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Physical Review A

Self-induced transparency and the soliton lattice

B. Horovitz, N. Rosenberg

Published Nov 1, 1982• 2 citations
Publisher

Abstract

The propagation of a step-function light pulse in a nondamping resonant medium is studied. Numerical analysis and conservation laws show that the asymptotic solution cannot be described by a single solition-lattice-type solution. As the pulses propagate in space they become narrower and with larger amplitude, but the time separation between pulses remains constant. This result is explained by an asymptotic area theorem which restricts the time average of the electric field to be equal to that of the imput field.

DOI
10.1103/physreva.26.2799

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Physical Review A•Apr 1, 1993•9 citations
The propagation of a step-function light pulse in a nondamping resonant medium is studied. Numerical analysis and conservation laws show that the asymptotic solution cannot be described by a single solition-lattice-type solution. As the pulses propagate in space they become narrower and with larger amplitude, but the time separation between pulses remains constant. This result is explained by an asymptotic area theorem which restricts the time average of the electric field to be equal to that of the imput field.
The results of theoretical research of the propagation effect of an ultrashort optical pulse under the conditions of coherent interaction with two resonant transitions are presented. One of the transitions is tuned to exact resonance with the pulse carrier frequency; the other is distanced by much more than the pulse spectral width, thus the adiabatic following approximation may be applied to it. The interaction between the pulse and the far-from-resonance transition is described by the wave equation with the nonlinear index of refraction in the form of the Lorentz function of the field amplitude. This wave equation is solved together with the Bloch equations for the exact resonant transition. The solution in the form of the stationary phase-modulated pulse is obtained, and (pi) -pulse of self-induced transparency is the special case of this solution. The obtained solution has the area under the pulse envelope less than (pi) , and with increasing the absorption coefficient of the far-from-resonance transition the pulse area decreases. The effect can be used for reducing the transparency threshold of the resonant medium below (pi) .
Exact single-soliton solutions of the modified system of Maxwell-Bloch equations, in which the dipole-dipole interactions of the atoms of a dense resonant medium are taken into account, are obtained. Two propagation regimes are analyzed: “coherent,” where the pulse duration is much shorter than both relaxation times ( T p ≪ T 1 , T 2 ), and “incoherent,” where the pulse duration falls between the relaxation times ( T 2 ≪ T p ≪ T 1 ). It is predicted, for the first time, that soliton propagation of an ultrashort pulse is possible in a dense resonant absorbing medium in an incoherent interaction regime. The differences between the amplitude and phase characteristics of the solitons considered and the corresponding characteristics of the solitons for McCall-Hahn self-induced transparency are noted.
The standard self-induced transparency (SIT) phenomenon represents the lossless propagation of a long multicycle stable soliton in an absorbing two-level resonant medium, if the soliton's envelope area equals $2\ensuremath{\pi}$ ($2\ensuremath{\pi}$ pulse). In this case, the leading half of the $2\ensuremath{\pi}$ pulse excites the medium, while the trailing half completely brings the medium back to the ground state. In this work, we theoretically demonstrate an analogous phenomenon for subcycle pulses propagating in an arbitrary multilevel resonant medium. The duration of the obtained solitary pulses must be smaller than the periods of resonant transitions in the medium. The obtained subcycle solitary solutions first turn an initially unexcited multi-level resonant medium to excited levels and then fully return it back to the ground state.
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.
The self-induced transparency in the Kerr host medium has been studied beyond the slowly-varying-envelope approximation. An analytic solution is obtained. There exists a hyperbolic-secant soliton whose power, pulse width, group velocity, and propagation constant are uniquely determined for given parameters of the medium. The reductions of the group velocities of the solitons by the self-induced-transparency effect in the resonant medium with homogeneous and inhomogeneous broadening are also studied. It is found that there is negative dispersion induced by the self-induced transparency, which is not predicted by the theory with the slowly-varying-envelope approximation. The amount of the induced dispersion is determined by the total dispersion of the system required to be compensated by the Kerr effect. Numerical examples of the self-induced transparency in an erbium-doped fiber are shown. In a typical erbium-doped fiber, the soliton solution has not been found because the inhomogeneous-broadening linewidth of the medium is too large.