Introduction

Surface plasmon polaritons (SPPs) [1], the localized surface electromagnetic waves, propagate along the metal-dielectric interface. Because they can overcome the traditional diffraction limit and tame the electromagnetic fields into desired spatial patterns, various metallic plasmonic devices have been comprehensively demonstrated in a variety of systems, such as the waveguides [2], all-optical switchers [3], and filters [4]. Despite these extensive applications, the functionality of some metal-based plasmonic devices has been constrained due to the difficulty in controlling permittivity functions of metals and the existence of material losses. For example, the quality of the plasmon resonance is degraded and the propagation length of SPP waves is limited. Because of these drawbacks, graphene [5–7], a single layer of carbon atoms gathered in a honeycomb lattice, has recently drawn a great deal of interest due to its advantageous optical properties including extreme confinement, low losses, and dynamic tunability [8]. Especially, the gate-voltage dependent feature [9, 10] that the chemical potential of graphene can be controlled in real time by external gate voltages, which has been confirmed in experiment, is opening up an opportunity for electrically controlled plasmonic devices. Therefore, it is easy to tailor the SPP waves by varying the chemical potential of graphene without re-optimizing and re-fabricating the nano-structures. The corresponding waveguides constructed on a single flake of graphene [11] by non-uniform conductivity patterns, optical spatial switches [12], splitters [13], and directional couplers [14] have been reported in detail and exhibit indeed better tunability than that consisting of noble metals. On the other hand, the investigation of the SPP modes supported by graphene ribbons [15–17] is emerging as a research focus. Two kind of plasmonic modes, waveguide modes and edge modes [18–20], have been found. Of particular interest is the edge mode that the field is concentrated on the rims of graphene ribbons. It has not only tighter confinement but also benefits enhancing the electromagnetic coupling between objects [18]. The edge mode, without doubt, will serve as a new platform for the operation of plasmonic devices.

In combination with the two noticeable facts mentioned above, we are impressed to find out that it is feasible to devise desired planar plasmonic devices by using different values of gate voltages at different locations across a monolayer graphene, and that these devices could operate perfectly based on the edge mode propagation. Hence, we propose a structure of two coplanar graphene strips coupling system supported on substrates. It also can be created on a single graphene sheet by spatially varying external gate voltages. It is investigated numerically by using the finite-difference time-domain (FDTD) method. Simulation results reveal that the edge mode enhances indeed the electromagnetic coupling between ribbons. In particular, despite of no traditional resonators (rings, disks, or rectangular geometry) used in the most metallic plasmonic filters [4, 21–23], this structure based on the novel side-coupled resonator formed by the parallel graphene strips presents an original, ultra-narrowband band-stop filtering effect in the mid-infrared region. Only by engineering the locations of gate voltages, the overlapping length and coupling distance between two coplanar graphene ribbons can be varied, without re-optimizing and re-fabricating the structure. Correspondingly, the transmission spectrum is tuned and optimized. Thus, the way of gate voltages exhibits better tunability than that used in metal-based devices. In addition, the change of substrates also tunes effectively the transmission spectrum for practical applications. The FDTD results are verified by the theoretical analysis. Our studies hence will enrich further the fabrication of ultra-compact planar plasmonic devices in the mid-infrared region for optical communication.

