Active Multiple Plasmon-Induced Transparency with Graphene Sheets Resonators in Mid-Infrared Frequencies

A multiple plasmon-induced transparency (PIT) device operated in the mid-infrared region has been proposed. The designed model is comprised of one graphene ribbon as main waveguide and two narrow graphene sheets resonators. The phase coupling between two graphene resonators has been investigated. The multimode PIT resonances have been found in both cases and can be dynamically tuned via varying the chemical potential of graphene resonators without optimizing its geometric parameters. In addition, this structure can getmultiple PIT effect by equipping extra two sheets on the symmetric positions of graphenewaveguide. The simulation results based on finite element method (FEM) are in good agreement with the resonance theory. This work may pave new way for graphene-based thermal plasmonic devices applications.


Introduction
Surface plasmon polaritons (SPPs), coupled modes of plasmons and photons, propagate along the metal-dielectric interface [1,2].SPPs provide an intriguing approach to localize and guide light in subwavelength metallic structures.In recent years, a great variety of efforts have been devoted to applications based on SPPs, which promotes the development for high density integration of photonic circuits.Electromagnetically induced transparency (EIT) is a sharp transparency window within a broad absorption spectrum in the medium, which is due to the coherent interaction between atomic upper levels and applied optical field [3,4].Recently, a novel analogy phenomenon to EIT, known as plasmonically induced transparency (PIT), has been manifested in various systems based on metamaterial structures [5][6][7][8][9][10][11][12].The PIT effect has been applied in various applications such as nonlinear optical processes, ultrafast switching, slow light, and signal processing [13,14].However, the most of PIT effects are based on MIM (metal-insulator-metal) structure, and the adjustment of PIT effects must rely on changing the geometrical parameter of structures.Recently, multimode PIT effects have been achieved by dual coupled grapheneintegrated ring resonators [15,16].The MIM was replaced by graphene structure to achieve PIT effects without modifying parameter of structure.The dynamically tunable PIT planar hybrid metamaterials have been proposed, in which the VO 2 stripes are filled in the grooves of the device surface.The simulation results indicate that the PIT effects can be achieved by changing the temperature of VO 2 stripes [17].To compare with [18], the structure of our proposal is simpler and more convenient in modulating PIT effects.
In recent years, graphene has aroused extensive attention for its unique and fantastic properties in optics and electronics [19].A monolayer graphene has been widely researched as a promising platform for plasmonics for well supporting SPPs on graphene [20,21].Most importantly, the surface conductivity of graphene could be dynamically tuned by chemical potential via gate voltage, electric field, magnetic field, and chemical doping [22], which increasingly promotes the development of active plasmonic devices including polarizers [23], high-speed optical modulators [24], absorbers [25], and transformation optical devices [26].Zeng et al. have Figure 1: Schematic of plasmonic structure consists of two graphene ribbon and two graphene sheets.(a) Three-dimensional schematic of the plasmonic device.(b) Side view of the device configuration. = 200 nm is the length of two graphene sheets;  = 50 nm is the coupling distance between graphene waveguide and two graphene sheets;  = 300 nm is the separation between two graphene sheets;  G0 ,  G1 , and  G2 , respectively, represent the gate voltages on graphene waveguide and two graphene sheets.studied a graphene-integrated Fabry-Perot (F-P) microcavity to efficiently modulate plasmon-induced transparency effect [27].Shi et al. numerically demonstrated a plasmonic analog to EIT based on graphene nanostructures [28].Therefore, the monolayer graphene is a promising platform to design a dynamically wavelength tunable PIT planar device.
In this paper, we have proposed and investigated a PIT device operating in the mid-infrared region.The designed model consisted of one graphene ribbon as main waveguide and two narrow graphene sheets as resonators.Firstly, a graphene waveguide side-coupled with single graphene resonator has been studied as a filter.Next, the phase coupling between two graphene resonators side-coupled to the main graphene waveguide has been investigated.In order to modulate multipeak PIT, four sheets were inserted into the symmetric position of graphene waveguide.The multimode and multipeak PIT resonances can be actively controlled via changing the chemical potential (Fermi energy) of graphene.Our designs have been well verified by numerical simulation based on finite element method (FEM).Hence, the proposed structures may have great potential in ultra-compact graphene thermal plasmonic devices in the infrared range.

