  Hot atomic vapors are widely used in non-linear and quantum optics due to
their large Kerr non-linearity. While the linear refractive index and the
transmission are precisely measured and well modeled theoretically, similar
characterization remains partial for the $\chi^{(3)}$ non-linear part of the
susceptibility. In this work, we present a set of tools to measure and estimate
numerically the non-linear index of hot atomic vapors both in the steady state
and during the transient response of the medium. We apply these techniques for
the characterization of a hot vapor of rubidium and we evidence the critical
role played by transit effects, due to finite beam sizes, in the measurement of
the non-linear index.
