Dual nonlinearity Controlling of Mode and Dispersion Properties in Graphene-Dielectric Plasmonic Waveguide
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We study the mode and dispersion properties of graphene-dielectric nonlinear plasmonic waveguide considering the dual nonlinearity of dielectric and graphene. For TM polarization, the mode distribution, the permittivity distribution, and dispersion relation were obtained by numerically solving the Maxwell equations. Compared with the case considering only the nonlinearity of dielectric, the initial field intensity to excite plasmon modes reduces obviously when introducing the dual nonlinearity. In addition, the influence of dual nonlinearity on dispersion relation is discussed, and we find that the graphene’s nonlinearity affects strongly the dispersion properties. The introduction of dual nonlinearity leads to the decrease of the initial field intensity, which has potential application in all-optical switches with low threshold.
KeywordsDual nonlinearity Graphene Plasmonic waveguide Dispersion
PACS number(s)78.67.Wj 42.65.Wi
The graphene plasmonics have attracted widespread attention [1, 2, 3, 4] due to the unique electronic and optical features of graphene compared with metals. At THz and far-infrared frequency range, the intraband transition of electrons dominates and graphene behaves like a metal. Therefore, the surface plasmon polaritons (SPPs) could be supported by graphene. For the graphene-dielectric multilayer composite structure, the modes excitation, coupling, and propagation of SPPs have been investigated. The quasi-transverse electromagnetic mode was found in a graphene parallel-plate waveguide . The coupling of SPPs were studied [6, 7] in a graphene-dielectric multilayer structure. For the monolayer graphene sheet periodic array structure, strong coupling between SPPs emerges when the graphene sheets are arranged tightly.
Considerable efforts have been devoted to investigating the optical properties of graphene-dielectric nonlinear composite structures [8, 9, 10, 11, 12] for their great potential in controlling light propagation at the micro-and nano-scales. For the single layer graphene case, the surface plasmons at the interface between graphene and kerr-type nonlinear substrate were discussed . It is shown that the wavelength of graphene plasmons can be tuned by adjusting the nonlinear permittivity of substrates. For the graphene-nonlinear dielectric multilayer structure, the propagation and localization properties of graphene plasmons were explored, and the exact dispersion relations for TM surface plasmons of a graphene parallel plate waveguide were obtained . The propagation and localization length are remarkably affected by adjusting nonlinear permittivities. Recently, the dispersion relation for the symmetric and antisymmetric plasmon modes has been derived in a graphene-coated kerr slab structure . Except for the typical forward-propagating mode, the symmetric, and antisymmetric modes were found.
Based on the graphene’s strong nonlinearity, several nonlinear optical effects have been predicted [13, 14, 15, 16, 17]. Nesterov et al.  studied the nonlinear propagation of light in a graphene monolayer, and found that graphene monolayer can supports TE and TM spatial optical soliton at optical frequencies due to the intrinsic nonlinearity of graphene. More recently, replacing monolayer graphene by multilayer graphene, Smirnova et al.  investigated the nonlinear properties of a multilayer stack of graphene sheets, and derived the nonlinear equations describing spatial dynamics of the nonlinear plasmons. The previous studies mainly focused on the influence of single nonlinearity on control of light properties in graphene-dielectric structures. The idea of dual nonlinearity control was introduced in the graphene-based photonic superlattices [18, 19], in which the electrical and all-optical control of photonic beams with deep-subwavelength accuracy was achieved. However, the dual nonlinearity control of mode and dispersion properties in graphene-dielectric plasmonic structure still leaves open many questions. Therefore, in this paper we consider the graphene’s and dielectric’s nonlinearity simultaneously in the graphene-dielectric-graphene waveguide, and study the influence of the dual nonlinearity on modes coupling and dispersion properties.
where p = c − b 2/3, q = d − bc/3 + 2b 3/27. Using Eq. (10) we can obtain the ε. Replacing the ε in Eq. (2) and (3) by the real solution, the ordinary differential equations can be solved numerically by a relaxation method.
