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We propose and numerically investigate a gate-controlled on-chip graphene metasurface consisting of a monolayer graphene sheet and silicon photonic crystal-like substrate, to achieve an electrically-tunable induced transparency. The operation mechanism of the induced transparency of the on-chip graphene metasurface is analyzed. The tunable optical properties with different gate-voltages and polarizations have been discussed. Additionally, the spectral feature of the on-chip graphene metasurface as a function of the refractive index of the local environment is also investigated. The result shows that the on-chip graphene metasurface as a refractive index sensor can achieve an overall figure of merit of 8.89 in infrared wavelength range. Our study suggests that the proposed structure is potentially attractive as optoelectronic modulators and refractive index sensors.
© 2016 Optical Society of America
1. Introduction
Electromagnetically induced transparency (EIT) has first demonstrated in the experiment of strontium atomic vapor by S. E. Harris [1]. It is a result of quantum interference between different excitation paths in three-level atomic gases system. A transparency window in a broad absorption spectrum can be formed via interference among atomic level transitions. This phenomenon has potentially attractive for highly sensitive sensors [2] and ultrafast switching [3]. Meanwhile, the demands of stable atomic gases and low temperature environment severely hinder practical applications, particularly with respect to on-chip integration. These obstacles can be overcome via the metamaterial-based devices. Recently, the classical EIT effect was also studied in the artificially designed metamaterials as a new research direction, such as metamaterial arrays of split-ring meta-molecules [4,5] and other nanostructures [6–13], which was alluded to as metamaterial induced transparency (MIT) [14]. However, those above devices are functional-fixed, and cannot achieve the electrically-modulated function after those devices are fabricated.
Graphene is a two-dimensional material, and a monolayer of carbon atoms arranged in a hexagonal lattice. It is found that graphene can be used as a candidate material for many tunable photonic devices at mid-infrared frequencies due to unique physics properties [15–17]. For instance, its conductivity can be easy controlled by an external gate-voltage. Recently, the capability of the transformation optics using graphene is reported by N. Engheta [18], many graphene-based plasmonic structures and devices have been proposed theoretically and experimentally, such as modulators [19], metamaterials and metasurfaces [20–24], photonic crystal circuits [25]. Importantly, it has also been reported that the induced transparency effect can be realized in graphene-based metamaterial [26, 27] and nanostrips [28–30] and waveguide [31], which rely on the regulation of graphene by shifting the Fermi levels via chemical doping. In those cases, the induced transparency effect can be achieved via the optimization to the geometry of the devices, which is imperfect in traditional metasurfaces and other graphene-based integrated photonic devices. The quality factors of most of graphene-based EIT-like systems [32] is generally unsatisfactory, which is unpractical for the future highly sensitive sensors. In addition, those graphene nanostrips very easily cause the quantum confinement effects associated to the type of edges when the nanostrip width is under 20nm [33].
In this article, to address the above issues, we propose a flexible design of on-chip graphene metasurface. Based on the new design, the MIT effect can be actively tuned via only an external gate-voltage, to achieve the electrical-modulating function. The operation mechanism of the induced transparency of on-chip graphene metasurface is also analyzed. In addition, the figure of merit (FOM) of on-chip graphene metasurface system can reach as high as 8.89, which is comparable to recently reported results realized in other systems. Being different from the previous studies, the designed on-chip graphene metasurface is not required to reconstruct the geometry of a device, but it can avoid the quantum confinement effects of graphene nanostrips. Our work may inspire interest in developing optoelectronic modulators.
