostatic tunability by the connect structure. On the other hand, based on the Babinet equivalence rule, the complementary graphene metamaterials can offer a simple means to actively control the electromagnetic wave propagation. However, a few results are reported in this respect [17].
In this paper, we presented a dynamically tunable reflection window by using planar complementary graphene metamaterial for terahertz region. The unit cell of the proposed structure is composed of the wire-slot and SRRs-slot structures. A reflection window can be obtained by adjusting the near field coupling strength between the elements of the unit cell. The underlying mechanisms of the reflection window are well explained by the induced field distributions. Further investigations show that this window response can be dynamically tuned by varying the Fermi level of the graphene without reoptimizing and refabricating structure. Moreover, there exist very large group delays within the reflection window. Therefore, the proposed complementary graphene metamaterial structures with tunable window response have potential applications in design of tunable slow light devices, tunable sensors and switchers.
2. Design and simulation of complementary structure
Figure 1 shows the unit cell schematic of the proposed tunable EIR planar structure based on the complementary graphene metamaterial. This unit cell structure is composed of the graphene-based SRRs-slot structure symmetrically placed on the left and right side of a graphene-based wire-slot structure (as shown in Fig. 1(a)), which are fabricated on the light-doping silicon substrate covering with the thin SiO2 layer (as shown in Fig. 1(b)). In this structure, the lateral displacement of the SRRs-slot structure with respect to the symmetry axis of the wire-slot structure is defined as y, and the other structural parameters are as follows: Px = 80µm, Py = 120µm, L = 85 µm, a = 27 µm, s = 7 µm, and g = w = 5 µm, while the thicknesses of the SiO2 layer and silicon substrate are h2 = 300 nm and h1 = 30 um, respectively. However, in according to the Babinet’s principle, a sharp reflection window within the broad spectral profile will be established in this complementary metamaterials, instead of enhanced transimission. Therefore, accurate terminology should suggest using the expression “EIR-like effect”, as we would introduce electromagnetically induced reflection (EIR).
Fig. 1 The proposed tunable EIR structure based on the complementary graphene metamaterial: (a) schematic of unit cell structure and (b) section view of unit cell.
In order to explore EIR response properties of the complementary grphene metamaterial, numerical calculations based on finite difference time domain (FDTD) method were performed, where periodic boundary conditions are used for a unit cell in the x- and y-directions, and the z plane has a perfectly matched layer boundary condition. The plane waves polarizing along x-direction are normally incident to the structure along the -z direction, as shown in Fig. 1(a). In our calculations, the permittivity of the SiO2 and Si substrate are taken as 3.9 and 11.7 respectively. To simplify the numerical calculations, we further assumed the graphene to be an effective medium of thickness t = 1 nm with a relative complex permittivity of εr (ω) = 1 + j σ(ω) / (ωε0t), in which the conductivity σ(ω) can be described as [18]:
Where ε0 is the permittivity of vacuum, ω is the frequency of incident wave, EF is the Fermi energy, Г is the scattering rate (Г = 2.4 THz), T is the temperature of the environment (T = 300 K), e is the charge of an electron, kB is the Boltzmann’s constant, ħ = h / 2π is the reduced Planck’s constant.When a bias voltage Vg is applied between the top and back gates, the carrier density and the position of the Fermi energy level in graphene can be dynamically controlled. This in turn leads to a voltage-based tuning conductivity of the graphene and dynamically controlled propagation of THz wave. An approximate closed-form expression that relates EF to Vg is given by [19]:
Where vF = 106 m/s is the Fermi velocity, and α = 7.56 × 1014/m2V is the capacitance per unit area per charge of the SiO2 oxide. Here, the corresponding relation of EF and Vg is shown in Fig. 2.
3. Results and discussions
To clarify the underlying forming process of this EIR response, three different structures are calculated and analyzed, respectively. The first structure consists of a wire-slot array, the second consists of a SRRs-slot structure array and the third (EIR metamaterial structure) consists of both the wire-slot and a SRRs-slot structure array. The calculated reflection spectra of three structures are shown in Fig. 3, where the Fermi energy of graphene is fixed as 0.5 eV. We observed from Fig. 3 that the sole wire-slot array shows a broad LSP resonance at 0.91 THz in the reflection spectrum when incident magnetic field H is oriented along the y-axis, while the sole SRRs-slot array supports a narrow LC resonance at the same frequency when incident magnetic field H is oriented along the x-axis. As a result, for incident magnetic field excitation along the y-axis, the slot-wire and SRRs-slot in the EIR structures serve as the bright and dark modes, respectively.
