## Abstract

To achieve plasmonically induced transparency (PIT), general near-field plasmonic systems based on couplings between localized plasmon resonances of nanostructures rely heavily on the well-designed interantenna separations. However, the implementation of such devices and techniques encounters great difficulties mainly to due to very small sized dimensions of the nanostructures and gaps between them. Here, we propose and numerically demonstrate that PIT can be achieved by using two graphene layers that are composed of a upper sinusoidally curved layer and a lower planar layer, avoiding any pattern of the graphene sheets. Both the analytical fitting and the Akaike Information Criterion (AIC) method are employed efficiently to distinguish the induced window, which is found to be more likely caused by Autler-Townes splitting (ATS) instead of electromagnetically induced transparency (EIT). Besides, our results show that the resonant modes cannot only be tuned dramatically by geometrically changing the grating amplitude and the interlayer spacing, but also by dynamically varying the Fermi energy of the graphene sheets. Potential applications of the proposed system could be expected on various photonic functional devices, including optical switches, plasmonic sensors.

© 2016 Optical Society of America

## 1. Introduction

Two phenomena of coherent light-matter interactions can dramatically modify the optical response of an absorbing system in the presence of an additional field: electromagnetically induced transparency (EIT) and Autler-Townes splitting (ATS). The process known as EIT is the result of Fano interferences that require coupling of a discrete transition to a continuum [1], which will create a narrow transparency window by eliminating a resonant absorption and, during the past decades, has a rich variety of important applications such as slow light propagation [2], optical storage [3], plasmonic switching [4], holographic imaging [5] and so on. While ATS is due to the strong-coupling induced splitting of resonant modes and is not associated with interference effects [6,7]. It also creates a transparency window through the doublet structure in the absorption profile, which has been used for quantum control of spin–orbit interactions [8] and measuring transition dipole moments [9]. Both of these two quantum effects are quantified by a transparency window in the absorption or transmission spectrum via different pathways in various three level systems [10]. This similarity of the spectrum attracts much confusion and discussions on the distinction between EIT and ATS. It is crucial for applications and for clarifying the physics involved. Therefore, various studies have been made to discern the EIT and ATS effects using different systems [6,7,10–13]. Especially, Anisimov *et al*. had proposed to use the Akaike Information Criterion (AIC) as an objective method to discern ATS from EIT in experimentally obtained absorption or transmission spectra [6], which has been successfully applied to identify whether the obtained spectra is resulted from EIT or not in systems such as whispering-gallery-mode microresonators [7], coupled mechanical oscillators [11], cold cesium atoms [12], plasmonic devices [13], and superconducting flux quantum circuits [14].

Among these systems, plasmonic devices have received much attention due to their interesting physics, important applications, and the inherent property of strong light-matter interactions. Very recently, there have been numerous researches in the realization of a novel phenomenon analog to EIT or ATS transmission, called plasmon-induced transparency (PIT). This plasmonic phenomenon will result in a transparency window in the absorption or transmission spectra when the coherent interference of coupled resonators occurs, bringing the original quantum phenomena into the realm of classical optics [15]. PIT has been found mainly in systems such as in metamaterials [15–19] and waveguide structures [13,20–22]. Notably, most of the PIT effects result from the interaction between the superradiant (radiative) mode and subradiant (dark) mode, or the symmetric breaking of the metamaterial system [13,15,17,18,22], which is commonly known as an EIT-like phenomenon. So far, most of the transparency windows observed in metal-based PIT systems have been realized at a fixed working wavelength once the devices are fabricated. To address this issue, approaches that can achieve dynamic tuning of the transparency windows have emerged by integrating materials with tunable permittivity, including phase-change media [23] and nonlinear metamaterials [24]. Among these the most intriguing one is to use graphene. Since that the optical properties of graphene can be dynamically tuned by changing the Fermi energies via a chemical or electrostatic gating, or magnetic field [25], it allows one to effectively tune the PIT devices at different working frequencies. Therefore, the combination of graphene and metamaterials is a promising platform to design a dynamically wavelength tunable PIT devices [16,18,19]. However, the implementation of most of PIT systems rely heavily on the shaped nanostructures that need to pattern the resonators and gaps between them to an extremely small dimensions [13,15,18,19]. This requirement, on the one hand, poses great challenges for the synthetic processes associated with making these materials. On the other hand, it cannot keep the integrity of a natural graphene because the quality of the sheet is affected by the lithographic process. At the same time, most of the transparency windows observed from the absorption or transmission spectra in the plasmonic devices are due to the EIT effect, instead of ATS. And most importantly, in these plasmonic systems, only several studies have objectively discerned whether the induced transparency window is resulted from an EIT/ATS effect or not.