2. Induced transparency properties of on-chip graphene metasurface
The characteristics of the tunability in graphene have been analytically and experimentally researched in-depth [34–36]. This property is achieved by changing the Fermi level, which is based on the fact that the carrier density can be changed by chemical doping or applying electrostatic or magnetostatic field. Under the condition of the random-phase approximation, its complex surface conductivity dominated by interband and intraband transitions is estimated as:
Where
Here kB is the Boltzmann constant, ω the optical angular frequency, is the reduced Planck’s constant, τ the carrier relaxation time, T is the temperature (T = 300 K in this paper). Ef is the Fermi level. In the whole simulations, the thickness of graphene is much smaller than the wavelength at THz frequencies. As the result, graphene is modeled as a thin anisotropic material film with the locally two-sided surface characterized by a surface conductivity σ(ω). Based on the Ampere’s law in stationary regime and Ohm’s law, the 2D-effective in-plan permittivity of graphene is defined as follows [18]:
where t is the thickness of graphene, which is estimated as 0.6 nm with τ = 0.25 ps, and ε0 is the permittivity in vacuum. The SPP modes at subwavelength scale can be supported and spread in a 1D graphene layer sandwiched between the two medium layers above and below for frequencies ranging from THz to the visible [15, 37], known as graphene Plasmon wave (GPW). The dispersion relationship of the SPP modes at subwavelength scale can be expressed in the following form [37]:Where ksp is the in-plane wave vector in graphene, k0 is the wave vector in vacuum. ε1 and ε2 are the dielectric constants of the top and bottom surface materials of the graphene sheet. The graphene Fermi level can be changed by applying a gate voltage, which can be calculated according to Ref [38]. It can be seen that, one can adjust the properties (metal or medium) of graphene sheet via a suitable voltage to vary Fermi levels. In frequency domain of interest, a graphene layer effectively behaves as a very thin “metal” because that the imaginary part of its conductivity is positive as the Fermi level is greater than 0.07 eV. Therefore, we propose a novel on-chip graphene metasurface for manipulating the induced transparency on the 2D plane, which will be described in more details in the following.
The designed on-chip graphene metasurface is shown in Fig. 1(a). A monolayer graphene sheet is placed on top of a photonic crystal-like (PC-like) dielectric layer, which consists of periodically arranged double tangent cylindrical silicones in SiO2 dielectric layer. A doped-silicon substrate is attached to the PC-like dielectric layer and is used for an electrode and to support the on-chip graphene metasurface. Figure 1(b) shows the unit cell structure of the design, the graphene in area I is on top of Si, and the graphene in area II is on top of oxidized silicon. The voltages on regions I and II are equal, but the graphene film is alternately placed on top of silicon disks in region I and on top of SiO2 in region II, which form two different kinds of plate capacitors (combined with the bottom doped-silicon electrodes) with different capacitances of C1,2 in parallel connection. Therefore, carrier/electrons are respectively and locally charged or stored on graphene regions I and II of the capacitors. The Femi level of each graphene region (I or II) is determined by the capacitor-inducing carrier concentration of ng, and thus can be different even at the same voltage, based on Ef = ħvf(πng)1/2 [38] with the Fermi velocity of vf = 106 m/s. The capacitor-inducing carrier concentration is obtained fromng1,2 = C1,2Vg/(eS1,2) = ε0εd1,2Vg/(ed), which Vg is the external voltage loaded on graphene, εd1,2 are receptively the dielectric constants of silicon and SiO2. S1,2 and d are receptively the areas and thickness of insulating layers in regions I and II. Although the plate capacitors in regions I and II are in parallel connection, the doping of graphene in regions I and II is different due to different dielectric constants of εsi and εsio2, as well as different electric fields within the capacitors of regions I and II. As the result, the Femi levels of graphene in regions I and II are also different. Hence, one can adjust the graphene to metal properties. As the result, one can use the graphene with the metallic properties to replace the noble metals in the traditional metasurface, which can realize a kind of ultrathin metasurface with the electrically-tuned function. Based on this approach, one can realize the tunable on-chip double graphene nanodisk metasurface (DGNM) by only an external gate-voltage. In our numerically calculate process, the Fermi energies of graphene in areas I and II are fixed as 0.9 eV and 0.3 eV, respectively. The dielectric constant of the doped-silicon is set to be 4.
Fig. 1 (a) 3D schematic of the on-chip graphene metasurface, H1 = H2 = 50 nm. (b) The unit cell structure of our design, the periods of the x and y direction are Px = 1.1 μm and Py = 0.6 μm, respectively. The radius of circular Si is R = 0.25 µm.
