Perturbation theory for graphene-integrated waveguides: Cubic nonlinearity and third-harmonic generation

We present perturbation theory for analysis of generic third-order nonlinear processes in graphene-integrated photonic structures. The optical response of graphene is treated as the nonlinear boundary condition in Maxwell's equations. The derived models are applied for analysis of third-harmonic generation in a graphene-coated dielectric microfiber. An efficiency of up to a few percent is predicted when using subpicosecond pump pulses with energies of the order of 0.1 nJ in a submillimeter-long fiber when operating near the resonance of the graphene nonlinear conductivity $\hslash \omega =(2/3)E_F$.

Figure 1

Graphene-induced surface current: local coordinates.

Figure 2

Graphene-coated silica microfiber: (a) effective indexes of pure fiber modes (without the coating) at the fundamental frequency $\omega$ (black curve) and third harmonic $3\omega$ (solid red curve) as functions of the fiber diameter $D, \lambda =2\pi c/\omega =2\mu \text{m}$; (b) profile of the pure fiber $H{E}_{11}$ mode at the fundamental frequency, $P_1=1W, \lambda =2\mu \text{m}$. The red dashed curve indicates corrections to the shape of the mode when the fiber is fully coated with graphene. (c) and (d) Corrections to the propagation constants (solid black curve) and attenuation constants (dashed red curve) in the fully coated fiber for the fundamental (FH) $H{E}_{11}$ and third harmonic (TH) $H{E}_{31}$ modes as functions of wavelength, computed from Eq. (28). Solid squares and open diamonds indicate corresponding values computed with the help of FEM Maxwell solver comsol. In (b)–(d) the Fermi level of graphene is set to $E_F=0.93\mathrm{eV}$.

Figure 3

THG efficiency in a graphene-coated microfiber of diameter $D=0.98\mu \mathrm{m}$: (a) maximal efficiency and the optimal length $z_0$ for the case of pure phase matching $\mathrm{\Delta }\beta =0$ (corresponding pump wavelength $\lambda \approx 2\mu \mathrm{m}$); (b) efficiency for different propagation distances as a function of pump wavelength, $E_F=0.93\mathrm{eV}$.

Figure 4

THG efficiency with a pulse excitation: (a) normalized efficiency for different input peak powers and pulse durations; the thin solid line indicates efficiency for the CW case [see Eq. (51)]. (b) Efficiency as a function of the input pulse duration for different levels of input pump energy $W_1(z=0)=E$ and a fixed propagation distance. Fiber and input parameters: $D=0.98\mu \mathrm{m}, E_F=0.93\mathrm{eV}, {\lambda }_0=2\mu \mathrm{m}$.