1. Introduction

Plasmon induced transparency (PIT) in metallic metamaterials [1, 2], in analogy to the electromagnetic induced transparency (EIT) effect in cold atomic systems [3,4], has focused enormous attentions. Due to the inherent radiative-dark mode coupling, PIT underlies very useful applications, such as slow light [5], plasmon sensing [6, 7] and optical switching [8]. However, the active tuning of the plasmon resonance in noble metal is difficult except for the accurately control the geometry of structures, which limits the performance and repeatability of designated PIT samples. Fortunately, these drawbacks can be overcome by using graphene instead of traditional noble metal as plasmon materials. Graphene plasmons (GPs) have become of extreme importance in plasmonics due to their extremely strong local field enhancement, relatively long propagation length and controllable optical properties through electrostatic gating or chemical doping [9–11]. These unique properties open a route to numerous practical applications, such as plasmonics waveguides [12, 13], optical antennas [14] and optical near-field switches [15]. Nowadays, GPs are theoretically and experimentally investigated in many different graphene structures. Specifically, scattering-type scanning near-field optical microscope (s-SNOM) is utilized to generate and mapping the GPs in real-space [16, 17]. By applying different bias voltages between adjacent ribbons, graphene heterogeneous structures can be conveniently realized [18]. So it is naturally for us to consider using graphene structures to realize PIT effect for the following reasons. Firstly, the tunability of graphene plasmons offers us a way to control the PIT effect, and the poor performance of designed PIT devices result from the inaccuracy of the geometry parameters can be amended. Secondly, the tuning of PIT effect gives us an active way to control coherent phenomena in plasmonics, which can be learned for further coherent effect control in quantum optics [19].

In this paper, tunable PIT effect is proposed in heterogeneous graphene ribbon pairs. In our scheme, the heterogeneous graphene structure is much simpler than traditional structures, where only one gap distance needs to be accurately controlled. In contrast, in a typical dolmen structure for PIT [1, 8], at least three gap distances must be accurately controlled. As is well-known, the PIT effect comes from the near-field coupling between dark and bright modes, and the reduction of need to be accurately controlled parameters is important. Moreover, the PIT effect are physically explained as the coherent coupling between dipolar and quadrupole modes in graphene ribbons, and how the PIT effect is affected by dark and bright modes is discussed independently. At last, we show the PIT effect dependence on the gap distance between these ribbons. Our work not only explained the PIT effect in graphene ribbons, but also proposed a kind of way to actively control the coherent phenomena in plasmonics.

2. PIT in graphene ribbon pairs

Fig. 1(a) illustrates the scheme employed in our design. Two graphene ribbons with a gap distance of 10 nm are placed on a substrate. The width of each ribbon is 100 nm. Different bias voltages are applicable to realize unequal Fermi energies for these two ribbons. An x-polarized normally incident plane wave is utilized to excite the GPs. As a result, different plasmon modes can be generated in the ribbons. Here, $E_F^l$ is set as 0.2 eV, and $E_F^r$ is set as 0.41 eV, where the $E_F^l$ and $E_F^r$ denote the Fermi energies of the left and right ribbon, respectively. The in-plane conductivity of the graphene is computed within the local-random phase approximation (LRPA) [20–22] with an intrinsic relaxation time $\tau =\mu E_F/e{v}_F^2$, and a measured DC mobility μ = 10000 cm²/Vs, v_F ≈ c/300 indicate the Fermi velocity in graphene [9]. Commercial software COMSOL Multiphysics based on FEM method is adopted to solve the Maxwell equations. It is worthy to point out that the extinction of nano-structured graphene is mostly come from absorption instead of scattering, because the size of the graphene structures is far away smaller than the incident wavelength. For example, our proposed structure is about 200 nm, while the wavelength of the incident light is in the mid-infrared range (around 7.7μm). Therefore, the transmission of our structures can be equally described by the extinction cross section per unit length. The extinction cross section of the structure with and without considering the left ribbon are illustrated in Fig. 1(b). One can see that a transparency window appears at about 39 THz with the left ribbon. In contrast, a extinction peak appears in the same frequency with the right ribbon only. From the near-field plot in Fig. 1(a), most of the electromagnetic fields lay in the left ribbon near the extinction valley, which makes it transparency for the incident wave.

 

Fig. 1 (a) Schematic of two graphene ribbons of the width 100 nm lay on the substrate and an x-polarized plane wave normally irradiates. The gap distance between the ribbons is d=10 nm. The Fermi energies of the left and right ribbon are $E_F^l=0.2\hspace{0.17em}\text{eV}$ and $E_F^r=0.41\hspace{0.17em}\text{eV}$, respectively. (b) The solid (dash) line indicates the extinction cross section of the ribbon pair with (without) the left ribbon.