In this paper, we propose an easily implemented and dynamically tunable PIT system that is composed of a sinusoidally curved and a planar graphene layers, avoiding any of the pattern of the graphene sheet. We will show that the induced transparency can be drastically tuned by changing the coupling strength between the two layers through the varying of the grating amplitude and the interlayer spacing. Meanwhile, by using the coupled-mode theory, the physics of the proposed PIT system can be understood by examining the analogy between our system and atomic EIT systems. Furthermore, analytical fitting is employed and AIC-based method is used to distinguish whether the obtained absorption spectra is resulted from EIT or ATS. Finally, the sensitivities and stabilities of the proposed system are discussed.

## 2. Description of the designs and materials

The system under study is schematically depicted in Fig. 1. The minimum distance between the plane layer (PL) and the grating layer (GL) of graphene sheets is *d*, the amplitude and the period of the GL are assumed to be *A* and *Λ*, respectively. The dielectric permittivities of the substrate, the material between the two graphene sheets, and dielectric constant in the space above the GL are *ε*_{3}, *ε*_{2}, and *ε*_{1}, respectively. An *x*-polarized plane wave with wave number *β*_{0} and an incident angle *θ* strikes the surface of the periodically structured graphene system. Different from the other metamaterial-based PIT systems, the proposed designs avoid any pattern of the graphene sheet. Technologically, one possible way to create such graphene-dielectric-graphene hybridized structure is to use the following fabrication process. Firstly, high-quality large-area graphene films can be fabricated either by using an optimized liquid precursor chemical vapor deposition method before transferred onto a planar ITO/oxide/Si substrate [26] or directly by laser ablation on the structured dielectric surface demonstrated recently [27]. Then, the interbedded sinusoidal dielectric grating with specified parameters can be fabricated by using the nanoimprinting process similar to the method described in [28]. Finally, the upper graphene layer can be fabricated either by transferring the graphene sheet or grown directly onto the dielectric grating surface. In this paper, the transmission properties of proposed structures are numerically performed with the finite element method with COMSOL Multiphysics using a refined triangular mesh and the frequency-domain solver. The two-dimensional simulations were performed for a single unit cell with *x*-polarized TM beam incident in the -*y* direction, and periodic boundary conditions are imposed in the *x* direction. In the numerical experiments, the dielectric constants are set as *ε*_{3} = *ε*_{2} = 2.5, and *ε*_{1} = 1, which are taken from experimental data [29]. The period of the grating is *Λ* = 250 nm. These parameters remain unchanged unless otherwise specified.