Simulations and Theory

Our proposed scheme is demonstrated in Fig. 1. Two coplanar graphene ribbons with same width W are placed on the substrate with the refractive index n. It is well known that the chemical potential (μ c) of the graphene can be controlled by external gate voltages, and the relationship between the chemical potential and the gate voltage (V) is governed approximately by the equation [17]: \( {\mu}_c=\hslash {v}_f\sqrt{\pi {\varepsilon}_0{\varepsilon}_dV/ eD} \), where the ε d and D are respectively the dielectric constant and the thickness of the insulating layer; v f is the Fermi velocity in graphene; −e is the electron charge; ħ = h/2π is the reduced Planck’s constant. If the chemical potential is lesser than half the photon energy, the SPP guided surface wave is no longer supported by the graphene sheets because the interband transition of electrons takes place at this condition, and the graphene shows strong absorption of light. On the contrary, intraband transitions at the conduction band dominate, and the graphene behaving as a metal supports the transverse magnetic SPPs, when the chemical potential is greater than half of the photon energy [13]. Hence, the proposed structure can be constructed on a single graphene sheet by means of a split gate device to apply different bias voltages to different locations for achieving inhomogeneous conductivity patterns. As shown in Fig. 1, on the flake of graphene, only the red areas with the bias voltage value of V 1 behave as metals and support the SPP waves. The SPP waves are prohibited on the others with the gate-voltage V 2. The coupling distance between two graphene strips is d and the overlapping length is L. Certainly, in addition to the way of a split gate device to apply different bias voltages to different locations, the other ways to realize inhomogeneous conductivity patterns across a single graphene layer have also been achieved, which include an uneven ground plane holding the monolayer graphene, and non-uniform permittivity distribution underneath the graphene sheet [11].

Fig. 1
figure 1

Schematic of two coplanar graphene ribbons supported on the substrate with the refractive index n. It is constructed by spatially varying external gates acting on a graphene sheet where only the red areas with the gate-voltage value of V 1 support SPPs and the others with V 2 do not. The width of the ribbon is W and the distance between two monitors (P 1 and P 2) is H. The coupling distance between two graphene strips is d and the overlapping length is L. The excited SPP waves propagate along the x direction

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To start with, the characteristics of SPP modes propagating along a single graphene nano-ribbon are investigated by using the 3D FDTD method (Lumerical FDTD Solutions) with a perfectly matched layer (PML) absorbing boundary conditions. The graphene ribbon with a narrow width of W = 10 nm is modeled as an ultra-thin film with a thickness of Δ (in the z direction). The surface conductivity (σ g) of the graphene is governed by the Kubo formula [24, 25] which depends on the momentum relaxation time τ, temperature T, chemical potential μ c, and incident wavelength λ (frequency ω). At room temperature, the Kubo equation is reduced to [13]

$$ {\sigma}_g=\frac{i{e}^2{\mu}_{\mathrm{c}}}{\pi {\hslash}^2\left(\omega +i{\tau}^{-1}\right)}, $$

where the intraband transition dominates. The equivalent permittivity of graphene is calculated by the equation [11]: ε eq = 1 + iσ g η 0/(k 0Δ), where η 0 (≈377 Ω) is the intrinsic impedance of air, and k 0 = 2π/λ. The Δ is assumed to be 1 nm in our calculations, although other extremely small values lead to similar results because the ε eq is thickness-dependent. The single graphene ribbon is well known to support two types of SPP modes: the waveguide mode, in which the field is concentrated along the whole area of the ribbon (in the y direction), and the edge mode, in which the field is concentrated on the rims of the ribbon. The waveguide mode is mainly used for information transmission and the edge mode may be more conducive to enhance the EM coupling between objects. The number of the SPP modes supported by the nano-ribbon increases as the ribbon width increases [19]. However, if the width is ultra-thin, such as only dozens of nanometers, the ribbon just supports fundamental edge modes in the mid-infrared region. In the implementation, the microscopic details of the edges of the graphene ribbon are ignored [19, 20], and the parameters of τ and μ c, which directly affect the surface conductivity of the graphene, are chosen as 0.5 ps, 0.3 eV, respectively. The SPP waves are excited by a dipole point source placed 2 nm over the ribbon in z direction and have the form \( {\overrightarrow{E}}_{\left(r,\omega, t\right)}={\overrightarrow{E}}_{\left(y,z\right)} \exp \left(i{k}_x\left(\omega \right)x- i\omega t\right), \) where k x (ω) is the wave vector in the propagation direction x. The corresponding SPP wavelengths calculated by λ spps  = 2π/Re(k x (ω)) are illustrated in Fig. 2. At the same time, it is demonstrated clearly in the inset where the contour profiles of |E| 2 in y-z cross section of the graphene ribbon is shown that the most field energy is concentrated on the rims of the graphene ribbon. In the inset shown in Fig. 2, the incident wavelength λ is 10 μm and n = 1.0. Therefore, the ultra-narrow graphene ribbon with W = 10 nm only supports indeed the fundamental edge mode.

