Second order optical nonlinearity of graphene due to electric quadrupole and magnetic dipole effects (original) (raw)

We present a practical scheme to separate the contributions of the electric quadrupole-like and the magnetic dipole-like effects to the forbidden second order optical nonlinear response of graphene, and give analytic expressions for the second order optical conductivities, calculated from the independent particle approximation, with relaxation described in a phenomenological way. We predict strong second order nonlinear effects, including second harmonic generation, photon drag, and difference frequency generation. We discuss in detail the controllability of these effects by tuning the chemical potential, taking advantage of the dominant role played by interband optical transitions in the response. Graphene is being enthusiastically explored for potential applications in plasmonics, optoelectronics, and photonics 1 , due to its unique optical properties. They arise from the linear dispersion of gapless Dirac fermions as well as the ability to tune the Fermi energy with relative ease, by either chemical doping 2 or applying a gate voltage 3,4. With the large optical nonlinearity predicted theoretically 5-8 and observed experimentally 9 , graphene is also a potential resource of optical nonlinear functionality for photonic devices, including saturable absorbers, fast and compact electo-optic modulators, and optical switches. Taking into account the maturing integration of graphene onto silicon-based chips, the utilization of the optical nonlinearity of graphene opens up new opportunities for the realization of nonlinear integrated photonic circuits. Due to the inversion symmetry of the graphene crystal, the first nonvanishing nonlinear effect is the third order nonlinearity. In spite of the one-atom thickness of graphene, strong third order nonlinear effects have been demonstrated 6,7 including parametric frequency conversion, third harmonic generation, Kerr effects and two photon absorption, and two color coherent current injection. The extracted effective nonlinear coefficients are incredibly large, with values orders of magnitude larger than those of usual semiconductors or metals. When fundamental photon frequencies ω i are much smaller than the chemical potential, as occurs in THz experiments on doped graphene, the nonlinear optical response is dominated by the intraband transitions 5,7 , occurring mostly around the Fermi surface, and the third order optical conductivities have a typical frequency dependence 5,7 of ∝ (ω 1 ω 2 ω 3) −1 in the absence of relaxation. For photon energies in the near infrared to visible, the nonlinear processes are dominated by the interband transitions and the mixing of interband and intraband transitions 7. In the presence of an energy gap induced by a suitable chemical potential 6 , which behaves as an energy gap in semiconductors, novel features arise in the nonlinear optical response that cannot be easily found in semiconductors or metals. These nonlinearities are both large and tunable, and promise a new functionality in the design of the nonlinear optical properties of integrated structures. The theoretical results based on the independent particle approximation predict third order optical nonlinearities of graphene orders of magnitude smaller than the experimental values 6,7 , and the reason for the discrepancy has not been identified. The second order nonlinear optical response of graphene is forbidden in the usual dipole approximation. However, it can arise due to a number of effects 7,9,10 : (1) When the variation of the electromagnetic field over the graphene is taken into account, contributions analogous to those due to magnetic dipole and electric quadrupole