Besides, in our simulations, the graphene film is modeled within the random-phase approximation, thus the dynamic optical response of graphene can be derived from the linear-response theory (Kubo formalism) in a form consisting of interband and intraband contributions: *σ* = *σ _{inter}* +

*σ*, where

_{intra}*σ*is the optical conductivity of graphene. Both interband and intraband conductivities are closely correlated with the frequency of incident light and the chemical potential of graphene (also Fermi level

*E*). For the doped graphene considered here, the optical conductivity can be approximated for

_{F}*k*<<

_{B}T*E*,

_{F}*ћω*as [30]

*k*is the Boltzmann constant,

_{B}*T*is temperature,

*ћ*is the reduced Planck’s constant,

*ω*is the angular frequency,

*e*is the elementary charge, and

*τ*is the carrier relaxation time, which satisfies the relationship

*τ = μE*(

_{F}/*ev*

_{F}^{2}), while

*E*=

_{F}*ћν*(

_{F}*n*)

_{g}π^{1/2}(where

*μ*= 10000 cm

^{2}/(V∙s) is the measured dc mobility,

*ν*= 10

_{F}^{6}m/s is the Fermi velocity,

*n*is the carrier concentration). The first term on the right-hand side of Eq. (1) is the contribution from the intraband, while the second term is from the interband. Besides in our simulations, both of the flat and the curved graphene sheet are modeled as a thin layer with an isotropic dielectric constant [31] ${\epsilon}_{g}=1+i\sigma \left(\omega \right)/\left({\epsilon}_{0}\omega t\right),$ where

_{g}*ε*

_{0}is the vacuum permittivity,

*t*is the thickness of the graphene and is modeled with a well-known van der Waals thickness of 0.34 nm.

Note that both of the isotropic and the anisotropic models are widely used to describe the optical response of graphene. The former has been commonly used in the model with a flat monolayer graphene [25,32], and, especially, in almost all of the situations with a curved graphene sheet [31,33–35]. In addition, our analyses have reasonably neglected any effects that may arise because of the possible lattice discreteness and distortion (tensile and shear strains) in a real curved graphene, since the effects of curvature along the sinusoidal surface are expected to play a negligible role in that limit [36]. We emphasize that in this work we focus exclusively on the optical properties of the geometrical potential, thus assuming that graphene is homogeneously doped and has uniform distribution of the Fermi energy *E _{F}* over its surface. The effect of a small change in

*E*on trapped modes is the same as that for conventional plasmons on a flat graphene layer.

_{F}## 3. Theoretical model of the coupled plasmonic systems

The physics of the proposed PIT system can be understood by examining the analogy between our system and atomic EIT systems. To explore the physical origin of the PIT transmission observed, the widely used three-level plasmonic system is adapted to elucidate the analogy between atomic and photonic coherence effects leading to EIT and ATS. By using the coupled-mode theory, the equations of motion for the complex plasmonic modes with field amplitudes *Φ*_{1} and *Φ*_{2} in the steady state can be expressed as [7]

*δ*

_{1}=

*ω*–

*ω*

_{1}and

*δ*

_{2}=

*ω*–

*ω*

_{2}denote the detuning between the frequency

*ω*of the incident field and the resonance frequencies

*ω*

_{1}and

*ω*

_{2},

*η*

_{1}=

*η*

_{1}ʹ +

*η*

_{c}and

*η*

_{2}indicate the total losses of resonant modes, respectively, where

*η*

_{1}ʹ is the intrinsic loss of the mode with resonant frequency

*ω*

_{1}and

*η*

_{c}is the coupling loss of the incident field,

*κ*is the coupling strength between the resonant modes,

*Φ*is the incident field. In addition, the normalized absorption can be expressed as ${A}_{absorption}=2{\eta}_{c}{\chi}_{i}$, where

_{in}*χ*is the imaginary part of the susceptibility on the transition for a control field on resonance, which is given by [7]

_{i}Note that *χ* has a form similar to the response of an EIT medium (three-level atom) to a probe field. In the ideal case of zero damping factor, the imaginary part of the susceptibility disappears, leading to an optically transparent effective medium. Thus, it is sufficient to analyse the behavior of *χ _{i}* to understand the conditions leading to the energy dissipation in the system. Next, we will discuss how

*χ*responses with different coupling strengths.