Results and Discussions
where ν F = 0.95 × 108cm/s is the Fermi velocity.
For the graphene, only in THz and far-infrared frequency range can its surface conductivity be simplified to a simple Drude type; therefore, we choose the incident wavelength as λ = 10 μm. Other parameters are fixed to the values ε 1 = 1, ε L = 2.25, α = 5 × 10− 16(m/v)2  E F = 0.27 ev, τ = 1.5 ps. It is well known that there are two modes in graphene-dielectric-graphene linear structures, which are symmetric and antisymmetric modes, respectively. In the following, we will discuss the influence of nonlinearity on modes distribution in the graphene- dielectric composite structures.
For the branch of black dashed curve, the corresponding permittivity and fields associated with A are plotted in Fig. 3a, b, in which the distribution of the permittivity and the electric field E z is symmetric. Therefore, this branch represents the symmetric mode. For the branch of red dashed curve, the permittivity and fields associated with B are given in Fig. 3c, d. Distribution of permittivity is still symmetric; however, the distribution of electric field E z is antisymmetric which implies this branch is an antisymmetric mode. The distribution of permittivity and field associated with C and D are plotted in Fig. 3e–h. It is noted that the distribution of corresponding magnetic field and electric field associated with C and D is asymmetric; therefore, the branch of blue dashed curve represents asymmetric mode. Meanwhile, the asymmetric distribution of electric field leads to the asymmetric distribution of permittivity.
In the following, we introduce simultaneously the nonlinearity of dielectric and graphene, and discuss the influence of nonlinearity of graphene on dispersion relation with a fixed nonlinear coefficient of dielectric α = 5 × 10− 16(m/V)2. The results are shown in Fig. 4d-f. Compared Fig. 4d with Fig. 4c, it is noticed that the fold-back phenomenon of dispersion relation appears in all three branches. From Eq. (13), we know that the graphene’s nonlinearity can be controlled by adjusting the chemical potential. As the nonlinearity of graphene further increases from μ c = 0.27 eV to μ c = 0.16 eV, as shown in Fig. 4e, the fold-back point of dispersion relation moves up. For a larger nonlinearity of graphene (with small chemical potential μ c = 0.10eV), as shown in Fig. 4f, only the symmetric mode appears and forms a closed loop. From Fig. 4, we know that considering only the nonlinearity of dielectric, the dispersion relation shows three branches which are almost unchangeable as the nonlinear coefficient of dielectric increases. However, when we further introduce the nonlinearity of graphene, the fold-back phenomenon of dispersion relation appears. For the specified initial magnetic field H 0 and chemical potential the dispersion relation only shows a symmetric mode with a closed loop.
In summary, we have investigated the mode and dispersion properties of graphene-dielectric nonlinear plasmonic waveguide. The mode distribution, permittivity, and dispersion relations were obtained by numerically solving Maxwell equation for TM polarization. Compared with the case considering only the dielectric’s nonlinearity, the initial field intensity reduced apparently when considering the nonlinearity of dielectric and graphene simultaneously. In addition, the dual nonlinearity affects the dispersion properties of the waveguide significantly. Especially, as the graphene’s nonlinearity increases, the antisymmetric and asymmetric modes merge into one and gradually disappear. Therefore, only the symmetric mode can be found in the case of strong nonlinearity.
This work was supported by the Program for Innovation Research of Science in Harbin Institute of Technology, the National Basic Research Program of China (2013CB328702), and the Fundamental Research Funds for the Central Universities.
XQJ and JLB conceived the idea, calculated the results, and made the conclusions. All the authors read and approved the final manuscript.
The authors declare that they have no competing interests.
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- 20.Janson S. Roots of polynomials of degrees 3 and 4, available at https://arxiv.org/abs/1009.2373. Accessed 13 Sept 2010
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