Analytical expressions for the optical response of active semiconductor waveguides are derived, taking into account the effects of propagation, gain, and index dispersion as well as pulse chirp. The dynamics of the gain and the index that are due to carrier density changes, carrier heating, and spectral holeburning are derived from semiclassical density-matrix equations, and instantaneous two-photon and Kerr effects are included phenomenologically. It is shown that dispersion of the gain strongly affects short-pulse pump-and-probe measurements, since it induces a coupling between the phase and the intensity of the probe pulses. Good agreement with published experimental data is obtained.
J. Mørk and A. Mecozzi, "Theory of the ultrafast optical response of active semiconductor waveguides," J. Opt. Soc. Am. B 13, 1803-1816 (1996)