_{i}In the weak-driving regime *κ* < *κ _{T}*, where

*κ*= (

_{T}*η*

_{1}-

*η*

_{2})/4 is the threshold coupling strength [7], the absorption of the system can be written as

*κ*>>

*κ*, we have the absorption of the system as

_{T}*j*= 1, 2. And ${\chi}_{\pm}=\mp \left({\omega}_{\pm}+i{\alpha}_{2}\right)/\beta $. The subscripts

*r*,

*i*represents the corresponding real and imaginary parts, respectively.

*C*

_{+},

*C*

_{−},

*C*

_{1},

*C*

_{2}are the amplitudes of the Lorentzian curves,

*δ*is the detuning from resonance,

*γ*

_{+},

*γ*

_{-},

*γ*

_{1}, and

*γ*

_{2}are the respective widths. Equation (5) describes a Fano interference and corresponds to the EIT model, while Eq. (6) corresponds a strongly driven regime with a splitting of the excited state corresponding to ATS. Note that Eqs. (5) and (6) have been normalized respect to the forms in [7]. In addition the intermediate-driving regime quantified by

*κ*>

*κ*

_{T}is not concerned in this paper, and more detailed discussions can be found from previous work [7]. For a three-level system, the various parameters introduced in the two above expressions can be calculated from Eq. (4). Conversely, in the proposed system, we use functions

*A*

_{EIT}and

*A*

_{ATS}to fit the obtained absorption curves, adjusting all the parameters in Eqs. (5) and (6). In this paper, these fitting procedures are directly provided by the NonlinearModelFit function in MATHEMATICA. We demonstrate the physical mechanism by fitting the absorption data to the

*A*and

_{EIT}*A*models, from which we can calculate AIC weights.

_{ATS}## 4. Results and discussions

#### 4.1 Plasmonically induced transparency (PIT)

The main challenges for the further development of graphene in the fields of plasmonics is the large momentum mismatch between incoming free-space electromagnetic waves and plasmonic waves in graphene. To end this, several mechanisms enabling the excitation of graphene surface plasmons (GSPs) have been proposed. One possibility consists of patterning the graphene sheet to coplanar configurations such as the unshaped [25] and shaped [37,38] nanoribbons, disks [39] and circles [27]. Another possibility is to modulate the position-dependent conductivity of a continuous graphene sheet, either by using the diffractive gratings in the substrate [40], or through patterned gates to locally modify the conductivity via the electric field effect [41] or the metallic gratings [42,43]. Besides, more options of exciting GSPs include using a periodic corrugation of the graphene surface to form the diffractive gratings [31,33]. By placing the graphene layer on a periodically structured conducting substrate or exciting from one end by a mechanical vibrator, a sinusoidal grating can be formed, which allow the incident light to convert into surface plasmonic waves and subsequently being absorbed by the graphene layer. The reason why the GSPs can be excited by a periodic corrugation can be understood by the fact that the corrugation can provide the GSP momentum with an extra reciprocal lattice vector which is needed to compensate the wavevector mismatch. That is: Re(*β _{SPP}*) =

*β*

_{0}sin

*θ*+

*MG*, where

*β*is the wave number of GSP,

_{SPP}*M*is an integer that represents the excited mode order,

*G*is the reciprocal grating wavevector [31,33]. In this paper, we propose to use two layers of graphene to create a PIT system, as shown in Fig. 1. By sinusoidally pattern the substrate to generate a periodic sinusoidal surface, the graphene sheet placed on it can form periodic gratings that will efficiently couple the incident light into GSPs. However, the planar layer placed under the grating layer with a certain distance cannot be excited directly by the incident field but can coherently couple with the upper layer.

