Nonlinear interaction of propagating short pulses in optically dense media

The optical density of a resonant medium significantly affects the output signal in time resolved nonlinear optical interactions. The resonant interaction with a medium reshapes any propagating short pulse, and the mutual interaction of several such pulses, as in four-wave mixing, must be treated self-consistently. We present a theoretical framework for the proper handling of the propagation of all pulses, input as well as generated, in the small area limit. For optically thick resonant absorbers, negative time delay signals are observed, and the apparent decay rate of the induced polarization is faster than the rate observed for thin samples. We experimentally measure degenerate four-wave mixing in an atomic medium as an example, and demonstrate the quality of the theoretical model by the excellent fit to measured signals over several orders of magnitude. The improved understanding enables us to provide a simple, but surprisingly accurate, estimate for the apparent decay rate in homogeneously broadened optically thick media: If the absorption is given by α, the propagation length is L, and the transverse relaxation time is ${\mathit{T}}_2$, the apparent decay rate 2/${\mathit{T}}_{\mathit{a}}$ of a time resolved four-wave mixing signal is given by 2/${\mathit{T}}_{\mathit{a}}$=(2/${\mathit{T}}_2$)(1+αL/2).