Fig. 2
figure 2

The dependence of SPP wavelengths (λ spps) on the incident wavelengths with different substrates: the blue, green, and red lines correspond to the substrates with refractive indices of 1.0, 1.2, and 1.4, respectively. Inset displays the contour profiles of |E| 2 in y-z cross section of the graphene ribbon with W = 10 nm

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Based on the edge mode, we investigate two coplanar graphene strips coupling system embedded in air by using the FDTD method. In our calculations, we use non-uniform mesh. The minimum mesh size inside the graphene layer equals 0.1 nm and gradually increases outside the graphene sheet to reduce the storage space and computing time. Similarly, a dipole point source is placed 2 nm over the left ribbon shown in Fig. 1, in order to excite the SPP waves. Two monitors are, respectively, put at the points of P 1 and P 2 for detecting the incident power P in and transmitted power P out. The distance between P 1 and P 2 is H = 500 nm. The transmission is defined to be T = P out/P in. The material parameters are unchanged and the structural parameters of W, L, and d, are chosen to be 10, 100, and 10 nm, respectively. Simulation results are presented in Fig. 3. In Fig. 3a, one can find obviously that three transmission dips corresponded respectively to incident wavelengths of λ = 5.47, 6.41, and 8.14 μm appear in the transmission spectrum. The minimum FWHM (full width at half maximums) of the transmission spectrum is 10.5 nm. It exhibits a novel ultra-narrowband band-stop filtering effect in the mid-infrared region. Unusually, the filtering effect is achieved without a traditional resonator (such as rings, disks, and rectangular structure), as would be needed usually for common metallic filters. Moreover, Fig. 3b displays the contour profiles of field H z of the coupling system at different incident wavelengths of λ = 5.47, 6.41, 8.14, and 8.42 μm, respectively. Clearly, the edge modes with high confinement propagate effectively along the graphene ribbon, and the novel side-coupled resonator formed by the parallel graphene strips is achieved. For the transmission dips, the edge mode resonance forms in the overlapping zone resulting from the effect of reflected waves from the end of the input graphene strip on the incident SPP waves, and the most power hence is confined to the overlapping section. As a result, there is little power transferred to the output graphene ribbon. The SPP wavelengths of transmission dips satisfy the resonance equation 2L = mλ spps (m = 1, 2, 3 …). The transmission dips with incident wavelengths of 8.14, 6.41, and 5.47 μm shown in Fig. 3a correspond to m = 2, 3, and 4, respectively. They are confirmed approximately by the corresponding theoretical values of 8.24, 6.1, and 5.0 μm shown in Fig. 2, where the dependence of SPP wavelengths (λ spps) on the incident wavelengths is demonstrated. On the contrary, the incident power can transfer effectively to the output graphene strip related to those non-resonant wavelengths, owing to the strong coupling feature of edge modes. For example, λ is 8.42 μm shown in Fig. 3b.

Fig. 3
figure 3

a Transmission spectrum of the proposed structure. b The contour profiles of field H z of the coupling system at different incident wavelengths of λ = 5.47, 6.41, 8.14, and 8.42 μm, respectively

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Next, utilizing the gate-voltage dependent optical property of the graphene, there is no need to re-fabricate structures for tuning the transmission spectrum. Only varying the locations of gate voltages can we tune the transmission spectrum. As shown in Fig. 1, the overlapping length L is changed by displacing horizontally the locations of red areas with gate-voltage V 1. Based on the above resonance equation, the transmission spectrum will be modified correspondingly. Simulation results are tidied in Fig. 4a, where the coupling distance is kept as 10 nm. The transmission spectra plotted in blue, green, and red lines correspond to the two graphene ribbons coupling systems with overlapping lengths of 60, 80, and 100 nm, respectively. With the overlapping length increase, transmission dips with same value of m tend to exhibit a red shift. It is consistent indeed with the above resonance equation. Certainly, it is also verified by the theoretical values shown in Fig. 2. Thus, this structure exhibits better tunability than metallic devices which need to re-fabricate structures for tuning transmission spectra.