To find out how the proposed PIT system work, electromagnetic simulations of the setup shown in Fig. 1 are carried out, where a normally incident plane wave (*θ* = 0) with electronic field polarized along the *x*-direction is used to excited the GSPs. For comparison, a simulation without the PL is carried out firstly for the following analysis, as shown in Fig. 2(a). Two peaks corresponding to the first two modes occurring at 39.34 THz and 56.35 THz are clearly identified. The first one is the fundamental mode (M = 1), which is dominant with absorption reaching 38.21% and a quality factor (Q) of ∼166. This mode associates with a phase shift of 2*π*, as the *H _{z}* component of magnetic field demonstrates in Fig. 2(b). While the second mode (M = 2) is dominant with absorption reaching 37.84% and Q ∼255, and is associated with a phase shift of 4

*π*, as it demonstrates in Fig. 2(c). However, the phenomenon will be totally different after a flat monolayer graphene is placed under the GL with a distance

*d*to form the coupled system. Figure 2(d) shows the obtained transmission, reflection, and absorption spectra of the proposed PIT system under a normal incidence. One can indeed see four sharp notches from the spectra. These four peaks are corresponding to the resonant frequencies of 32.64 THz (with absorption 33.73% and Q ∼203), 45.05 THz (with absorption 32.62% and Q ∼275), 53.97 THz (with absorption 34.38 and Q ∼286), and 59.72 THz (with absorption 34.94% and Q ∼294), respectively. Note that the observed quality factors are much larger than other graphene based plasmonic devices [40,44]. To get more insight into the physics of the transparency spectra observed, the distributions of magnetic field

*H*components are plotted in Figs. 2(e)-2(h) for the four notches, respectively. According to the spatial distributions of the

_{z}*H*components in

_{z}*x*-

*y*plane, the first and third peaks (count from left to right) show the same phases between the layers. Considering the sinusoidal curvature of the GL, these two modes are called quasi-symmetric mode 1 (QSM1) and quasi-symmetric mode 2 (QSM2), respectively. While the spatial distributions of the

*H*components of the second and fourth peaks show antiphases between the layers, which are called quasi-asymmetric mode 1 (QAM1) and quasi-asymmetric mode 2 (QAM2), respectively. In addition, these figures clearly show the structure of these modes: The

_{z}*H*components of the magnetic field of the modes QSM1 and QAM1 show a 2

_{z}*π*phase shift in a periodicity, thus these two modes are the fundamental modes. While the modes QSM2 and QAM2 show a 4

*π*phase shift in a periodicity, which corresponds to the second-order modes. Strong enhancements of electromagnetic fields is observed for all of these four modes, with absorptions reaching more than 32%. What is interesting for these modes is that for the same order of the QSM and QAM modes, the phases of the fields keep the same as that of the situation with a single GL (as can be seen from the comparison of fundamental modes among Fig. 2(b) and Figs. 2(e) and 2(f), or the second-order modes among Fig. 2(c) and Figs. 2(g) and 2(h)). While for the PL, the phases of the induced fields distribute inversely from each other for the same order of modes. It is indeed the inverse of the phase distribution that define two forbidden transitions at separated resonances while induce an allowed transition window at the transmission spectrum relative to the original situation without the PL, as shown in Figs. 2(a) and 2(d).

Since that the induced transparency window is caused by the strong-coupling of the induced resonance modes, what plays an important role in controlling the coupling strength will serve as a tuning parameter of the transparency window. To find out this, we conducted a parametric study by varying the grating amplitudes *A* from 10 nm to 60 nm with *d* = 20 nm. The calculated absorption spectra (dots-line) are displayed in Fig. 3(a). It is shown in the picture that as the grating amplitude increases, the absorption of the system increases. This indicates that the grating amplitudes are responsible for the absorption of the whole system. Big Amplitude will contribute to the absorption until it reaches about 40%. At the same time, the absorption of QSM1 increases as the increasing of grating amplitude (left peaks in Fig. 3a).

