Nonlinear plasmonic dispersion and coupling analysis in the symmetric graphene sheets waveguide

www.nature.com/scientificreports OPEN received: 14 June 2016 accepted: 22 November 2016 Published: 15 December 2016 Nonlinear plasmonic dispersion and coupling analysis in the symmetric graphene sheets waveguide Xiangqian Jiang, Haiming Yuan & Xiudong Sun We study the nonlinear dispersion and coupling properties of the graphene-bounded dielectric slab waveguide at near-THz/THz frequency range, and then reveal the mechanism of symmetry breaking in nonlinear graphene waveguide. We analyze the influence of field intensity and chemical potential on dispersion relation, and find that the nonlinearity of graphene affects strongly the dispersion relation. As the chemical potential decreases, the dispersion properties change significantly. Antisymmetric and asymmetric branches disappear and only symmetric one remains. A nonlinear coupled mode theory is established to describe the dispersion relations and its variation, which agrees with the numerical results well. Using the nonlinear couple model we reveal the reason of occurrence of asymmetric mode in the nonlinear waveguide. 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 plasmonics2–5. Considerable effort has been devoted to investigating the mode propagation 6–10, localization 11,12 and coupling13–21 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 predicted24–28. 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 Department of Physics, Harbin Institute of Technology, Harbin 150001, China. Correspondence and requests for materials should be addressed to X.J. (email: ) Scientific Reports | 6:39309 | DOI: 10.1038/srep39309 1 www.nature.com/scientificreports/ Figure 1. Schematic diagram of nonlinear symmetric graphene sheets plasmonic waveguide, ε1 = ε3 = 1, ε2 = 2.25. 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 σ L = σ intra = e 2 µc i π 2 ω + iτ −1 (1) 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 σ g = σ L + σ NL E τ 2 where Eτ is the tangential component of the electric field and σ NL (2) denotes nonlinear conductivity 2 σ NL = − i 3 e 2  eν F  µc   8 π 2  µc ω  ω (3) where νF =  0.95 ×  10 cm/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)ŷ and E = (E ±, x xˆ + E ±, z zˆ)exp(iβz ± K x x ) in the dielectrics or air, respectively, where K x = (β 2 − k 02 ε)1/2 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., 8 E 2, +, z e−K 2, x d + E 2, −, z − E1, +, z = 0 H 2, +, y e−K 2, x d + H 2, −, y − H 1, +, y = σ g E1, (...truncated)