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Next, let us to turn to explain the physical mechanism of PIT in the heterogeneous graphene ribbons. The PIT effect can be explained as Fano interference in the system [7]. As is well-known that Fano interference usually comes from coherent coupling between a bright mode and a dark mode. Though the dark mode couples weakly to the plane wave, a localized two-dimensional dipole can excite the subradiant mode as well [23, 24]. In the paper, the nonradiative enhancement factor Γ_nr/Γ₀ is used to probe the property of the subradiant mode, where Γ₀ and Γ_nr are isolated radiative decay rate and the non-radiative decay rate, respectively [23]. The extinction spectrum and non-radiative enhancement factor of a single graphene ribbon with the Fermi energy 0.2 eV, and the extinction spectrum of a single ribbon with the Fermi energy 0.41 eV are shown in Fig. 2(a). Only two lowest order modes are analyzed in the paper. When the ribbon is excited by a localized dipole source, both mode A at 27 THz and mode B at 38.8 THz are excited, while only the mode A (at 27 THz for E_F=0.2 eV or 38.8 THz for E_F=0.41 eV) could be excited by a normally incident plane wave. From the definition of bright and dark modes, one can know that the mode A is a bright mode, it can strongly interact with a plane wave. In contrast, the mode B is a dark mode, which interacts more strongly with a dipole than a plane wave. The electric near fields and charge distributions of modes A and B are shown in Fig. 2(b), the figure certifies in further the properties of these modes. Specifically, the mode A is a dipolar mode, while the mode B is a quadrupole mode. As a result, in Fig. 1, one can understand that the left graphene ribbon with $E_F^l=0.2\hspace{0.17em}\text{eV}$ is served as the dark mode while the right ribbon with $E_F^r=0.41\hspace{0.17em}\text{eV}$ served as the bright mode at 38.8 THz, and the coupling between these modes results in the transparency window.

 

Fig. 2 (a) The thick solid line indicates the nonradiative enhancement of graphene ribbon with a length of 100 nm, and the thin solid line indicates the extinction cross section of the same ribbon with the Fermi energy of 0.2 eV. Meanwhile, the dash line means the extinction length of 100 nm ribbon with the Fermi energy of 0.41 eV. (b) The electric near field distribution of modes A and B, and these two modes are bright and dark modes in this paper.

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3. PIT effect dependence on dark and bright modes

Naturally, the Fano dip usually appears near the frequency of the dark mode, so the resonance frequency of the dark mode is of the most important parameter to the position of the transparency window. Next let us investigate how the dark mode affects the PIT effects. Using $E_F^l$ as the variable parameter, the photon frequency and $E_F^l$ resolved extinction spectra are shown in Fig. 3(a). The figure shows two extinction peaks near the transparency position. When $E_F^l$ is far away from 0.2 eV, the extinction peak approaches 38.8 THz, which is the resonance frequency of the bright mode. Fig. 3(b) shows the extinction spectra with three different $E_F^l$ near 0.2 eV. One can see clearly that the transparency position blue-shifts as the $E_F^l$ increasing.

 

Fig. 3 The PIT extinction dependent on the dark mode. (a) The photon frequency and $E_F^l$ resolved extinction length of the ribbon pair. (b) The extinction spectra for Fermi energy $E_F^l=0.196\hspace{0.17em}\text{eV}$, 0.2 eV and 0.204 eV, respectively. S₁ and S₃ are labeled as the first and the second peak, while S₂ labeled as the valley of the extinction spectra. (c) Near-field and charge distribution of modes S₁, S₂ and S₃.

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This is reasonable due to the position of dark mode blue-shifts. Though the dark mode does not interaction with the plane wave directly, it contributes a lot to the extinction length of the ribbon pair through the near field coupling with the bright mode. The electric near-fields E_y and electric charge distributions are presented in Fig. 3(c). One can see that the left ribbon possesses quadrupole momentum (zero net momentum), while the right ribbon possesses dipole momentum. The extinction peaks S₁ and S₃ are the coupled modes of the graphene pair, the electromagnetic energy lays in both ribbons. The extinction valley S₂ means that most of the energy stored in the quadrupole mode, which makes it transparency for the incident light wave.