While for the QAM1, the absorption increases firstly and then dramatically decreases (right peaks in Fig. 3(a)). These changes of the absorption ability can be attributed to the variation of the distances between the two layers. Although the minim distance *d* is fixed, the maximum distance (*d* + 2*A*) increases as the amplitude increasing. Note that on the one hand, the increase of the amplitude will generate more efficient coupling between the incident light and the upper graphene layer and further stronger induced coupling (to distinguish, called indirect coupling) between the two graphene layers. On the other hand, the increase of the amplitude will directly lead to decreased coupling (direct coupling) between the two layers. When the amplitude is small (i.e., *A* < 40 nm), the indirect coupling effect increases faster than the decreases of the direct coupling. Thus the coupling gets stronger, leading to the increased coupling of the system. While the amplitude is further increased beyond about 40 nm, the direct coupling effect reduces faster than the increases of the indirect coupling. Therefore the coupling decreases as the amplitude increasing, approaching the situation without the flat graphene layer. Meanwhile, we observed that the two modes in the absorption are not symmetric (Fig. 3(a), upper and lower panels), and they have different absorption dips. This can be attributed to the unequal distribution of the supermodes in the two resonators. However, as we fix the grating amplitude and then increase the distance *d* between the two layers, the coupling strength monotonically decreases, leading to reduced absorption of the resonant modes and further resulting in the asymmetric absorption lineshape, as it is shown in Fig. 3(b). And at the same time, the two modes approach each other. At the stage of sufficiently large distance (e.g., > 200 nm), there are not exchange energies between the two layers and the absorption shows single resonance with the same lineshape as the case of single grating layer.

#### 4.2 Discerning ATS from EIT using AIC

Since that in the theoretical simulations and the experimental observations, EIT and ATS both display a transparency window in the transmission spectrum of coupled plasmonic modes, it is crucial to discern whether a transparency window in the absorption spectrum of the coupled modes system originates from ATS or EIT. Figure 4(a) shows the simulated absorption as a function of the detuning *δ* (green dots/curve) together with the fits to *A _{EIT}* (blue curve) and

*A*(red curve) by using Eqs. (5) and (6). Parameters

_{ATS}*C*

_{+},

*C*

_{−},

*C*

_{1},

*C*

_{2}are in dimensionless units, representing the amplitudes of the absorption curve, while the parameters

*δ*

_{1},

*δ*

_{2}, and

*γ*

_{+},

*γ*

_{−},

*γ*

_{1},

*γ*

_{2}are all in THz, representing detunings and widths of the absorption curve, respectively. As expected, the ATS model fits much better than the EIT model. Though Fig. 4(a) demonstrates how well these EIT and ATS models fit calculated absorption data, an objective criterion is needed to discern the best model or whether the data are inconclusive.

AIC provides a method to identify the most informative model from a set of models based on the Kullback–Leibler (*K*–*L*) divergence between the model and the truth [45]. AIC quantifies the amount of information lost when the model *M _{i}* with

*N*obtained parameters is used to approximating the actual data ${I}_{i}=2\mathrm{log}\left({\widehat{\sigma}}_{i}^{2}\right)+2{K}_{i}$ (

_{i}*i*= EIT or ATS), where

*K*denotes the total number of unknown parameters including ${\widehat{\sigma}}_{i}^{2}$ and ${\widehat{\sigma}}_{i}^{2}={\displaystyle \sum {\widehat{\epsilon}}_{i}^{2}}/2$ is the variance of the likelihood function obtained from the considered model, ${\widehat{\epsilon}}_{i}$ are the estimated residuals for a particular candidate model. We demonstrate the AIC-based testing by fitting the simulated absorption data to quantify the models of

_{i}*A*and

_{ATS}*A*. In our study, the relative likelihood of model

_{EIT}*M*out of the two models is given by Akaike weight

_{i}*w*and

_{EIT}*w*obtained at two different parameter settings. As the amplitude

_{ATS}*A*or the distance

*d*change, ATS dominates both of the situations. The models assigned using AIC to the simulated data agree very well with the requirements to observe EIT or ATS. According to the AIC,

*w*>

_{ATS}*w*, thus the ATS is the best model that could describe very well the observed spectra. That is, the induced transparency window in the spectrum observed from the proposed system is more likely to be caused by ATS, rather than EIT. This conclusion can also be confirmed by the fitting curves shown in Figs. 3(a) and 3(b), where the

_{EIT}*A*model fits the simulated absorption data pretty well for all cases of the parameter settings.

