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Comprehensive analysis of the optical Kerr coefficient of graphene

We present a comprehensive analysis of the the nonlinear optical Kerr effect in graphene. We directly solve the S-matrix element to calculate the absorption rate, utilizing the Volkov-Keldysh- type crystal wave functions. We then convert to the nonlinear refractive index coefficients through the Kramers-Kronig relation. In this formalism, the source of Kerr nonlinearity is the interplay of optical fields that cooperatively drive the transition from valence to conduction band. This formalism makes it possible to identify and compute the rates of distinct nonlinear processes that contribute to the Kerr nonlinear refractive index coefficient. The four identified mechanisms are two photon absorption, Raman transition, self coupling, and quadratic AC Stark effect. We also present a comparison of our theory with recent experimental and theoretical results.