To further explore the dependence of the extinction valley and resonant frequency on the bright mode, we set $E_F^l=0.2\hspace{0.17em}\text{eV}$ unchanged, which means that the resonant frequency of the quadrupole mode remains unchanged, the evolution of extinction spectra as the changing of $E_F^r$ or the width of the right ribbon l_r is shown in Fig. 4. In Fig. 4(a), different $E_F^r$ values with l_r=100 nm are used to obtain the dependence of PIT effect on the dipolar plasmon resonance. One can know clearly that as the dipolar mode gets through the resonant frequency of the quadrupole mode (38.8 THz), the extinction valley appears near the frequency position of the dark mode. If the dipolar mode peak lays far away from the quadrupole mode, large extinction length is obtained and the PIT effect cannot be achieved. Moreover, in order to overlap the spectrum of dipolar and quadrupole resonances, increasing the Fermi energy and decreasing the width are two optional methods to make the frequency of the dipolar mode blue-shift [12]. If the two ribbons are homogenous, which means that both ribbons have the same Fermi energy with E_F=0.2 eV, the dipolar mode at 38.8 THz can also be achieved by setting l_r=41 nm instead. In this case, the dependence of extinction spectrum on ribbon width is shown in Fig. 4(b), which performs similarly to the tuning of Fermi energy. Thus, tuning the bias voltages can take place of traditional geometry size tuning in PIT effect. This will break up the unrepeatable drawback of traditional geometry control.

 

Fig. 4 The PIT extinction dependent on the bright mode. (a) The dependence of extinction length on the dipolar plasmon resonance with different $E_F^r$. The width of the right ribbon is kept as 100 nm. (b) Different width of the right ribbon are used to obtain the PIT effect. In the case, $E_F^l$ and $E_F^r$ are both set as 0.2 eV.

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4. PIT effect dependence on the gap distance

Due to that the coupling of modes comes from spatial field overlap, the distance between the ribbons must be a critical parameter to determine the coupling strength. Attribute to large field localization of the GPs, the spatial prolongation is usually sub-200nm, even though the wavelength in vacuum may be as long as 10 μm [11]. Different distances range from 10 to 60 nm are adopted to obtain the coupling effect of the system. As the distance decreasing, the extinction dips become deeper and wider as the result of the generation of stronger quadrupole mode. In Fig. 5(a), $E_F^l=0.2\hspace{0.17em}\text{eV}$ and $E_F^r=0.41\hspace{0.17em}\text{eV}$ are applied, respectively, the width of the ribbons both are 100 nm. The extinction dip disappears when the distance is larger than 60 nm. In Fig. 5(b), $E_F^l=E_F^r=0.2\hspace{0.17em}\text{eV}$ are applied and l^r=41 nm is adopted. The extinction dip disappears when the distance is larger than 50 nm.

 

Fig. 5 The effect of the gap distance for PIT effect. The dependence of extinction length on the gap distance in the cases (a) $E_F^l=0.41\hspace{0.17em}\text{eV}$ with l_r=100 nm and (b) $E_F^l=0.2\hspace{0.17em}\text{eV}$ with l_r=41 nm. The dash line indicate the Fano dip.

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5. PIT effect dependence on the substrate

In fact, the calculations above are based on self-standing graphene pairs, while the graphene for a real experiment usually is placed on substrates. In theory, the effect from the dielectric permittivity of substrate ε can be approximately described by homogeneous environment with (ε + 1)/2 [13]. As a result, the substrate just changes the resonance frequency of graphene structures into a new frequency. The extinction lengths with ε=1, 2 and 4 are shown in Fig. 6(a), respectively. In these calculations, $E_F^l=0.41\hspace{0.17em}\text{eV}$ while $E_F^r=0.2\hspace{0.17em}\text{eV}$, 0.191 eV and 0.184 eV to make the resonance of quadrupole mode coincides with the resonance of dipolar mode. Different values of $E_F^r$ come from the different environmental sensitivity for dipolar and quadrupole modes. The frequency of the mode A and corresponding $E_F^r$ for resonance coincidence are shown in Fig. 6(b). As the ε increasing, the dipolar resonance blue-shifts and the $E_F^r$ has to decrease to realize the PIT effect.

 

Fig. 6 The substrate effect for PIT effect. (a) The dependence of relative permittivity ε on the substrate. The black, red and blue solid lines (dash lines) indicate the extinction with (without) the dark mode for ε=1, 2 and 4, respectively. (b) The black line marked as squares is the dipole plasmon resonance frequency as a function of ε. The blue line marked as triangles is the corresponding $E_F^r$ to make the quadrupole resonance coincide with the dipole plasmon resonance.

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6. Conclusions

In this paper, we have proposed a kind of very simple plasmon induced transparency systems in mid-infrared range. The transparency effect is understood by coherent coupling of dipolar and quadrupole mode of graphene ribbons. Both tuning the bias voltage and the width of ribbon are considered in realizing the mode overlapping. The plasmon induced transparency is very sensitive to resonance frequency and the gap distance, which has great potential in nano-ruler or spectroscopic applications. More important, the PIT effect not only can be realized by the coupling between a dipolar mode and a quadrupole mode, but also can be obtained by the coupling between a dipolar mode and a more higher-order mode, such as an octupole mode. In further, the coherent phenomena, not only PIT effect, can be realized in graphene structures.