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General actively tunable near-field plasmon-induced transparency (PIT) systems based on couplings between localized plasmon resonances of graphene nanostructures not only suffer from interantenna separations of smaller than 20 nm, but also lack switchable effect about the transparency window. Here, the performance of an active PIT system based on graphene grating-sheet with near-field coupling distance of more than 100 nm is investigated in mid-infrared. The transparency window in spectrum is analyzed objectively and proved to be more likely stemmed from Aulter-Townes splitting. The proposed system exhibits flexible tunability in slow-light and electro-optical switches, promising for practical active photonic devices.
© 2016 Optical Society of America
1. Introduction
Graphene has shown various interesting properties in mechanical, electric, and optical applications [1]. Interestingly, graphene is also a promising candidate plasmon material in the terahertz to mid-infrared spectral range. Graphene plasmons (GPs) have been extensively inves tigated theoretically and experimentally [2–13 ]. Compared to metal plasmonics, GPs possess higher confinement, longer relative propagating distances and easier electrostatic tunablility. The wavelengths of the GPs are about 102 times smaller than the wavelength in free space [3]. Therefore, the huge momentum mismatch between free space photons and GPs should be compensated in experiments, such as a metallic antenna like a near-field tip [6, 7] or a resonant rod on graphene [12] can be utilized to launch propagating GPs. Apart from propagating GPs, localized GPs and their coupling have also attracted considerable attention in recent years. In particular, plasmon-induced transparency (PIT) [14–17 ], as an analog of electromagnetically induced transparency (EIT) [18, 19] in atomic systems, has been demonstrated in graphene structures [20–23 ]. These active systems exhibit promising applications in integrated photonic circuits, including slow light devices, plasmonic switches and optical sensors.
Nevertheless, there are several drawbacks that are needed to be improved. Firstly, the proposed near-field coupling schemes [20–23 ] rely heavily on the strength of the coupling between constituent nanoantennas, which requires a distance control as small as 20 nm. Such spacings, not only require a precise lithographic control, but also restrict the access to the coupling medium between the antennas [16]. Besides, the transparency windows can only be tuned along spectrum instead of switchable, which blocks their practical applications in active photonic devices. As an alternative, phase coupled PIT schemes have also been proposed [24], where antennas couple with each other mediated by phase differences instead of near-field. However, in these systems, maximum transparency only occurs at certain separations on the order of light wavelength and the transparency windows are spectrally narrow, limiting their applications in compact photonic circuits. On the other hand, strong coupling between localized and propagating surface plasmons can also take place in graphene systems [25], which provides us a platform to investigate PIT in these systems.
Both EIT and Aulter-Townes splitting (ATS) [26–28 ] can lead to transparency windows in spectra. While EIT is the result of Fano resonance among different transition pathways, on the other hand, ATS corresponds to the splitting of energy levels caused by strong field-driven interactions. Clarifying whether a transparency window is due to EIT or ATS is important for understanding the physical process behind and for specific applications. The Akaike Information Criterion (AIC) [27–29 ] has been proved as an objective test to discern EIT from ATS in experimentally obtained transmission spectra of atom systems [27] and coupled microresonators [28]. Detailed analysis on the origin of transparency windows in graphene PIT systems is needed.
In this paper, we achieve PIT effects in mid-infrared based on the strong coupling between localized and propagating plasmonic modes in layered graphene ribbon-grating and continuous sheet systems. The large evanescent decay length of propagating GPs enables the observation of transparency window for the separation of 120 nm at 31.7 THz. In addition, analytical fitting is employed and the AIC method is proved efficiently to distinguish the transparency window, which is found caused by ATS instead of EIT. EIT-like coupling only occurs when the two coupled modes endure quite varied damping rates. The presented system is tested for slowing light, exhibiting large and spectrally tunable group delays at the transparency window. Benefit-ting from the detachment of coupling components, the transparency window can be electrically switched with large on/off ratio. Therefore, this flexibly tunable system promises practical applications in electro-optical switches and integrated photonic circuits.