In the following, we investigate the optical performance of the proposed structure using the 3D finite-difference time-domain (FDTD) method (Lumerical FDTD solutions software) [39] to simulate of the unit cell structure surrounded by periodic boundary conditions in the x and y directions and the perfectly matched layer absorbing boundary conditions in the y direction. The above dispersive permittivity of graphene is applied or embedded to our FDTD simulations. The mesh sizes used for graphene simulation in x, y and z directions are respectively set to be 1, 1 and 0.1 nm. The simulation results are stable as long as the mesh is not larger than 1 × 1 × 0.1 nm3. To analyse the formation mechanism of the induced transparency of the proposed structure, we firstly analyse the spectral features of the single graphene nanodisk metasurface (SGNM) in the frequency domain of our interest by using a normally incident plane wave with the polarization direction parallel to the diameter direction of cylindrical silicone, as shown in Fig. 2(a). The transmission spectrum of the SGNM for the different radii of graphene nanodisk is plotted in Fig. 2(b). It is found that the resonance dip appears and becomes sharper when the radius of graphene nanodisk is increased from 0.1 µm to 0.25 µm. It means that the quadrupole mode of the graphene nanodisk is excited by the incident light.
Fig. 2 (a) The unit cell structure of the SGNM; the periods in the x and y directions of Px = Py = 0.6 μm, R = 0.25 µm. (b) Calculated transmission spectra for single graphene nanodisk metasurface, with Ef(I) = 0.9 eV, Ef(II) = 0.3 eV.
Figures 3(a)-3(b) show the unit cell structure of the SGNM and DGNM, and their transmission spectra. The doping and the conducting properties of graphene in regions I and II can be different, due to different dielectric constants of εsi and εsio2, even at the same and suitable voltage Vg and the same thickness of the Si and SiO2 layers. The normal incident light with the polarization E perpendicular to the symmetric axes (x-axes, red dashed line) can excite powerful plasmon oscillations at the side face of marginal-points C and D of a graphene disk, as shown in Fig. 3(a), similar to the surface plasmons on the interface between of metal and dielectric. The E2-component of the light field (perpendicular to the margin at M point in Fig. 3(a)) can also excite plasmon oscillations at other M points of graphene disk. These plasmon oscillations can drive current circulating in opposite directions in the left and right parts of the disk, and induce the (high-order) quadrupole modes of the graphene disk in the diagonal distribution, as shown in Figs. 3(c) and (d). It can be seen that the transmission spectra (star shape line) exhibit a transmission resonance dip with the Lorentz symmetric line shape, as shown in Fig. 3(b). The plasmon oscillations of graphene disk can be described with a quasi-static analysis, which the plasmon frequency is calculated from ω0 & (Ef/λsp)1/2 (where λsp∼D, λsp is resonant wavelength of the SPs and D is the diameter of the disks) [40].
Fig. 3 (a) The unit cell structure of the SGNM. (b) Transmission spectra of numerical calculations by FDTD for single/double graphene nanodisk metasurfaces. (c)(d) Real part of the normal component of the electric field (Ex) distributions and the electric field intensity distributions (|E|2) at the transmittance dip λ = 7.44 µm for the unit cell structure of the SGNM. (e)(f) Real part of the normal component of the electric field (Ex) distributions and the electric field intensity distributions (|E|2) at the transmittance dip λ = 7.43 µm for the unit cell structure of the DGNM. (g) (h) The calculated Ex distributions for the DGNM at 7.32 µm and 7.63 µm wavelengths. The black arrows indicate the directions of the currents along the nanodisk edge.
Next, the transmission property of the DGNM and the corresponding electric field distributions for different transmittance tips or peak are also investigated. In the DGNM system, the symmetry is broken by adding another graphene nanodisk. As a result, the transmission spectrum of DGNM manifests an EIT-type resonance profile, as shown in Fig. 3(b) (circle shape line). The transmittance tips and peak are obvious as indicated by blue arrows at the resonant wavelengths of λ = 7.32 µm, 7.63 µm, and 7.43 µm. The quadrupole mode each of graphene nanodisk is not excited but coupled to each other at λ = 7.43 µm, as shown in Fig. 3(e). Notably, these antiparallel electric dipoles due to the suppression of radiative damping can drive current circulating in opposite directions in the left and right nanodisks, which corresponds to the highly enhanced near-field is mostly concentrated in the gap between two graphene nanodisks as shown in Figs. 3(e) and (f). Figure 3(g) shows the electric field distribution in one period at the first resonant dip of λsp = 7.32 μm. One can see that the E-field concentrated mostly in the outer edges of two graphene disks as result of the subradiant state, due to the destructive interference of two antisymmetric modes of the disks. On the other hand, because of the constructive interference of two symmetric modes of the graphene disks, the superradiant state exists at the second resonant dip of λ = 7.63 µm, which results in the highly enhanced near-field in the gap between two disks as shown in Fig. 3(h).
