Second harmonic generation in graphene-coated nanowires

We study second harmonic generation in a pair of graphene-coated nanowires. We show that the phase matching condition for harmonic generation can be engineered in a wide range of frequencies by tuning the spacing between graphene nanowires. We derive coupled mode equations describing the process of second harmonic generation using an unconjugated Lorentz reciprocity theorem. We show that the highest harmonic generation efficiency can be achieved by phase matching the fundamental mode with the two lowest order symmetric modes at the second harmonic frequency. Despite losses in graphene, we predict that the efficiency can be further enhanced by reducing the radius of nanowires due to larger mode overlap and lower propagation loss.