Nonlinear coupling in graphene-coated nanowires

We propose and analyze nonlinear coupler based on a pair of single mode graphene-coated nanowires. Nonlinear wave interactions in such structure are analyzed by the coupled mode equations derived from the unconjugated Lorentz reciprocity theorem. We show that the routing of plasmons in the proposed structure can be controlled by the input power due to the third order nonlinear response of graphene layer. Our findings show that graphene nonlinearity can be exploited in tunable nanoplasmonic circuits based on low-loss, edgeless cylindrical graphene waveguides.

where e is electron charge, μ c is chemical potential, k B is Boltzmann constant, τ is scattering time, and T is temperature. We assume that μ c = 0.5, τ = 10 ps, corresponding to low loss high quality graphene 16,20,21 , and T = 300 K. The third order nonlinear conductivity 22 , where v f ≈ c/300 is the Fermi velocity. A single graphene coated nanowire can support different order modes, which can be classified by azimuthal mode profile 6 . For a GNW of a given radius, the fundamental mode does not have a cut-off frequency, while all the higher order modes do. Alternatively, for a given frequency, existence of the higher order modes requires certain minimum wire radius, while thinner GNWs support only fundamental mode. For the case of directional coupler, we choose both wires in a single mode regime. Figure 2(a) shows the radius-dependent mode index of the first two modes of a single wire at 30 THz. When the nanowire radius reduces to below about 50 nm, the second order mode experiences cut off, allowing us to work in a single mode regime.
When two single-mode GNWs are placed parallel and are close enough, the hybridisation of the modes creates antisymmetric (odd) mode and symmetric (even) mode, as shown in the inset in Fig. 2(b). Figure 2(b) shows the mode index of odd and even modes in GNW pair as a function of distance between wires d. The odd mode has larger mode index than its even counterpart. Any wave propagation in these coupled waveguides in linear regime can be represented as a superposition of the odd and even modes. If both waves are excited, then one will observe beating of the waves, and the beat length is defined by the difference between propagation constants of the two modes: L B = π/Δ β with Δ β = β a − β s , β a and β s are propagation constants of odd and even modes, respectively. Nonlinear coupling. The linear modes in GNW pair can be described as where β m is the wavenumber, α m is the attenuation of the m-th mode, {e m , h m } are transverse profiles of electric and magnetic fields, and m is either "a" or "s", denoting the antisymmetric or symmetric mode.
To describe the nonlinear interactions, we aim to derive coupled mode equations 23,24 . Considering relatively high loss in plasmonic waveguide, as compared to dielectric waveguides, we employ unconjugated Lorentz reciprocity theorem to derive the coupled mode equations for the nonlinear coupler. For z-invariant graphene plasmonic waveguide, the reciprocity theorem 23 gives where x 0 denotes the position of graphene layer on the cross section of waveguide, J NL is the nonlinear surface current in graphene, {E, H} are the electromagnetic field in the presence of nonlinear current J NL , {e q , h q } is the q-th eigen mode of the waveguide, and S is the entire cross-section of the waveguide where the fields are present.
Assuming that the presence of J NL does not modify the mode profile but only its amplitude, we can write electromagnetic field {E, H} in such waveguide as a superposition of linear modes. Considering the fact that GNW pair in this paper only supports two modes, we have in which A s and A a are the amplitude of even and odd modes with a dependence on z. Expressions for the magnetic field H can be written in a similar way.
Since the even and odd modes originate from the hybridization of TM modes in a single GNW 25 , the z component of electric field is much larger than other tangential field components. Then, the induced nonlinear surface current has only z-component and it is expressed as We first consider the amplitude evolution of even mode A s (z). Substituting (4) and (6)  , which corresponds to the even mode propagating along -z-direction, we have amplitude equation for A s as   Due to the field symmetry of even and odd modes, γ 2 , γ 4 and γ 5 vanish after integration. The amplitude equation for the odd mode A a could be derived in a similar way. As a result, the coupled mode equations for the nonlinear coupling can be derived as where η i can be calculated from (8) by replacing e −s,z by e −a,z . To show a particular example, we consider a pair of graphene coated nanowires with R = 50 nm, d = 150 nm and the working frequency is 30 THz. We calculate eigen mode profiles and their mode indices numerically using commercial simulation package COMSOL. We found that the mode indices of odd and even modes are 21.50 and 20.08, respectively, and this leads to the beat length L B of 3.52 μm. Figure 3 illustrates the spatial distribution of energy density on a plane of the GNW axes. We assume that the input wave of the power P 0 is launched into the left nanowire. The input power in Fig. 3(a) is P 0 = 0.01 mW, which is low enough so that all nonlinear terms in equations are negligible, i.e. at least 10 times smaller than linear terms. As the wave propagates, the energy couples from the left nanowire to the right one, with complete energy transfer after propagating a distance of L B . We note that the total energy decreases with propagation due to intrinsic loss in graphene. When the input power is large enough, the coupling changes due to graphene nonlinear response. Figure 3(b) shows the energy density distribution for P 0 = 2.9 mW. After propagating a distance of L B , most of the energy still flows in the left nanowire with little coupling to the right waveguide. At the distance about L B /2, a small portion of energy is coupled into the right wire, and then coupled back to left wire while propagating toward L B .

into (3) and letting
To further demonstrate the power dependence of the coupler performance, Fig. 4(a) shows transmitted energy in the left and right waveguides for different input powers coupled to the left wire. With the input power increasing, nonlinear self action switches the output power from the right nanowire into the left nanowire. At the critical power P 0,c = 2.29 mW, equal output power from both wires can be achieved. When P 0 further increased to 2.9 mW, the power is predominantly localized in the left nanowire. It is worth noting that in our simulations, when the input power is 3 mW, the maximum electric field in graphene is 3 × 10 7 V/m, which is well below the breakdown threshold of graphene 26 .
The spacing d has an important impact on the switching power. Figure 4(b) shows the change of critical power with spacing d. With d increasing, critical power monotonously decreases. This could be explained as follows: larger d results in a larger L B and vice versa, which can be inferred from Fig. 2(b). As the length of the whole process considered in this paper is L B , a longer coupler length means longer nonlinear interaction region, leading to larger nonlinear phase shift in the coupler and as a result smaller input power is required to achieve the nonlinear switching. At the same time, the length of the coupler cannot be too large, because of graphene absorption and finite plasmon propagation length. Thus we choose an intermediate spacing d = 150 nm in this work, which gives L B that is shorter than the graphene plasmon decay length, and long enough so that the nonlinear effects allow to observe the switching in the coupler.

Conclusion
In conclusion, we have studied the nonlinear plasmon coupling in a pair of single mode graphene coated nanowires. Based on unconjugated Lorentz reciprocity theorem, we derived the coupled mode equations describing nonlinear wave propagation in such a structure. We demonstrated that the nonlinear switching of plasmons between the waveguides can be achieved by changing the input power, and due to the large third order nonlinearity and tight field confinement, relatively low power is sufficient for achieving efficient switching. Our results shows nonlinearity of graphene could be exploited as a new degree of freedom for designing active plasmonic devices based on cylindrical graphene structure.

Methods
Lorentz reciprocity theorem for graphene waveguide. Electromagnetic wave excited by surface current J and guided by a GNW pair satisfies Maxwell's equation  By integrating both parts of this equation over a suitable volume and applying the divergence theorem, taking into account the bounded nature of guided wave, we derive Eq.