An expression is derived for the third-order polarization of a two-level electronic system induced by a plane polarized light pulse, including the effects not only of recombination and dephasing, but also of the rise time of the pulse. It is shown that the expression encompasses all three main electronic mechanisms that lead to an intensity-dependent refractive index, namely the blocking of interband optical transitions by free carriers (the band filling effect), virtual interband transitions, and the motion of free carriers in the non-parabolic band. It also shows explicitly the regimes in which each mechanism dominates. It is also shown (i) how the transition from the virtual interband transition regime to the band filling regime is brought about by the frequency dependence of the dephasing time, (ii) that near resonance the virtual interband transition mechanism is just a casual consequence of the optical Stark effect and (iii) that the intensity-dependent refractive index resulting from the motion of free carriers is the causal consequence of the intensity-dependent shifts in the energy levels, leading to an intensity-dependent effective mass. How the non-linear response of the semiconductor changes as the frequency of the light pulse is changed is discussed in physical terms.
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