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Physical Review A

Propagation of Step-Function Light Pulses in a Resonant Medium

Michael D. Crisp

Published Mar 1, 1972• 17 citations
Publisher

Abstract

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 $\mathcal{E}=1.25 {\mathcal{E}}_{0}dn[(\frac{1}{\ensuremath{\tau}})(t\ensuremath{-}\frac{z}{V}); \ensuremath{\lambda}]$, where the modulus of the elliptic function $\ensuremath{\lambda}$ 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 $\ensuremath{\tau}$ 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.

DOI
10.1103/physreva.5.1365

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A theoretical model for the propagation of coherent light pulses in resonant media where a large number of atoms cooperate in spontaneous emission (superradiance) is discussed. The medium is supposedly free of inhomogeneous broadening, and no lossy relaxation mechanisms are taken into account. The conditions for the emission of all spontaneous radiation in the direction of pulse propagation are described. A semi-classical model is used and, at each point in the medium, the propagation field and the spontaneous damping field are considered separately. The equation governing the evolution of a superradiant system in the presence of an applied pulse is derived. A mechanism of energy transfer from the traveling pulse to the medium is described which includes superradiant spontaneous emission effects. As a result of the neglect of all lossy relaxation processes, all energy stored in the medium is eventually returned to the pulse by coherent spontaneous emission. Therefore, in this theory, all pulses travel through the medium without energy loss. Numerical results for square input pulses are discussed which show two possible types of propagation behavior: (a) Low input intensities produce a negative tail to the input (positive) pulse and sharpening of the tail with the advance in the medium; (b) higher input intensities produce, after sufficient penetration, a steady-state pulse envelope somewhat similar to the $2\ensuremath{\pi}$ pulses of self-induced transparency. The steady-state solutions of the equations of motion are found numerically and the pulse shapes are computed for different degrees of superradiance. In the limit of strong superradiance, very asymmetric pulse envelopes of large area are found. For negligible superradiance, approximately $2\ensuremath{\pi}$ hyperbolic-secant pulses are obtained. The effects of inhomogeneous broadening are briefly considered, and the consequences of important superradiant damping in photon-echo experiments are qualitatively discussed.
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We develop a general method allowing one to construct the consistent theory of light pulse propagation through an atomic medium in arbitrary nonlinear regime with respect to the field strength, taking into account the light polarization, temporal (frequency), and spatial dispersions. The method is based on the reduced Maxwell equation, atomic density matrix formalism, and adiabatic approximation. In order to demonstrate the efficiency of our method we investigate in detail the case of propagation of the pulses under conditions of coherent population trapping in a medium of two-level atoms with degenerate energy levels. Equations describing the evolution of various field parameters are derived. It is shown that pulses of ellipticity and spatial orientation of polarization ellipse propagate with slowing. Analytical expressions for the slowing factor are obtained. Previously unknown effects of the stimulated phase modulation and the generation of a phase pilot pulse by the variation of spatial orientation of the polarization ellipse are predicted.
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