Fig. 3 Simulated reflection spectra for sole wire-slot structure, sole SRRs-slot structure and the EIR structure composed of the wire-slot and SRRs-slot structures.
When two slot-type resonators are integrated within a unit cell and form a complementary EIR structure with the lateral displacement of y = −29µm, we observe that a reflection peak with the amplitude of 96% at 0.97THz appears between two sharp resonance dips located at 0.85THz and 1.04THz respectively. This reflection peak can attribute to the near field coupling between the bright and dark modes. To better elucidate the occurrence of the reflection peak, surface current density distributions at different resonance frequencies are shown in Fig. 4. For the reflection peak at 0.97THz, the surface currents mainly concentrate on near SRRs-slot structures, while there are no currents near wire-slot structure (as shown in Fig. 4(b)). Such current density distributions show that the power from bright mode is efficiently transferred to the dark mode by the near field coupling. Here, the dark mode in the EIR structure is indirectly excited by the near field coupling, while the bright mode is completely suppressed, giving rise to a destructive interference between the bright and dark modes, as a result, inducing a sharp reflection peak (as shown in Fig. 3). In contrast, both SRRs-slot and wire-slot structures are simultaneously excited at the resonance dips due to near field coupling (as shown in Figs. 4(a) and 4(c)). Here, the interference is constructive at resonance dips and the coupling effects in both elements would be enhanced. But, the depths of two resonance dips are different due to the intensity difference of near field coupling, as a result, resulting in the reflection peak in EIR strucutre shown in Fig. 3 is asymmetric. Moreover, the similar characteristic is also observed in the traditional terahertz metamaterials [20].
Fig. 4 Surface current density distributions of EIR structure at different resonance frequencies: (a) 0.85 THz, (b) 0.97 THz, and (c) 1.04 THz.
To demonstrate the amplitude modulation of the reflection peak, next, the reflection spectra of the designed EIR structure with a fixed Fermi energy (EF = 0.5eV) but different lateral displacements are numerically calculated for the incident H field excitation along the y-axis, where the symmetry of the whole EIR structure along the y-axis is unperturbed and the lateral distance between the slot-wire and SRRs-slot structures remains constant as well. We observed clearly that the EIR window undergoes an on-to-off modulation as the lateral displacement is gradually varied from y = −29µm to y = 14µm, as shown in Fig. 5. For example, for y = −29µm a reflection window with a maximum amplitude of 96% at 0.97THz is obtained (as shown in Fig. 5(a)). With change in the lateral displacement from −29µm to 12µm, the amplitude of the reflection peak gradually narrows and decreases without obvious frequency shift, which indicates that the reflection window undergoes strong modulations. Such EIR structure could be used as a terahertz modulator. When the lateral displacement is y = 14µm, the reflection peak completely disappears and only a single broad resonance dip with a minimum of 0.016 at 0.90 THz is observed, thus realizing an on-to-off EIR peak modulation (as shown in Fig. 5(d)).
Fig. 5 Simulated reflection spectra of the designed EIR structure with different lateral displacements at EF = 0.5eV: (a) y = −29µm, (b) y = 0µm, (c) y = 12µm, and (d) y = 14µm.
To better understand the mechanism of the amplitude modulation, the distributions of the magnetic field at the reflection peak frequency are calculated for different lateral displacements. Figure 6 shows the magnetic field (Hz) distributions of the EIR structure with different lateral displacements (such as y = −29µm, 0µm, 12µm, and 14µm) for the incident H field excitation along the y-axis, corresponding to a pronounced EIR peak, a weaken EIR peak, a disappeared EIR peak, and only a LSP resonance respectively (as shown in Fig. 5). For y = −29µm (as shown in Fig. 6(a)), the magnetic field is primarily concentrated near the SRR-slot gaps, whereas the magnetic field near the wire-slot is completely suppressed, which is characteristic of the typical EIR effect (as shown in Fig. 5(a)). As the lateral displacement is gradually varied from y = −29µm to y = 14µm, the magnetic fields are gradually translated from the SRRs-slot structure to wire-slot structure, as a result, resulting in a redistribution of the magnetic fields in the EIR structures. This is verified in Figs. 6(b) and 6(c) where the SRRs-slot and wire-slot structures are both excited by the near field coupling when the lateral displacements are y = 0µm and 12µm, respectively. But, the field intensity is obviously different at two cases. Moreover, for y = 14µm, the magnetic field is mainly focused on the LPS resonance of wire-slot structure, while the magnetic field near the SRRs-slot gaps is completely suppressed, as a result, leading to the disappearance of the EIR peak. Therefore, the nature of the EIR peak modulation, as shown in Fig. 5, depends on how well the wire-slot structure couples to the SRRs-slot structure. Moreover, this coupling in turn depends on the constructive and destructive interactions between Hxslot-wire and Eslot-wire excited in wire-slot structure.
