▎ 摘　　要

The previous studies on graphene were primarily focused on the effect of photon energy to optical nonlinearities but less on that of doping. By the perturbative treatment of external field, we theoretically illustrate the dependence of the second order refractive and absorptive indices on the Fermi energy of doped graphene with the effects of relaxation and temperature. Some new phenomena are observed. Due to thermal fluctuations at finite temperature, photon with energy below twice the Fermi energy could also induce nonlinearities as the ones above that. And when the Fermi energy is around half that of incident photon, giant enhancement of nonlinear refraction and negative nonlinear absorption could be obtained. We also find interband relaxation dominates in nonlinear decrements rather than intraband one. By comparison between the results of simulation with those of other experiments, the perturbation theory of optical nonlinearities of graphene is confirmed. (C) 2017 Elsevier GmbH. All rights reserved.