Nonlinear TE-polarized surface polaritons on graphene

We analyze the propagation of electromagnetic waves along the surface of a nonlinear dielectric medium covered by a graphene layer. We reveal that this system can support and stabilize nonlinear transverse electric (TE) plasmon polaritons. We demonstrate that these nonlinear TE modes have a subwavelength localization in the direction perpendicular to the surface, with the intensity much higher than that of an incident wave which excites the polariton.

Figure 1

Schematics of a graphene layer separating the upper linear and lower nonlinear dielectric media with the dielectric permittivities ${\varepsilon }_2$ and ${\varepsilon }_1^{\text{NL}}$, respectively, shown with a spatial profile of the nonlinear surface polariton.

Figure 2

(a) Surface polariton dimensionless phase velocity $\omega \sqrt{{\varepsilon }_2}/c{k}_x$ versus frequency and parameter $\Delta$. Inset in (a) depicts zoom in the vicinity of $\Delta =0$. (b) Dispersion relation $k_x\left(\omega \right)$ (solid line) for fixed $\Delta =-0.05$ [extracted from (a) along the respective horizontal line] and the light line in the linear dielectric $k_x=(\omega /c)\sqrt{{\varepsilon }_2}$ (dashed red line). (c) and (d) Spatial profiles $|{\mathcal{E}}_y{\left(z\right)|}^2$ of surface polaritons for $\Delta =0$, $\omega =0.39$ eV [curve C in (c)] and $\Delta =-0.05$, $\omega =0.275$ eV, or $\omega =0.3$ eV [curves A and B in (d), correspondingly]. The curves A–C correspond to the respective points in (a) and (b). The interface between the linear and nonlinear dielectrics is depicted by a vertical dash-dotted black line. In all panels the dielectric permittivities of the nonlinear and linear media are ${\varepsilon }_1=2.89$, ${\varepsilon }_2=3$, while the relaxation rate and the chemical potential of graphene are $\Gamma =0$ and $\mu =0.2$ eV.

Figure 3

(a) Surface polariton dimensionless phase velocity $\omega \sqrt{{\varepsilon }_2}/c{k}_x$ versus frequency and parameter $\Delta$. (b) Dispersion relation $k_x\left(\omega \right)$ (solid line) corresponding to the horizontal line $\Delta =0.1$ in (a) and the light line in the linear dielectric $k_x=(\omega /c)\sqrt{{\varepsilon }_2}$ (dashed red line). (c) Spatial profiles $|{\mathcal{E}}_y{\left(z\right)|}^2$ of surface polaritons for the parameters: $\Delta =0.1$, $\omega =0.35$ eV, or $\omega =0.36$ eV (curves A${}^{\prime }$ and B${}^{\prime }$, correspondingly). Curves A${}^{\prime }$ and B${}^{\prime }$ correspond to the respective points in (a) and (b). The interface between the linear and nonlinear dielectrics is depicted by a vertical dash-dotted black line. In all panels the dielectric permittivities of the nonlinear and linear media are ${\varepsilon }_1=2.89$, ${\varepsilon }_2=2.8$, while the relaxation rate and the chemical potential of graphene are $\Gamma =0$ and $\mu =0.2$ eV.

Figure 4

(a) Soliton norm $N$ versus wave vector $k_x$ for $\omega =0.3$ eV (dashed line) and $\omega =0.35$ eV (dotted line). (b)–(e) Regions of stability and instability of the nonlinear surface waves (domains shaded in blue and red, respectively) in the $(\Delta ,\omega )$ [(b)], $(\Delta ,{\varepsilon }_2)$ [(c) and (d)], or $(\Delta ,\mu )$ [(e)] planes. Other parameters are ${\varepsilon }_1=2.89$, ${\varepsilon }_2=3$ [(b) and (e)], $\mu =0.2$eV [(b)–(d)], $\omega =0.3$eV [(c)], or $\omega =0.35$eV [(d) and (e)]. Dashed and dotted vertical lines in (b)–(e) correspond to the respective curves in (a).

Figure 5

Evolution of the spatial profile of the nonlinear surface polariton as it propagates along $x$. In (a), (b), and (e) solid and dashed lines correspond to the initial (at $x=0\mu$m) and final (at $x=500\mu$m) wave profiles, respectively. Parameters of (a), (c) and (b), (d) correspond to points A in Fig. 2 and B${}^{\prime }$ in Fig. 3 of the paper, respectively, while those of (e) and (f) are $\Delta =-3.8\times {10}^{-4}$, $\omega =0.3$ eV, $\mu =0.2$ eV. Shown is the beam intensity $|{\tilde{\mathcal{E}}}_y{(x,z)|}^2$ normalized to unity. The dash-dotted line marks the interface between the linear and nonlinear dielectrics.