A typical three-level atomic system can be used to in demonstrations of EIT. Figure 4 shows the prototype system for EIT. The ground state (|0〉) is coupled to the excited state (|1〉), where absorption occurs via a probe beam. A pump beam couples the metastable state (|2〉) to the excited state, while transitions from the metastable to the ground state are not allowed. Interference between the two transitions leads to a vanishing probability for the atoms to be found in the excited state; consequently, absorption is minimized. One can also use the three-level atom system to analyze the near-field interaction of the quadrupole modes in the DGNM system. The DGNM system involves a superradiant state (excited state) |1〉 (bright mode) and a subradiant state (metastable state) |2〉 (dark mode) and a ground state |0〉, which have resonant frequencies ω01, ω02, respectively. Herein, we recognize that a graphene nanodisk may function as a quadrupole that could serve to support the superradiant state (bright mode) or “bright atom” in the EIT-like system. The resonance frequency of the graphene nanodisk can be readily tuned by varying its radius, as shown in Fig. 2(b). Therefore, we define |0〉→|1〉 as a dipole-allowed transition, which is analogous to single quadrupole mode (bright mode) be excited and correspond to the quadrupole mode resonance in the SGNM system, as shown in Fig. 3(c). Similarly, the DGNM system consists of double tangent graphene nanodisks supports a subradiant state (dark mode) or “dark atom” in the EIT-like system. In the DGNM system, two quadrupole modes can be excited when the symmetry of DGNM system is borken due to introducing another graphene nanodisk, which leads to a EIT-like resonance peak at the centerof the transmission tip, as shown in Fig. 3(b). As a result, the transition between states |1〉 and |2〉 is related to the coupling between the two quadrupole modes, which confirms the EIT-like destructive interference between the two pathways: |0〉→|1〉 and |0〉→|1〉→|2〉→|1〉. However, this interference can be described by the coupled Lorentz oscillator model in the following form [27]:
where E1, E2, r1 and r2 are the amplitudes and the damping factors of the bright and dark modes, respectively. k is the coupling coefficient between states |1〉→|2〉, and g is a geometric parameter indicating the coupling strength of the bright mode with incident electromagnetic field E0.Fig. 4 The energy diagram for typical three-level atomic system. |0〉, |1〉, and |2〉 are the ground state, excited state, metastable state, respectively. The two possible pathways, namely, |0〉→|1〉 and |0〉→|1〉→|2〉→|1〉, interfere destructively and lead to the EIT-like phenomena.
3. Tunable properties of the gate-controlled DGNM for refractive index sensing
The spectral variation of the DGNM as a function of the external voltage is one of the most fascinating properties expected for the DGNM system, which should be designed effectively and flexibly towards realizing tunable on-chip graphene optoelectronic devices. In this section, we will focus on discussing the refractive index sensing and tunable optical properties of the gate-controlled DGNM. Different from previous studies that the induced transparency effect has to be tuned by changing the geometry of devices, herein we demonstrate numerically that the induced transparency effect of the DGNM can be tuned via an external voltage. Figure 5(a) shows the transmission spectrum of the DGNM for the external voltage of 50 V, 22 V and 12.5 V, which correspond to Ef(I) = 0.9 eV and Ef(II) = 0.3 eV, Ef(I) = 0.7 eV and Ef(II) = 0.2 eV, Ef(I) = 0.5 eV and Ef(II) = 0.15 eV, respectively (based on the formula in Ref [38].). As can be seen, the resonance dip is becoming shallower and wider with the decrease of the external voltage. Meanwhile, the metallic properties of graphene nanodisk become weak due to the decrease of the external voltage, which results in the near-field interaction between two graphene nanodisks weaker. As a result, the induced transparency effect is almost disappeared when the voltage is reduced to 22 V. It indicates that an optical switch of EIT can be achieved. One can also find that the shift of the transparency window is sensitive to the external voltage. Therefore, it is possible to realize electrically-switching of the slow light.
