Introduction

At THz and far-infrared frequency range, the electrons transition of intraband dominates primarily and the metallic conductivity of Drude type makes the graphene surface plasmon be supported. Based on its unique electric and optical properties1 graphene has been suggested as an alternative to conventional metal-based structures to confine light and guide surface plasmon polaritons. Electromagnetic properties of graphene-dielectric composite structures have attracted special attention in the past years, leading to the rapid development of a new branch of plasmonics known as graphene plasmonics2,3,4,5.

Considerable effort has been devoted to investigating the mode propagation6,7,8,9,10, localization11,12 and coupling13,14,15,16,17,18,19,20,21 of graphene plasmon in the linear graphene-dielectric composite structures. The propagation properties of guided graphene plasmon in individual and paired graphene ribbons were studied6, and the features of low loss, large confinement of light and flexible tunability were found. To manipulate the energy flow of light, Wang et al.7 proposed a graphene plasmonic lens7, this lens can be used to focus and collimate the graphene plasmon waves propagating along the graphene sheet. The confinement of plasmon in very small regions has potential applications in optoelectronics, the surface plasmon resonance in graphene sub-nanometre scale has been explored11.

The coupling effects of graphene plasmon have attracted wide interest. The demonstration of surface plasmon excitation in graphene based on the near-field scattering of infrared light has been reported13,14. Recently, Constant et al.15 presented an all-optical plasmon coupling scheme which takes advantage of the intrinsic nonlinear optical response of graphene, and found that surface plasmons with a defined wavevector and direction can be excited by controlling the phase matching conditions. To realize ultra-high contrast optical modulators, the phase-coupling scheme of localized graphene plasmon resonances has been proposed to replace the original near-field coupling17. Moreover, the tunable multiple plasmon induced transparencies based on phase-coupling has been demonstrated by the same group18. For the graphene-dielectric multilayer structure, the mode coupling properties and its control are useful for designing compact and tunable nanophotonic devices. It is shown that the graphene-dielectric-graphene waveguide can support both symmetric and antisymmetric modes19,20. When the graphene sheets are arranged periodicly and tightly, the strong coupling between surface plasmon polaritons emerges21.

As was shown, graphene is a strongly nonlinear material22,23. Several nonlinear optical effects based on graphene’s nonlinearity were predicted24,25,26,27,28. A novel class of nonlinear self-confined modes originated from the hybridization of surface plasmon polaritons with graphene optical soliton is demonstrated to exist in graphene monolayers25. In order to increase the nonlinearity of photonic structures with graphene, the graphene multilayer structure is presented. The nonlinear switching and palsmon soliton based on graphene multilayer were demonstrated26,27. For the nonlinear graphene-dielectric-graphene structure26, the symmetric, antisymmetric and asymmetric mode were found in the structure. The occurrence of asymmetric mode means the symmetry breaking phenomenon. However, the mechanism of symmetry breaking is still unclear although the phenomenon was found in nonlinear plasmonic waveguides. Therefore, the purpose of this article is to study nonlinear plasmonic dispersion and coupling properties in symmetric graphene sheets waveguide, and reveal the mechanism of symmetry breaking phenomenon.

Results

Nonlinear modes and dispersion properties

The nonlinear graphene plasmonic waveguide is illustrated in Fig. 1. The dielectric slab waveguide of ε2 is bounded by the graphene layers at x = ±d/2 with the surrounding dielectric (ε1 = ε3). According to the Kubo formula29, the linear conductivity of grapheme σL contains the interband and intraband transition contributions. In the THz and far-infrared frequency range, the intraband transition dominates the linear conductivity of graphene which can be reduced to the Drude form29

Figure 1
figure 1

Schematic diagram of nonlinear symmetric graphene sheets plasmonic waveguide, ε1 = ε3 = 1, ε2 = 2.25.

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where e is the electron charge, μc is the chemical potential of graphene, ω is the frequency, and τ is the momentum relaxation time. This model is applicable in low temperature limit (kBT ≪ μc) at low frequency (ħω ≤ μc).

For the strong field condition, the nonlinear part of the conductivity must be considered and the total conductivity of graphene reads27

where Eτ is the tangential component of the electric field and σNL denotes nonlinear conductivity

where νF = 0.95 × 108cm/s is the Fermi velocity.