Method
The structure of proposed device is schematically depicted in Figure 1.It consists of one graphene ribbon as main waveguide and two uniform graphene sheets as resonators.For experimental consideration, the designed graphene system is assumed in the SiO 2 medium.It should be noted that the SiO 2 medium is replaced with air in simulation for simplicity.The Kubo formula has governed the optical conductivity of graphene including the interband and intraband transition contributions [29] In our simulation, the employed incident light is in the mid-infrared region where the intraband transition contribution dominates in monolayer graphene [30].Under this condition, the optical conductivity is given by Here,  is the electron charge and the carrier relaxation time  =   /(V 2  ) relates to the carrier mobility , chemical potential   , and Fermi velocity V  = 10 6 m/s in graphene.The equivalent permittivity of graphene is given by the equation [26] where  0 = 2/ is wave number in vacuum and  0 ≈ 377 Ω is the intrinsic impedance of air.The carrier mobility is reasonably chosen to be  = 20000 cm 2 V −1 s −1 from experiment results [25,31].The thickness of graphene is assumed as Δ = 0.5 nm in order to take the numerical simulation, but Δ is not the real thickness of the graphene [19,21].TM-(Transverse Magnetic-) polarized SPPs on graphene are only in consideration for the investigation.The dispersion relation of this TM SPP surface wave follows the equation where  SPP is the propagation constant of graphene SPPs in air (  = 1).

Result and Discussion
At first, we focus on one single graphene resonator sidecoupled to graphene waveguide as a filter.As shown in Figure 1, there are two graphene sheet resonators whereas we set the right resonator inactive by fixing its   at 0.05 eV.Under this condition SPPs will not be supported on graphene because the interband transition of electrons occurs at this Fermi energy [11].The Fermi level of graphene waveguide is fixed at 0.4 eV.When SPPs on graphene meets the resonant condition, the graphene resonator will influence the transmission of main waveguide by forming standing wave.The resonance wavelength is determined by the width and Fermi level of the graphene resonator, which can be deduced from quasi-static analysis and Drude-like model [19,32].It follows the following equation: Here,  is the velocity of light in vacuum and  parameter is introduced for a freestanding graphene sheet ( = 1/  = 1) [33].The fitting dimensionless parameter  can be derived from numerical simulation [19,31,34] and  = 200 nm is the width of graphene sheet.The resonance condition is given by 2Re( SPP ) + B = 2 [34].Here,  is an arbitrary positive integer, resonance mode number, and B is an additional phase shift because of the edge ends of the graphene sheets.
Actually, the influence of the edge ends is only effective when the structure is in extremely small size, so we can neglect the influence in most cases [33,35].Besides, another equation for propagation constant of SPPs on monolayer graphene has been derived by combining the Drude-like model with the dispersion relation [36]: By using the FEM, the transmission spectra have been calculated for different Fermi level   of left graphene resonator ranging from 3.6 m to 5 m, shown in Figure 3(a).The above discussion indicates that SPPs are better confined when Fermi level is lower.So the Fermi level is selected to be 0.42 eV, 0.43 eV, 0.44 eV, 0.45 eV, and 0.46 eV, respectively.The corresponding transmission spectra are denoted by blue, pink, green, purple, and red lines, respectively.As the Fermi level   decreases, an obvious red shift of the transmission dip can be found.Obviously, the larger Fermi levels correspond to the higher transmission dips at the same resonance mode, shown in Figure 3(a).It is because that the loss of propagation mode determined by the imaginary part of  SPP is inversely proportional to the Fermi level.As shown in Figures 3(b)-3(d), when SPPs on graphene meets the resonant condition, the energy can be confined in graphene sheet by forming a standing wave mode, which leads transmission dips.According to the above analysis, this structure can achieve filter effects by controlling Fermi level without optimizing geometric parameters.
Next, the influence of interval  has been investigated.The transmission spectra have been calculated for different values of  (45 nm, 50 nm, 55 nm, 60 nm, and 65 nm), shown in Figure 4.The transmission spectra of  selected to be 45, 50, 55, 60, and 65 nm are represented by blue, pink, green, purple, and red lines, respectively.Obviously, as the interval  increases, the coupling process becomes weaker.In order to investigate the coupling between main waveguide and graphene sheet concretely, the coupled mode theory is introduced to interpret the simulation results.By using coupled mode theory, the transmission can be expressed as Here,  and  0 are the incident frequency and the resonance frequency, respectively.1/  is the decay rate resulting from the intrinsic loss of graphene sheet resonator, and 1/  is the decay rate due to the power escaping from the graphene waveguide.With the growing of interval , the decay rate 1/  decreases while the decay rate 1/  remains the same.In this way, the energy confined in graphene resonator will become less for an increasing interval , which leads to a growing value of the transmission dip.At the same time, the spectra width Δ ≈ 4(1/  + 1/  )/ 2 0 is proportional to the decay rate 1/  .That explains that the spectrum width becomes narrower with a growing interval  at one resonance mode.This structure can achieve best filter effect when interval  is fixed at 50 nm.
In this section, the discussion focuses on the phase coupling between two graphene sheet resonators.To avoid the direct interference between two graphene sheets, the separation  has been chosen to be large enough.The other geometric parameters are selected as follows:  = 300 nm and  = 50nm.We know that the resonance wavelength can be tuned via gate voltage on graphene sheets through the resonant condition.The difference of resonant wavelength between two graphene resonators is defined as   =  1 −  2 ( = 1, 2, 3 for each mode).The   of left graphene sheet is fixed at 0.42 eV, which presents three resonance dips (4.655 m, 4.165 m, 3.8 m) corresponding to first-order, second-order, and third-order resonance modes depicted in Figure 3(a).The gradually increasing   leads to a broader gap between two resonance dips for three resonance modes as   of the right sheet grows from 0.425 eV to 0.44 eV.In Figures 5(a 7 shows the transmission spectra in the mid-infrared region of two peaks PIT based on the four graphene sheets structure.The upper and lower picture of Figure 7 show the magnetic distributions of the second-order mode under  0 = 4.105 m, 4.12 m, and 4.14 m, which correspond to the standing wave modes originating from the splitting of the second-order resonance.In addition, Figure 8 shows the transmission spectra of three peaks PIT based on the four graphene sheets structure.The upper and lower picture of Figure 8 show the magnetic distributions of the second-order mode under  0 = 4.05 m, 4.1 m, and 4.12 m.Due to the great difference of resonant wavelengths in first-order mode from Figure 3(a), the first-order mode of multipeak PIT has disorderly phenomenon because of the interference between two random resonance wavelengths.