2. Graphene PIT system
The geometry of the graphene grating-sheet hybrid system is illustrated in Fig. 1. The graphene grating has a period of Λ with ribbon width of W. In this paper, Λ is chosen as 300 nm and the distance between the grating and sheet is defined as D. The Fermi energies of the graphene grating and sheet are denoted by EF 1 and EF 2, respectively. And the carrier mobilities are assigned to be µ 1 = µ 2=10000 cm2/(V s).
Fig. 1 Scheme of the proposed graphene grating/sheet system. (a) 3D view shows a graphene grating consisted of ribbons, beneath which lies a layer of graphene. (b) Cross-section of the structure in x-z plane. The period of graphene grating is Λ, and the width of ribbons is W. A plane wave incidents normally on the grating, with the electric field polarized along x direction. The separation of the two graphene layers is D. Surrounding dielectric constants are ε 1, ε 2 and ε 3, respectively.
The optical response of the designed structures is numerically simulated using COMSOL, which is based on finite element method. The optical conductivity of graphene is related to the Fermi energy EF and carrier mobility µ, through random phase approximation (RPA) [30]. In the local limit, it follows
whereHere, is the intrinsic relaxation time, where vF ≈ c/300 is the Fermi velocity. Temperature is set as 300 K.
To simplify analysis of the mechanism of coupling between graphene layers, ε 1, ε 2, and ε 3 are assumed to be 1, while the separation D is changed.
The dispersion of surface plasmon waves in the graphene sheet satisfies [3]
where k 0 = ω/c is the wave vector in vacuum, and kGP is the wave vector for the plasmons. As ε 1 = ε 2 = ε 3 = 1, the dispersion in the nonretarded regime (kGP ≫ k 0) is simplified asIn Fig. 2(a), the dispersion relation of the real part of kGP (represented by q(ω) here) is shown for EF 2=0.3 eV. For smaller D, i.e. the upper graphene grating is getting closer to the bottom graphene sheet, the propagating GPs in the sheet can be excited by the localized plasmon resonance (LPR) in the ribbon layer once the momentum conservation law is fulfilled as a result of the large wave momenta of near fields. In this way, the two layers are coupled efficiently. Moreover, due to the periodicity of graphene ribbons, only plasmon waves whose wave momenta satisfy
can exist. Integer m denotes the order. The frequencies of the first four supported SPPs modes can be deduced from Fig. 2(a), where the first one (FSP) exists at 31.7 THz (m=1). As shown in Fig. 2(b), the corresponding width of graphene ribbons should be 104 nm for the strongest near-field coupling between the LPR and FSP modes.Fig. 2 (a) Dispersion relation of a suspended graphene sheet. The Fermi energy is 0.3 eV. (b) Extinction spectra for graphene grating of period 300 nm and Fermi energy of 0.3 eV. The widths of ribbons are changed from 50 to 250 nm. The dipolar plasmon resonance arises at 31.7 THz for W of 104 nm, as indicated by the dashed line.
Figure 3(a) displays the transmission spectrum for the designed configuration, where a transparency window is obtained. Detailed field distributions at the two dips and peak are shown in Fig. 3(b). At ν 1 and ν 2, electrical fields are localized at the edges of graphene ribbons. The Ez distributions at ν 12 in the bottom right panel of Fig. 3(b) imply the excitation of the FSP mode. Benefitting from the large decay length of the FSP mode [as depicted in Fig. 3(c)], the distance between graphene ribbon gating and graphene sheet is set as 120 nm, which is much larger than previous designs [20,21].
Fig. 3 (a) Transmission spectrum for the graphene grating-sheet system when the width of the ribbons is 104 nm. Here, D=120 nm. ν 1,ν 2 denote the frequencies at the left and right dip respectively. ν 12 is the frequency at the PIT peak. (b) Simulated field distributions (|Ez|2) for a unit cell at frequency of ν 1(top left), ν 2(bottom left) and ν 12(top right), respectively. A and B mark the planes of graphene sheet and grating, respectively. The bottom right panel shows the electric field distributions(Ez) at ν 12. (c) Normalized distributions of |E| along the vertical line indicated in (b) at ν 12.