Considering the transverse-magnetic (TM) surface plasmon polaritons mode that propagates along z direction with a propagation constant β, the magnetic and electric field should be in the form of H = H±,y exp (iβz ± Kxx) and in the dielectrics or air, respectively, where and k0 = ω/c. According to the boundary condition, the tangential component of electric field must be continuous while that one of the magnetic field has a discontinuity of σgE1,+,z, i.e.,

‘±’ in the subscript represents the field decrease and increase along upward direction of x. Similar boundary condition was also established at lower boundary. The Maxwell equation gives the relation

Applying Eq. (5) to region 1, 2 and 3, the dispersion relation equation was obtained with unknown variations of (β,ω,H1,−z  ≡ H0).

Dependence of the magnetic field H0 on the propagation constant β at wavelength λ = 10 μm is shown in Fig. 2, where other parameters are fixed to the values d = 100 nm, ε1 = 1, ε2 = 2.25, μc = 0.27 eV and τ = 1.5 ps. The propagation constant β is normalized by Fermi vector kF = (πn)1/2 30 with the carrier density of n = 6 × 1012 cm−2. There are three modes in the nonlinear plasmonic waveguide, which are symmetric mode, antisymmetric mode and asymmetric mode26. However, it is impossible to distinguish which branch denotes symmetric, antisymmetric or asymmetric mode. To verify the mode properties of these branches in Fig. 2 we plot electric field and magnetic field distribution associated with A, B, C and D, respectively.

Figure 2
figure 2

The initial magnetic intensity versus the propagation constant for the fixed wavelength λ = 10 μm, the horizontal black solid line is an auxiliary line.

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For branch I the fields are plotted in Fig. 3(a), in which distribution of electric field Ez is a symmetric. Therefore, the branch I represents the symmetric mode. For branch II distribution of electric field Ez shown in Fig. 3(b) is antisymmetric. It corresponds to the antisymmetric mode. The branches I and II represent symmetric and antisymmetric modes with respect to the linear conditions. They are caused by coupling of graphene plasmon on the upper and the lower air/graphene/dielectric interfaces. Another branch III is a novel mode which appears only due to nonlinearity. It yields to an interesting field distributions associated with C and D at branch III which are plotted in Fig. 3(c) and (d). Corresponding field distribution is asymmetric, and therefore branch III represents asymmetric mode.

Figure 3
figure 3

The field distribution for magnetic component Hy and electric component Ez, which correspond to points A, B, C and D in Fig. 2, respectively.

(a) (H0, β) = (400A/m, 0.0779kF), (b) (H0, β) = (400A/m, 0.0693kF), (c) (H0, β) = (400A/m, 0.0835kF), (d) (H0, β) = (1000A/m, 0.0805kF). Other parameters are the same as in Fig. 2.

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Next, we turn our attention to discuss the influence of nonlinearity of graphene on dispersion relation. In Fig. 4, the dispersion relations are depicted with the dotted curves in linear case (σNL = 0) and by the solid curves in nonlinear case. For the linear case only symmetric and antisymmetric modes exist. The black dotted curve and the red dotted curve represent the symmetric and antisymmetric modes, respectively. In Fig. 4(a–c) dispersion relation for fixed initial magnetic field (H0 = 1000 A/m) and different chemical potentials μc is given. As is shown in Fig. 4(a), for the larger nonlinearity, when μc = 0.19 eV, only symmetric mode represented by the solid curve is found. It is seen from Fig. 4(b) that at chemical potential is equal to 0.22 eV, antisymmetric (red solid curve) and the asymmetric (blue solid curve) modes appear in addition to symmetric mode of branch I. Further increase chemical potentia l (μc = 0.27 eV) leads to the intersection of antisymmetric and asymmetric modes, which is seen in Fig. 4(c). In Fig. 4(d), these results are compared to those obtained at constant value of chemical potential μc = 0.27 eV, and to decreased initial magnetic field H0. Decrease of H0 leads to consequent reduction of nonlinearity of graphene. In this case the fold-back point of the dispersion relations moves down. In addition, as is shown in Fig. 2, there is a intersection of the antisymmetric and asymmetric branch. Therefore, red and blue modes show an opposite trend when the wavelength of the insets in Fig. 4(c) and (d) is about 10 μm (ω/μc = 0.45). The lower branch of mode I is not plotted in Fig. 4(c) and (d), since it is too close to the lower branch of mode II and III. Nevertheless, it exists.

Figure 4
figure 4

The dispersion relations for various nonlinearity of graphene.

The dotted curves represent the dispersion relation of linear case, and the solid curves denote the dispersion relation of nonlinear case. The parameters (μc, H0) are chosen to (a) (0.19 eV, 1000A/m); (b) (0.22 eV, 1000A/m); (c) (0.27 eV, 1000A/m); (d) (0.27 eV, 400A/m).