Fig. 6 Magnetic field (Hz) distributions of the EIR structure with different lateral displacements at EF = 0.5eV: (a) y = −29µm, (b) y = 0µm, (c) y = 12µm, and (d) y = 14µm.
Based on above analysis, we notice that the reflection window is caused by the near field coupling, rather than the broken symmetry in a typical EIT system [21]. The reason is that when the incident H field excites along the y-axis, the symmetry of our EIR structure along the y-axis is unperturbed and the lateral distance between the slot-wire and SRRs-slot structures remains unchanged. In order to undstand the near field coupling underlying mechanism in integrating wire-slot and SRRs-slot EIR system, we further calculate the electromagnetic properties of two additional cases, such as, y = −56µm and y = −14µm, as shown in Fig. 7. As observed in Fig. 7, the structures at both cases exhibit the reflection peaks, but the amplitudes of the reflection preaks are different, as shown in Figs. 7(a) and 7(b). In order to explain the difference of two reflection peaks, we also calculate the field distributions of sole wire-slot structure, as shown in Figs. 7(c) and 7(d).
Fig. 7 Simulated reflection spectra of the designed EIR structure for (a) y = −56µm and (b) y = −14µm, respectively. Simulated distributions of (c) x component of magnetic field Hxslot-wire and (d) elecriic field Eslot-wire of the sole wire-slot pattern at 0.91THz.
Figures 7(c) and 7(d) show the simulated distributions of Hxslot-wire (x component of the magnetic field from the slot-wire structure) and of Eslot-wire (the electric field from the slot-wire structure) respectively, which indicates that the dark mode excitation mechanisms by the near field coupling for y = −56µm and y = −14µm are distinct. For y = −56µm, the SRRs-slot structure is near the bottom edge of the slot-wire structure, where Hxslot-wire is highly concentrated, while Eslot-wire is too weak to excite the dark mode. Thus, the dark mode results mainly from the Hxslot-wire excitation of the wire-slot bottom edge. For y = −14µm, the SRRs-slot structure is near the center of the wire-slot length, where Eslot-wire has the strongest electric field distributions while Hxslot-wire is extremely weak, as a result, the dark mode is mainly excited via the strong electric field coupling from the slot-wire structure to the SRRs-slot structure. As the SRRs-slot structure is translated along the wire-slot starting from both ends to the center of wire-slot structure, the coupling mechanism switches from being inductive via the magnetic field to be capacitive via the electric field of the wire-slot structure. Therefore, as the SRRs-slot structure is moved along the wire-slot from the bottom to the top (for example from y = −29µm to y = 14µm), the interaction between Hxslot-wire and Eslot-wire is gradually transformed from the constructive interference to the destructive interference because the electric field distributions about wire-slot structure remain unchanged but the magnetic field distributions at two ends of wire-slot change their directions (as shown in Figs. 7(c) and 7(d)), that is, a strong excitation of the LC resonance is switched to only the LSP resonance where the dark mode disappears, as a result, leading to the amplitude modulation of the reflection peak, as seen in Fig. 5.
