Second-harmonic generation in subwavelength graphene waveguides

We suggest a novel approach for generating the second-harmonic radiation in subwavelength graphene waveguides. We demonstrate that the quadratic phase matching between plasmonic guided modes of different symmetries can be achieved in a planar double-layer graphene structure when conductivity of one of the layers becomes spatially modulated. We predict theoretically that, owing to graphene nonlocal conductivity, the second-order nonlinear processes can be actualized for interacting plasmonic modes with an effective grating coupler to allow external pumping of the structure with the generated radiation at the double frequency.

Figure 1

Geometry of the problem. A free-standing waveguide created by two graphene layers is illuminated by an external wave being coupled to the phase-matched guided modes of the graphene waveguide. The red curve corresponds to an antisymmetric mode at the fundamental frequency, and the blue curve corresponds to a second-harmonic symmetric mode (shown with the tangential electric-field component). Conductivity of the upper layer is periodically modulated.

Figure 2

Plasmonic guide index vs frequency at the fixed separation between the graphene layers $d=62\mathrm{nm}$, calculated for the doping level of ${\mathcal{E}}_F=0.6\mathrm{eV},N=1$, and $\varepsilon =1$. Blue and red solid curves correspond to the symmetric and antisymmetric modes, respectively. Dashed gray and red lines show dispersion of an isolated graphene sheet and half-frequency antisymmetric mode for reference. Colored dots in the horizontal axis mark the fundamental and second-harmonic frequencies.

Figure 3

Operational fundamental frequency as a function of the separation between the layers for the exact phase matching at ${\mathcal{E}}_F=0.6\mathrm{eV},N=1$, and $\varepsilon =1$. The point corresponds to the parameters used in Fig. 2.