Fig. 5 (a) Transmission spectra of the DGNM with various the external voltages. (b) Evolution of the transmission response of the DGNM with different polarization angles, given Ef(I) = 0.9 eV and Ef(II) = 0.3 eV.
We also analyzed the polarization response of the DGNM excited at normal incidence with different polarizations. From Fig. 5(b), one can see two broad resonance dips due to the collective excitation of the quadrupole mode in each graphene nanodisk. Further, a blue-shift of the resonance dips was appeared with the increase of the polarization angle. As the polarization angle is larger than 30°, the resonance dip contrast is weak because the interaction between the quadrupole modes of the graphene nanodisks is also weakened. Therefore, we can still tune the EIT effect of the DGNM by changing the polarization state of the incident light for a given gate-voltage, which offering additional tunable degrees of freedom to operate light.
As shown in Fig. 6, the optical response of the DGNM with the changing of refractive index of local environment is investigated. To quantitatively evaluate the performance of a refractive index sensor, we can define the sensitivity as the shift of the resonance wavelength per refractive index unit (RIU). Another more accurate parameter to describe the sensitivity is the FOM which corresponds to the energy shift per RIU divided by the spectral linewidth. Figure 6(a) exhibits the transmission spectra of the DGNM embedded in various dielectric media. The resonance property of the DGNM system shows a distinct red-shift with an increase of the refractive index of the surrounding medium. The sensitivity of the DGNM system is 1.267 µm/RIU as shown in the inset of Fig. 6(b). As a result, one can obtain the slope (0.027eV/RIU) of the best-fitted line for the first peak position as a function of the refractive index. The average linewidth of the transmittance peak is calculated to be 0.003 eV. Hence, the overall FOM is 8.89. This value is not only higher than the well-known hybrid graphene metal gratings (FoM = 7) [32] but also higher than the nanohole quadrumers (FOM = 6.23) [41]. In addition, we have checked the performance of the structure for the higher losses of the graphene. Based on Ef = ħvf(πng)1/2 and ng = ε0εdVg/(ed), the Fermi level Ef will increase, when the external voltage Vg is boosted. Therefore, the imaginary part of the effective index of graphene and its losses decrease with the rise of the external voltage, derived from the calcluted result which the imaginary part of the effective index decreases with the increase of Ef. At the same time, the resonance dip of the transmission of the DGNM becomes sharp with the increase of the external voltage, as shown in Fig. 5(a). In another words, it reveals that the performance of the structure becomes worse for higher losses. The figure of merit also decreases with the increasing of graphene losses. The shift of the resonant wavelength is very sensitive to the change of refractive index, which will be very beneficial for the sensor study of graphene. Moreover, the induced transparency effect of the DGNM system can be tuned by only one gate voltage, which may open up a new path for sensing in the mid-infrared frequencies.
Fig. 6 (a) Transmission spectra of the DGNM with different refractive indices of local environment, given Ef(I) = 0.9 eV and Ef(II) = 0.3 eV, (b) Position of the first transmittance peak versus the refractive index (red solid circles). The inset shows the resonance wavelength versus the refractive index (black solid circles).
4. Conclusion
We have proposed a gate-controlled on-chip graphene metasurface in the mid-infrared or THz frequency regime and have analyzed the formation mechanism of the induced transparency property. We have also analysed the optical response of the SGNM with different radii of graphene nanodisk in frequency domain of interest. Based on the gate-voltage controllable characteristics of graphene, the tunable optical property of the DGNM as a function of external gate-voltage has been investigated, with an electrically-switched of the induced transparency. Meantime, the MIT of the DGNM with different polarizations at normal incidence and the refractive index of local environment have been investigated. These applications of the electrically-switched induced transparency and modulator have been presented. Moreover, this DGNM structure can potentially be useful for achieving the manipulation of terahertz signal for refractive index sensing, showing promise in future graphene-based integrated optics.
Funding
Project of Discipline and Specialty Constructions of Colleges and Universities in the Education Department of Guangdong Province (2013CXZDA012); Guangdong Natural Science Foundation (2014A030313446); Program for Changjiang Scholars and Innovative Research Team in University (IRT13064).
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