Compared with a traditional metal-based EIT structure, the most important advantage for the graphene is the capacity of dynamically tuning the conductivity through electrostatic gating or chemical doping. This is highly desirable in practical compact elements (such as tunable sensor, switches and slow light devices) since it is very difficult to change the physical structure after fabrication. To further demonstrate the frequency tunability of our structure, the reflection spectra of the designed EIR structure with a fixed lateral displacement (y = −29µm) but different Fermi energy are numerically calculated, as shown in Fig. 8. The calculated results from Fig. 8 clearly demonstrate that as increase in Fermi energy, the reflection peak can be blue shifted in the interested frequency range. For example, when the Fermi energy changes in the range of 0.1-0.5eV, the reflection peak can be modulated in the range of 0.79-0.97THz, and the corresponding frequency modulation depth (fmod = Δ f / fmax) of 19% could be obtained through a small variation of EF (0.4eV). With a large variation of Fermi energy, the frequency modulation depth can be further increased. This behavior can be interpreted by considering the resonance condition. Here, the wave vector of SPP along the graphene layer satisfies kspp ∝ (fr2) / (2α0EF c), where α0 is the fine structure constant and fr is the resonance frequency [14]. Thus, the resonance frequency can be written as fr ∝EF1/2. Therefore, the resonance frequency can be tuned by changing the Fermi energy of graphene without re-optimizing or re-fabricating the physical structure, which indicates that graphene-based metamaterials are more active for the EIT-like effect than metal-based ones. Based on the tunable capacity, moreover, the switches can be designed in the interested frequency range. Such as, the reflection amplitudes for EF = 0.1eV and EF = 0.3eV at f = 0.81THz are 0.90 and 0.04, respectively (as shown in Figs. 8(a) and 8(b)). This result indicates that the reflection could be switched between 90% and 4% by merely changing EF with a variation of 0.2eV.
Fig. 8 Calculated reflection spectra of the proposed EIR structure at different EF: (a) 0.1eV, (b) 0.3eV, and (c) 0.5eV.
As is well known, another remarkable characteristic of EIT response in atomic system is slow light effect, which could exhibit many potential applications in optical memory and quantum memory [22]. Currently, the slow light capacity of EIT in metamaterial structures is also extremely attractive for the practical applications such as strong light-matter interaction and enhanced nonlinear effects [9]. Here, the slow light effect is characterized by the group delay (tg = -dφ/dω, and ω = 2πf) of the terahertz wave packet through the EIR structure, instead of normally used group refractive index (ng), where φ and f are the phase shift and frequency of reflection spectrum respectively. Figure 9 shows the calculated group delays for the proposed graphene-based complementary EIR structure at different Fermi energy. As shown in Fig. 9, we observes the positive and negative group delays which correspond to slow and fast light respectively. In the vicinity of the transparency peaks, moreover, the large positive group delays are obtained, indicating slow light effect. For example, at the reflection peak, the group delays at different Fermi energy are more than 0.47ps (including the propagation in the substrate) which is equivalent to the time delay of a 135µm distance of free space propagation.
Fig. 9 Group delays for the graphene-based complementary EIR structure at different EF: (a) 0.1eV, (b) 0.3eV, and (c) 0.5eV.
The concept of the above group delay describes how slow the light is. However, for slow light device, only talking about tg is meaningless, because we must care about the bandwidth at the same time. The delay-bandwidth product (DBP) is a good indication of the highest slow light capacity that the device potentially provides [23]. Therefore, the DBP is defined as the product of the time group delay tg and the bandwidth Δf in slow light:
where Δf is the full width at half maximum bandwidth, tg is the maximum group delay at the transparency window [24]. For different Fermi energy of EF = 0.1eV, 0.3eV and 0.5eV, however, the corresponding delay-bandwidth products (DBPs) are about 0.12, 0.13 and 0.143, respectively. Moreover, the DBPs increase with the Fermi energy. In addition, we also notice that by tuning the Fermi energy, group delay similar to the reflection window could be blue shifted over a fairly wide frequency range. Therefore, we can achieve the capability of turning on and off the group delay and controlling the amount of the group delay by tuning the Fermi energy.Conclusions
In conclusion, we have demonstrated a tunable EIR response based on the graphene-based complementary metamaterials. The field distributions reveal that the EIR response origins from the dark mode excitation by the near field coupling. Moreover, by merely adjusting the Fermi energy of graphene, the reflection peak can be dynamically tuned and realize a broadband blue-shift effect, which could not be achieved by the metal-based EIT metamaterials. In addition, the group delays at different Fermi energy are more than 0.47ps at the reflection peak, and the corresponding DBPs increase with the Fermi energy. Therefore, the proposed complementary graphene metamaterials with tunable reflection window have potential applications in designs of tunable slow light devices, tunable sensors and switchers which can be operated at THz frequencies.
Acknowledgments
The work is supported by the National Natural Science Foundation of China (51402075 and 51005001), Heilongjiang Province Natural Science Foundation of China (F201309), the Postdoctoral Science-Research Developmental Foundation of Heilongjiang Province (LBH-Q11082), the Youth Academic Backbone Support Plan of Heilongjiang Province Ordinary College (1253G026), Special Funds of Harbin Innovation Talents in Science and Technology Research (2014RFQXJ031) and Science Funds for the Young Innovative Talents of HUST (2011F04).
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