Second-order nonlinear optical response of graphene

Although massless Dirac fermions in graphene constitute a centrosymmetric medium for in-plane excitations, their second-order nonlinear optical response is nonzero if the effects of spatial dispersion are taken into account. Here we present a rigorous quantum-mechanical theory of the second-order nonlinear response of graphene beyond the electric dipole approximation, which includes both intraband and interband transitions. The resulting nonlinear susceptibility tensor satisfies all symmetry and permutation properties, and can be applied to all three-wave mixing processes. We obtain useful analytic expressions in the limit of a degenerate electron distribution, which reveal quite strong second-order nonlinearity at long wavelengths, Fermi-edge resonances, and unusual polarization properties.

Figure 1

Sketch of the second-order nonlinear current generation in the graphene plane for obliquely incident light.

Figure 2

Nonzero components of the second-order nonlinear conductivity tensor for the process of SHG as a function of the fundamental frequency. The pump is incident in the $\left(xz\right)$ plane at ${45}^{\circ }$. The Fermi energy is 200 meV and all resonances are broadened by the same factor $\gamma$ equal to 5 meV.

Figure 3

Nonzero components of the second-order nonlinear conductivity tensor for the process of DFG as a function of one of the pump frequencies (${\omega }_2$). Frequency ${\omega }_1$ is fixed at 400 meV. Both pumps are incident in the $\left(xz\right)$ plane at ${45}^{\circ }$. The Fermi energy is 200 meV and all resonances are broadened by the same factor $\gamma$ equal to 5 meV.

Figure 4

Nonzero components of the second-order nonlinear conductivity tensor for the process of SFG as a function of one of the pump frequencies (${\omega }_2$). Frequency ${\omega }_1$ is fixed at 200 meV. Both pumps are incident in the $\left(xz\right)$ plane at ${45}^{\circ }$. The Fermi energy is 200 meV and all resonances are broadened by the same factor $\gamma$ equal to 5 meV.