_{ATS}#### 4.3 Electrical tunability, dielectric sensitivity, and resonant stability

The most intriguing property of graphene-based plasmonic devices (as compared with metal-based structures) is their ultrabroad and fast tunability, which can be achieved by chemical doping or electrostatic gating. Specifically, the optical response of the designed PIT systems can be changed by a potentially fast approach of electrical backgating so that they can work at different wavelengths (frequencies) without re-optimizing or reconstructing the physical structure [46]. As a result, the wavelength of transparency window of our proposed PIT system can be tuned dynamically by electrically changing Fermi energy. By employing an electrolytic gate, carrier concentration as high as 4 × 10^{18} m^{−2} in graphene sheet was observed, meaning *E _{f}* = 1.17 eV [47]. Using this method, the Fermi energy level of graphene could be experimentally modified from 0.2 eV to 1.2 eV after applying a high bias voltage [48]. Thus in this paper, we reasonably assume that

*E*can be dynamically tuned from 0.1 eV to 1.0 eV. Simulated spectra shown in Fig. 5(a) confirm the broad tuning range with changes in the Fermi level. With the increasing of the Fermi energy, the width of the transparency window shrinks. It is also clearly shown that the resonant wavelength can be tuned from 20.7 μm to 9.2 μm for the QSM1 and from 15.1 μm to 6.7 μm for the QAM1 when tuning

_{f}*E*from 0.2 eV to 1.0 eV. The physical mechanism behind this resonant shift is that the plasmon excitations correspond to collective oscillations of conduction electrons, electrostatically tunable carrier density in graphene allows for dynamic control over the optical conductivity of the graphene sheet, which will further changes the wave vector of GSPs. Meanwhile, as

_{F}*E*increasing, the width of the plasmon resonance is found to decrease. This can be easily understood by considering the allowed interband transitions of conducting electrons. Near the Dirac point most interband transitions are allowed, leading to wider resonance. At the same time, due to the lower electron density, the optical loss is lower. However, as the

_{F}*E*is increased some of the interband transitions are blocked and therefore, the width of the resonance is lowered. Because of the higher electron density, the optical loss is higher. In Fig. 5(b), we plot the transmission amplitude as a function of

_{F}*E*at a fixed working wavelength 10 μm. As the Fermi energy is dynamically tuned from 0.1 eV to 1.0 eV, four transmission dips are clearly identified from the transmission spectrum. Judging from the near field distribution from our simulation, we found that these four dips correspond to the four lower order modes, as it is marked in the figure. This property tells us that by dynamically tuning the Fermi energy of the graphene, the resonance of a specific mode can be tuned on and off, which is the capability to control and switch nanoscale optical fields

_{F}*in situ*. And on the other hand, the resonant mode can be continuously transformed from one to another. These appealing advantages can be explained by considering the relationship between the effective refractive index and the Fermi energy of graphene. The increase of the Fermi energy will result in the increase of the effective refractive index, which will further lead to the lower order of the resonant modes.