Fig. 4
figure 4

a Transmission spectra of the proposed structure with d = 10 nm for different overlapping lengths. b Transmission spectra with L = 100 nm for different coupling distances

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According to the same principle, varying the locations of gate voltage could give rise to the change of the coupling distance d between two coplanar graphene ribbons. The influence of the coupling distance on the transmission spectrum is further investigated and FDTD results are indicated in Fig. 4b, where the overlapping length is kept as 100 nm. The transmission spectra plotted in blue, green, red, and baby blue correspond to coupling distances of 8, 10, 12, and 14 nm, respectively. Clearly, the change of the coupling distance has little effect on the wavelengths of transmission dips. It is in excellent agreement with the above resonance equation where the resonance wavelengths are not related to the coupling distance. At the same time, the transmission reduces as the coupling distance increases because the coupling loss increases with the coupling distance increase. Therefore, the change of the coupling distance can optimize perfectly the ultra-narrowband transmission spectrum without changing the resonance wavelengths of transmission dips.

Finally, it is well known that the change of substrates affects directly the properties of SPP waves propagating along graphene strips. As shown in Fig. 2, for a certain incident wavelength, the SPP wavelength decreases with the refractive index of the substrate increase. In the practical applications, the two coplanar graphene strips coupling system should be supported by substrates. Hence, it is inevitable to discuss the effect of substrates on the operation of the proposed band-stop filter. The coupling distance and overlapping length are chosen to be 10 and 100 nm, respectively. FDTD results are demonstrated in Fig. 5a, where the transmission spectra plotted in blue, green, and red lines correspond to substrates with refractive indices of 1.0, 1.2, and 1.4, respectively. As the refractive index increases, the transmission dips with same value of m tend to exhibit a red shift, which is ascribed to that the SPP wavelength decreases with the refractive index increase. In combination with the above resonance equation, simulation results are also consistent with the theoretical values. As shown in Fig. 5b, the dependence of resonance wavelengths with m = 3 on refractive indices of substrates is demonstrated, and there is only a little difference between the theoretical results and the FDTD ones. Hence, for practical applications, we also can utilize the proper substrate to tune the desired transmission spectrum.

Fig. 5
figure 5

a Transmission spectra of the filter structure for different refractive indices of substrates. b The resonance wavelength of m = 3 as a function of the refractive indices of substrates. The blue and red lines correspond to theoretical data and FDTD simulation results, respectively

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Conclusions

To sum up, the two coplanar graphene strips coupling system supported on a substrate is proposed. It is constructed on a single graphene layer by spatially varying external gate voltages and is investigated numerically by using the FDTD method. Simulation results reveal that the novel side-coupled resonator formed by the parallel graphene strips is achieved, and this structure based on the edge modes propagation exhibits an original, ultra-narrowband band-stop filtering effect in the mid-infrared region. The transmission spectrum is tuned and modified only by varying the locations of the gate voltage to change the coupling distance and the overlapping length. It is unlike the traditional metallic devices which need to re-fabricate a new structure for modulating the transmission spectrum. The way of the gate-voltage hence presents better tunability. On the other hand, the changes of substrates also can tune effectively the transmission spectrum for practical applications. The FDTD results are consistent with the theoretical analysis. Undoubtedly, the proposed filter without traditional resonators is unusual and the way of gate voltages to tune the transmission spectrum is more convenient. Our studies will enrich the ultra-compact, versatile planar plasmonic devices for optical processing in the mid-infrared region.