3. Analytical fitting and AIC weight
In the coupling between the LPR and FSP modes, the LPR mode is excited by incident light, then electromagnetic energy is transported to the FSP mode. These interactions can be described by the coupled two-oscillator system as
here, xi, ωi and γi are the resonance amplitude, frequency and damping rate of the resonance mode (i = 1 for the LPR mode and i = 2 for the FSP mode), respectively. κ is the coupling coefficient between the two resonance modes, and a 1 is a geometric parameter indicating the coupling strength of the LPR mode x 1 with electromagnetic field E 0 eiωt.The susceptibility χ of the system is proportional to the amplitude x 1 [14] and can be described as
Exact expression for transmission in the two-resonantor model has been described in a developed radiating model [31]. However, for the graphene plasmon system, a predigestion can be employed. Due to the large wave vector mismatch between GPs and far-field, the scattering of plasmons back into photons is very weak. As a result, the extinction cross section is dominated by dissipated energy in the system [5]. As the energy dissipation is proportional to the imaginary part of susceptibility χi, the transmission T can be described as T(ω) = 1 − gχi(ω) [22], where g is a geometric parameter related to the coupling strength between the resonant mode and the incident electromagnetic field.
Figure 4 displays the transmission spectra at different distances D by numerical calculations and analytical fittings, which reveal excellent agreements. One can find that the transparency windows gradually appear and broaden as D decreases. The fitted parameters are shown in the top panel of Fig. 5(a). With the decrease of D, the coupling strength κ experiences rapid increase due to the stronger near-field coupling.
Fig. 4 Numerical (blue dots) and analytically fitted (red curves) transmission spectra for different separation of D.
Fig. 5 (a) Upper panel: The parameters κ, γ 1 and γ 2 used in fitting. Lower panel: AIC weights of the EIT (blue curve) and ATS models (green curve). (b) The same as (a), except that µ 2=30000 cm2/(V s) here.
Then, the AIC method is tested for the origin of the transparency windows at different coupling strengths. Given a parametric model, there is a unique set of parameters that minimize the Kullback-Leibler (K-L) distance between the model and the truth, where K-L distance defines the information lost when the model is used to approximate the truth. The AIC method defines a criterion to quantify information lost of the model gi and is explained as Ii = −2log(Li)+2ki, where Li is the the maximum likelihood for the candidate model and ki the number of estimable parameters. Then the relative likelihood of the model gi is expressed by the Akaike weight as , where n is the number of fitted models. In the case of least squares case, Akaikes information is , where is the estimated residuals from the fitted model and N is the number of data points. Here, the involved two models are explained as [28]
where Ωi, Γi and Ci (i = 1,2) are free parameters and represent resonant frequency, damping and strength, respectively.Following the standard method, the AIC weights of the two models are calculated and shown in the lower panel of Fig. 5(a). Clearly, the transmission spectra are ATS-like within all fitted coupling strengths. This is related to the similar damping rates of the the LPR and FSP modes. EIT-like spectra are expected for smaller damping of the FSP mode. This is confirmed by our simulations by choosing µ 2=30000 cm2/(V s). Corresponding fitting parameters and AIC weights are displayed in Fig. 5(b). In this case, γ 2 gets several times smaller than γ 1, as a result the spectra are EIT-like in the weak coupling region. This is consistent with the case in coupled WGM microcavities [28]. It can be further deduced that EIT is the dominant origins of transparency windows for all coupling strengths when γ 2 is far smaller than γ 1.
4. Electrical tunability
It is well known that graphene plasmons can be efficiently tuned by external voltages or chemical doping. As a result, the frequency of transparency window of our proposed PIT system can be tuned by electrically changed Fermi energy. Figure 6 displays the shift of the transparency window as Fermi energies of graphene components vary. One can see that the central frequencies of the transparency windows are kept consistent with the ones of the LPR/FSP modes, and the transparency windows broaden as the increasing of the Fermi energy of graphene. The maintenance between the agreement of the LPR and FSP modes with varied Fermi energies can be roughly understood below.
Fig. 6 Tunability of the transparency window with the change of Fermi energies EF of graphene ribbons and sheet (EF 1 = EF 2 = EF). D is 50 nm here. The white dashed line indicates the frequency of the LPR/FSP mode.