Conclusion
In this paper, we have numerically investigated a multimode plasmonically induced transparency (PIT) device operating in the mid-infrared region.The proposed model consisted of one graphene ribbon as main waveguide and two graphene sheets as resonators.In phase coupling systems, the simulation results evidently demonstrate the multimode PIT phenomenon.On the other hand, the multipeak PIT spectral responses emerge in four graphene sheets system.For phase coupling, the off-to-on PIT feature could be effectively achieved by varying the Fermi level of one graphene sheet resonator.Besides, by turning off one graphene resonator, this structure can achieve filter effects.The filter wavelength can be adjusted via varying the Fermi energy of graphene

Figure 2 :
Figure 2: (a) Real part of effective refractive index for graphene SPPs as a function of incident wavelength and Fermi level.(b) Dependence of propagation length  SPP on incident wavelength and the Fermi level.

Figure 3 :
Figure 3: The graphene system with the right graphene sheet turned off.(a) The transmission spectra correspond to different Fermi level   of left graphene sheet,   = 0.42 eV, 0.43 eV, 0.44 eV, 0.45 eV, and 0.46 eV denoted by blue, pink, green, purple, and red lines, respectively.The magnetic distributions |  | correspond to different resonance wavelengths when the Fermi level has been fixed at 0.42 eV.(b)   = 3.8 m.(c)   = 4.17 m.(d)   = 4.66 m.

Figure 4 :Figure 5 :
Figure 4: The graphene system with the right graphene sheet turned off.The transmission spectra correspond to different coupling interval  of graphene sheets;  = 45 nm, 50 nm, 55 nm, 60 nm, and 65 nm are expressed by blue, pink, green, purple, and red lines, respectively.

Figure 7 :
Figure 7: The transmission spectra of two-peak PIT based on the four graphene sheets structure.Magnetic field distributions of the 2nd mode under  0 = 4.105 m, 4.12 m, and 4.14 m, respectively.

Figure 8 :
Figure 8: The transmission spectra of three-peak PIT based on the four graphene sheets structure.Magnetic distributions of the 2nd mode under  0 = 4.05 m, 4.1 m, and 4.12 m, respectively.