Coherent pulse propagation: A comparison of the complete solution with the McCall-Hahn theory and others

The complete solution of the Maxwell-Bloch equations by the inverse scattering transform as given by Ablowitz, Kaup, and Newell is compared with the McCall-Hahn theory, and others. Not only it is shown that the earlier results are contained in the complete solution, but also several new results are obtained. Among these are an infinite set of "nonlinear moments" which evolve similar to the McCall-Hahn area, a closed-form solution for the nonlinear transmission, how one can determine absolute time delays, why the threshold area for lossless propagation is still exactly $\pi$ (even when the initial profile is off resonance but unchirped), and first-order effects of relaxation on $2\pi$ hyperbolic-secant pulse propagation. For the last-mentioned results, we find that in general if a $2\pi$ hyperbolic-secand pulse is initially off resonance, it will move away from resonance. Also, we find that the McCall-Hahn result for first-order effects on the time delay must be modified for small-amplitude $2\pi$ hyperbolic-secant pulses.