Except for the sensitivity to the doping concentration, the characteristic resonant wavelength is also extremely sensitive to the local dielectric environment, which can be effectively used to modulate the optical response of the resonance. In Fig. 5(c), we show the normal-incidence transmission spectra with different refractive indexes outside of the graphene layers (that is *ε*_{1} = *ε*_{3} = *n*^{2}) while keeping *ε*_{2} = 2.25. Because of the quasi-symmetric distribution of the dielectric in and out of the graphene layers, the transmission dips corresponding to the QSMs are lower than that of the QAMs, meaning that the coupling of the incident light with QAMs are obviously inferior to that of QSMs. Considering the high quality factors of these modes, as described before, these optical properties are useful to the design of refractive index sensor. Figure 5(d) shows the resonant wavelengths of QSM1 and QSM2 as a function of refractive index *n*. When the refractive index *n* is varied from 1.0 to 1.5, the resonant wavelengths of the QSM1 shift from 8860 nm to 9850 nm, while the QSM2 shift from 5025 nm to 6025 nm, respectively. It's necessary to note that according to our simulated results, the relationship between the resonant wavelengths *λ* and *n* can be fitted as a very good linear relation, as shown in Fig. 5(d), which can be expressed as *λ*_{QSM1} = 1986·*n +* 6851, *λ*_{QSM2} = 2004·*n +* 3009, respectively. The sensitivity of refractive index sensor is defined as *dλ*/*dn*, resulting in 1986 nmRIU^{−1} for the QSM1 and 2004 nmRIU^{−1} for the QSM2. This linear relation can be explained by considering the effective dielectric constant *ε _{eff}*. Since that the relationship between the resonant wavelength and the effective dielectric constant can be estimated as $\lambda \propto \sqrt{{\epsilon}_{eff}}$ [25,31,40], while ${\epsilon}_{eff}\propto {n}^{2}$, thus $\lambda \propto n$.

The above properties shows the sensitivities of the proposed PIT system to the changes of the electrical backgating and the dielectric environment, which does not mean that the resonant wavelength is sensitive to all the changes outside. In Fig. 5(e), we show the absorption spectra of the two graphene sheets for various values of incident angle *θ*. Firstly, we can see from the figure that even the incident angle has been changed for a large value, the absorptions of the system still keep stubbornly high. For example, when *θ* = 60 ̊, the absorptions of the two first order modes still remain higher than 30%, this value of the incident angle is much larger than that of the situation without the flat graphene layer at the same absorption level [31]. In addition, one may find that no matter how the incident angle changes, the resonant wavelengths of these two modes remain unchanged. To show this more clearly, we plot the resonant wavelengths of QSM1 (red-dotted line, left radial axis) and QAM1 (green-dotted line, right radial axis) as a function of incident angle *θ* in Fig. 5(f). It is clearly shown in this figure that the resonant wavelengths stay the same even the incident angle has been changed for more than 75 ̊, this incident angle stability is very conducive to ensure the reliability of the proposed PIT system.

## 5. Conclusion

In conclusion, we demonstrate that PIT can be achieved by using two layers of graphene sheet, which can avoid any of the pattern of the graphene. By placing a flat monolayer graphene under a sinusoidally curved graphene layer, two kinds of plasmon modes with field distributed symmetrically (QSM) and asymmetrically (QAM) can be excited, resulting in a transparency window in the spectrum. Meanwhile, we show that the induced transparency can be drastically tuned by changing the coupling strength between the two layers through the varying of the grating amplitude and the interlayer spacing. In addition, both the analytical fitting and the AIC method are employed efficiently to distinguish the transparency window, which is found to be caused by ATS instead of EIT. Our results show that the resonant modes at a fixed wavelength can be actively transformed from one to another by varying the Fermi energy of the graphene sheets without re-optimizing and refabricating the PIT system. Besides, thanks to the existence of the under layer graphene sheet, the resonant wavelengths are insensitive to the incident angle in a very large range. This proposed system demonstrates new scheme to design tunable transparency windows and can be used as either an optical switch, or a plasmonic sensors.

## Funding

National 973 Program of China (2012CB315701); National Natural Science Foundation of China (NSFC) (61505052, 11574079, 61176116, 11074069).

## Acknowledgments

The authors thank Qi Lin and Jian-Ping Liu for their technical assistance and discussions.

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