A theory for non-degenerate four-wave mixing in doped graphene

We present a theoretical study of the nonlinear optical (NLO) response of doped graphene to two coherent laser beams, of frequencies ω1 and ω2, resulting in the generation of radiation at frequency ωσ=2ω1−ω2. The two main ingredients of the developed theory are the interplay of interband and intraband electron motion, induced by the incident light waves, and the finite lifetime of excited electronic states, caused by electron scattering. Adopting a tight-binding approximation for the π-electronic band structure of graphene and the Genkin-Mednis formalism of the nonlinear conductivity theory of semiconductors, we calculate the third-order NLO susceptibility χ(3)(−ωσ;ω1,ω1,−ω2) responsible for the non-degenerate four-wave mixing process under consideration. Our calculations show the resonant enhancement of the |χ(3)| (up to a value of 2.8×10−7 esu) when the frequencies ω1 and ω2 of the input beams are mat"ched to provide a resonance for the output photon energy ℏωσ with an effective optical gap of 2EF in the π-electronic band structure of doped graphene (EF is the Fermi energy of charge carriers in the graphene, tunable by an external gate voltage). The results obtained may be of practical interest for generating mid-infrared radiation from doped graphene pumped with two near-infrared laser beams.