Ignoring the interactions between ribbons, the LPR resonance in suspended graphene ribbons can be described by a Fabry-Perot model [32–34 ]:
where integer n denotes the resonance order and Δφ is the phase shift of plasmon waves on ribbon edges, which is found to be π/4 for free-standing ribbons. For the identical Fermi energy, the graphene ribbons and graphene sheet shares the same wave vector. Comparing Eq. 4 with Eq. 9 and assigning values of m = 1,n = 1, the resonance condition at the same frequency is expressed as W = 3Λ/8=125 nm. Thus the agreement between the LPR and FSP resonance can be maintained and is independent on EF. This calculated width is larger than 104 nm, resulting from the redshift of spectrum when the interactions among ribbons are considered.5. Applications in slow light and electrical-optical switch
Next, to illustrate the applications of our proposed graphene PIT system, the dielectric environment is adopted as ε 1 = 1, ε 2 = 2 and ε 3 = 4. By redesigning the width of graphene ribbons to 183 nm, transparency windows with different Fermi energies of graphene are obtained, as shown in Fig. 7(a). The group delay times for the corresponding structures are displayed in Fig. 7(b). Positive group delay emerges at the transparency window, larger than 0.2 ps near 28.6 THz when EF is 0.7 eV. Besides, the large group delays can be actively tuned by different Fermi energies.
Fig. 7 (a) Transmission spectra for graphene grating-sheet system at different Fermi energies EF(EF 1 = EF 2 = EF). Width of graphene ribbons is 183 nm. Distance between grating and sheet is 150 nm. (b) Corresponding group delays with increased Fermi energies.
Moreover, the transparency window in Fig. 7(a) arises when the graphene ribbons and sheet are tuned at identical Fermi energy. Therefore, the divergence of Fermi energies can result in degeneration and even absence of the transparency window, revealing promising applications in active electro-optical switch. As demonstrated in Fig. 8(a), when EF 1 = EF 2=0.5 eV, transparency window emerges, thus light can pass through and have a transmission of 84% at 24.2 THz, which is ON state. While, when EF 2 is tuned to 0.3 eV, the transparency window disappears and the corresponding transmission is at the Lorentz dip with the lowest value of 11%, which is OFF state. Moreover, as shown in Fig. 8(b) and 8(c), the switch of ON/OFF state can be tuned by changing the Fermi energies of graphene ribbons, and the ON/OFF ratio can exceed 95% at EF 1=0.7 eV.
Fig. 8 The performance in active electric-optical switches. EF 1 are (a) 0.5 eV, (b) 0.6 eV and (c) 0.7 eV, respectively. At ON states (black solid curves), EF 2 have the identical values as EF 1 in all three panels. While at OFF states (red dashed curves, EF 2 are (a) 0.3 eV, (b) 0.35 eV and (c) 0.4 eV, respectively. The distance D is 100 nm.
6. Conclusions
In conclusion, the physical mechanisms and applications of graphene grating-sheet systems are demonstrated. Profited by the larger decay distance of the FSP mode, our PIT system relaxes the rigorous near-field coupling condition compared to previous designs. Specifically, the efficient coupling can be fulfilled at large separation of 120 nm. The response of the proposed system can be well fitted by two-oscillator model. Analysis employing AIC weight reveals that the transparency window is more like caused by ATS instead of EIT for low carrier mobility of graphene. Furthermore, EIT-like spectra are obtained for higher carrier mobility of graphene sheet. Then applications of the hybrid system in active slow light device and electro-optical switch are also demonstrated. Large group delay emerges at the transparency window and can be electrically tuned when the Fermi energies of graphene system are changed. The unique requirement on Fermi energies of the PIT system can be utilized in active optical shutter. The ON/OFF state of the switch can be electrically controlled by changing the difference of Fermi energies between the two graphene components, and the ON/OFF ratio can exceed 95%. Moreover the work frequency of the switch can be actively tuned by changing the Fermi energy of the graphene ribbons. The proposed configuration is demonstrated as an easy-fabricated and flexibly tunable PIT system, opening up new avenues for practical applications in photonic circuits, such as sensors, slow light devices and electro-optical switches.
Acknowledgments
This work was financially supported by the National Basic Research Program of China (2013CB328702), Program for Changjiang Scholars and Innovative Research Team in University (IRT0149), the National Natural Science Foundation of China (11374006) and the 111 Project (B07013).
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