Amplification of coherent optical pulses

The amplification of a coherent optical pulse is analyzed by employing an inverse scattering method. In the extreme Doppler limit, one obtains an analytical expression that exhibits the expected evolution toward a $\pi$ pulse with pulse amplification and compression. In the sharp-line limit, one obtains justification for previous usage of a similarity solution to describe $\pi$-pulse propagation in an amplifier. The analysis is carried out for an initial pulse profile (truncated hyperbolic secant) which yields a sufficiently simple scattering problem that the Marchenko equation of inverse scattering theory may be solved analytically.