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Nonlinear coupled mode theory

In the case of weak field without nonlinearity, the coupled graphene plasmonic waveguide shown in Fig. 1 are depicted in Fig. 5(a). According to the coupled mode theory31, the oscillation energies a1 and a2 satisfy the matrix equation

Figure 5
figure 5

(a) The scheme of the coupling between two graphene waveguides. (b) The dispersion relations of the linear bi-graphene waveguide with distance d. (c) The nonlinear induced propagation constant change of a single graphene waveguide. (d) The third solution a2 = f(a1) which satisfying Eq. (10) and Eq. (11).

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where β1 and β2 are the propagation constants of the single layer graphene waveguide without coupling and κ is the coupling coefficient. The weak field condition of symmetric structure without nonlinearity corresponds to β1 = β2 = β0. The propagation constants of the coupled mode are defined as the eigenvalues of the matrix

They could also be obtained from the mode analysis method. Figure 5(b) presents the dispersion relations of the same structure as is shown in Fig. 1. In this case the coupling coefficient is found to be κ = 4.3 × 10−3kF at wavelength λ = 10 μm.

When the graphene’s nonlinearity is considered, the propagation constant of each single graphene waveguide becomes a function of tangential component of electric field. The dispersion of the single graphene waveguide is32

where . Substituting Eq. (2) into Eq. (8) one gets the nonlinear propagation constant of the single layer graphene waveguide which is shown in Fig. 5(c). Replacing β1 and β2 in Eq. (6) with β1,2(|Eτ|2) and treating a1,2 as the tangential component of electrical field in amplitude, we obtain the coupled mode theory in nonlinear case

where |a1 + γa2|2 and |a2 + γa1|2 are the total field intensity with the similar meaning to the |E1,τ,+|2 and |E3,τ,−|2, respectively, and γ is an empiric factor related to β and d, which is fitted from the numerical data shown in Fig. 2. For propagation along the z direction (∂z = iβ), a1 and a2 must satisfy Eq. (10) and Eq. (11) simultaneously

The first two solutions are a20 = ±a10, and the third one can be only obtained numerically shown in Fig. 5(d). Thus, a20 = ±a10 and a20 = f(a10) represent three branches obtained from Eq. (10) (or Equation (11)). The theoretical result from the coupled mode theory in nonlinear case at a proper value of γ ~ −0.07is shown in Fig. 6. The relationship |H0|~|a1 + γa2|ωε0ε1/Kx,1 can be used. It is found that the theoretical result consistent with the numerical one shown in Fig. 2.

Figure 6
figure 6

The initial magnetic field H0 versus propagating constant β derived from nonlinear coupled mode theory after the transformation of |a10 + γa20| → |H0|.

The inset is β vs. |a1 + γa2| with κ = 4.3 × 10−3kF and γ ~ −0.07.

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The symmetric condition of a20 = ±a10 leads to the symmetric increase of β1 and β2, hence, equality β1 = β2 is always established. Corresponding branches are presented by black and red curves in Fig. 6 (and Fig. 2), i.e., symmetric and antisymmetric field distribution, respectively. For asymmetric condition a10 > a20 > 0 (or a20 > a10 > 0), the former term in Eq. (10) is larger (smaller) than that one in Eq. (11), but the latter term had an opposite order. When these two variation become equilibrium at a20 = f(a10), we have the blue branch as shown in Fig. 6. We can conclude that the asymmetric mode come from the equilibrium of the propagation constant (β1(|a10 + γa20|2)) increase caused by the nonlinearity and compensation (κa20/a10) due to the coupling.

Discussion

In summary, the coupled and dispersion properties of the graphene-dielectric-graphene structure are studied. The propagation constant is found to increase with the field intensity for both the symmetric and antisymmetric mode, whereas the antisymmetric mode splits off an asymmetric mode. When the nonlinearity of graphene is small (μc = 0.27 eV, 0.22 eV), the dispersion relations shows three branches, and there is a fold-back point in each branch. Continuing to increase the nonlinearity of grapheme (decreasing μc to 0.19 eV), the fold-back point disappears and there is only one branch corresponding to the symmetric mode. By introducing the nonlinear coupled mode theory, the features of the nonlinear plasmonic waveguide could be understood well. The reason for emergence of asymmetric mode is revealed. It is originated from the equilibrium of the propagation constant increase caused by the nonlinearity and the compensation due to the coupling.

Additional Information

How to cite this article: Jiang, X. et al. Nonlinear plasmonic dispersion and coupling analysis in the symmetric graphene sheets waveguide. Sci. Rep. 6, 39309; doi: 10.1038/srep39309 (2016).

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