Propagation of Step-Function Light Pulses in a Resonant Medium

A combined analytical and numerical analysis of the propagation of a step-function light pulse in a resonant medium is presented. It is found that the light pulse evolves into a steady-state amplitude-modulated wave train very closely approximated by the Jacobian elliptic function $\mathcal{E}=1.25\mathrm{} {\mathcal{E}}_{0}dn\left[\right(\frac{1}{\tau }\left)\right(t-\frac{z}{V}); \lambda ]$, where the modulus of the elliptic function $\lambda$ is equal to $\frac{4}{5}$ and ${\mathcal{E}}_0$ is the amplitude of the input pulse. This result is consistent with an earlier analysis which showed that the elliptic $\mathrm{dn}$ function represented one of two classes of functions which satisfy the equations of resonant-pulse propagation and correspond to the distortionless propagation of an amplitude-modulated wave train through an inhomogeneously broadened medium. Expressions for the period, velocity, and parameter $\tau$ of the elliptic function are found as functions of the intensity of the input pulse. The form of the steady-state wave train is found to be the same in either an attenuating or amplifying medium. It is found that the velocity of the pulse train becomes significantly less than the speed of light when the density of energy that can be stored in the resonant atoms becomes an appreciable fraction of the energy density of the electromagnetic field. The conditions necessary for observing the amplitude-modulated pulse train in gaseous S${\mathrm{F}}_6$ and Na vapor are discussed.