Propagation of Small-Area Pulses of Coherent Light through a Resonant Medium

This article presents a study of the propagation of small-area pulses of coherent light through matter. It is found that the small-area pulses exactly obey an area theorem which, in the case of an attenuating medium, requires that the pulse area decay to zero exponentially with increasing distance of propagation. This, however, does not necessarily imply that the pulse energy decays exponentially. Instead, it is found that pulses of duration comparable to or shorter than the transverse relaxation time of the medium $T_2$ (including both homogeneous and inhomogeneous broadening) propagate with low energy loss. The results are explained in terms of simple physical arguments which indicate that the pulse envelope should oscillate between positive and negative values, causing the area to decrease without a comparable decrease in the pulse energy. Analytic solutions are presented for the case of a pulse whose envelope varies as $t^k{e}^{-t}$ for $t>\mathrm{}0\mathrm{}$, a rectangular pulse, an ultrashort pulse (i.e., pulse width much less than $T_2$), and the leading portion of a step-function pulse. The propagation of a small-area Gaussian pulse is studied numerically.