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Optics Letters

Precursor propagation in inhomogeneous broadened media: experimental and numerical simulation

Federico Tommasi, Emilio Ignesti, Lorenzo Fini, Stefano Cavalieri

Published Oct 26, 2017• 2 citations
Publisher

Abstract

We report experimental results on the propagation temporal characteristics of the precursor in an inhomogeneous sample. The transient behavior of a step-like pulse in an atomic hot medium is two orders of magnitude faster than the radiative broadened case up to now presented in the literature. Moreover, we show the dependence on the resonant or nonresonant condition. Numerical simulations compare favorable to experimental results.

DOI
10.1364/ol.42.004498

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We report experimental results on the propagation temporal characteristics of the precursor in an inhomogeneous sample. The transient behavior of a step-like pulse in an atomic hot medium is two orders of magnitude faster than the radiative broadened case up to now presented in the literature. Moreover, we show the dependence on the resonant or nonresonant condition. Numerical simulations compare favorable to experimental results.
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 45\frac{4}{5}54​ and E0{\mathcal{E}}_{0}E0​ is the amplitude of the input pulse. This result is consistent with an earlier analysis which showed that the elliptic dn\mathrm{dn}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 SF6{\mathrm{F}}_{6}F6​ and Na vapor are discussed.
We discuss the transient effects associated with short-pulse propagation in a linearly dispersive medium by comparing optical precursors with 0π pulses. The classical and quantum effects were, in the past, treated independently due to the different time scales. Using results from recent studies [Phys. Rev. Lett.96, 143901 (2006)] as well as the discussions by Varoquaux et al. [Phys. Rev. B.34, 7617 (1989)] and Bürck [Hyperfine Interact. 123, 483 (1999)], we show analytically that precursors and 0π pulses describe the same physical phenomenon. We also show that most published results of transient pulse propagation can be expressed using one overarching equation. Finally, we give a physical interpretation of the analogy between precursors and 0π pulses as a medium-impulse response using a Green's function.
In a recent paper [Gorshkov et al., Phys. Rev. Lett. 98, 123601 (2007)] and in the two preceding papers [Gorshkov et al., this issue, Phys. Rev. A 76, 033804 (2006); 76, 033805 (2006)], we used a universal physical picture to optimize and demonstrate equivalence between a wide range of techniques for storage and retrieval of photon wave packets in homogeneously broadened $\ensuremath{\Lambda}$-type atomic media, including the adiabatic reduction of the photon group velocity, pulse-propagation control via off-resonant Raman techniques, and photon-echo-based techniques. In the present paper, we generalize this treatment to include inhomogeneous broadening. In particular, we consider the case of Doppler-broadened atoms and assume that there is a negligible difference between the Doppler shifts of the two optical transitions. In this situation, we show that, at high enough optical depth, all atoms contribute coherently to the storage process as if the medium were homogeneously broadened. We also discuss the effects of inhomogeneous broadening in solid state samples. In this context, we discuss the advantages and limitations of reversing the inhomogeneous broadening during the storage time, as well as suggest a way for achieving high efficiencies with a nonreversible inhomogeneous profile.