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Izvestiya VUZ Applied Nonlinear Dynamics

Self-induced - transparency soliton propagation under the influence of far-resonant transition line

Viktor V. Kozlov, Evald E. Fradkin

Published Nov 21, 1995• 0 citations
Publisher

Abstract

Theoretical research of propagation effect of ultrashort optical pulse under conditions of coherent interaction with two resonant transitions simultaneously is presented. One of the transitions is tuned in exact resonance with pulse carrier frequency; the other is distanced much more than spectral pulse width, thus the adiabatic following approximation may be applied. Interaction between the pulse and far-resonant transition is described by wave equation with nonlinear index of refraction in the form of Lorentz function for field amplitude. This wave equation is solved together with Bloch equations for the exact resonant transition. The solution in the form of stationary phase-modulated pulse is obtained, and 2π-pulse of self-induced - transparency is the special case of this solution. The obtained solution has a square under pulse envelope less than 2π, and with increasing of far-resonant transition absorption coefficient the pulse square decreases. The effect can be used for reducing of the transparency threshold of the resonant medium below pi.

DOI
10.18500/0869-6632-1995-3-6-112-119

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Physical Review A•May 1, 1978•6 citations
Theoretical research of propagation effect of ultrashort optical pulse under conditions of coherent interaction with two resonant transitions simultaneously is presented. One of the transitions is tuned in exact resonance with pulse carrier frequency; the other is distanced much more than spectral pulse width, thus the adiabatic following approximation may be applied. Interaction between the pulse and far-resonant transition is described by wave equation with nonlinear index of refraction in the form of Lorentz function for field amplitude. This wave equation is solved together with Bloch equations for the exact resonant transition. The solution in the form of stationary phase-modulated pulse is obtained, and 2π-pulse of self-induced - transparency is the special case of this solution. The obtained solution has a square under pulse envelope less than 2π, and with increasing of far-resonant transition absorption coefficient the pulse square decreases. The effect can be used for reducing of the transparency threshold of the resonant medium below pi.
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) .
Propagation of a pulse of self-induced transparency in a resonance medium is discussed for the case when it is necessary to take into account the direct electric dipole-dipole interaction between atoms. An equation for the envelope of a wave packet—the sine-Gordon equation—is obtained for the case of durations short compared to ω 0 −1 (ω0 is the transition frequency). The dependence of the velocity and the amplitude of the soliton on the magnitude of the dipole-dipole interaction of atoms is examined in the adiabatic approximation.
The phenomenon of self-induced transparency (SIT) in the exciton range of the spectrum consists of the propagation through the crystal of solitary wave packets of coherent excitons and photons with anomalous small losses. In the running frame of reference at the site occupied by the soliton a single aperiodic transformation of photons into excitons and a reverse motion of the energy into the field takes place. The solutions of the SIT equations have been received in the slowly-varying amplitude approximation. Taking into account the phase modulation, the authors have shown the appearance of the branch of the carrier wave frequency with anomalous dispersion not only at the given pulse width but at the given pulse amplitude. They have also found the exact solution of the above mentioned equations taking into account the effect of saturation of the dipole momentum transition from the crystal ground state to the exciton state. It leads to the increasing of the soliton time width, to the decreasing of its amplitude and worsens the SIT observation.
The coherent propagation of two optical short pulses through a resonant medium consisting of two three-level atoms in the configuration is investigated. A self-consisting analytical solution without steady state or adiabatic approximations is presented. The electric field in the model studied consists of two co-propagating plane waves, each of which is in near resonance with a transition in the absorber. The density matrix of two three level atoms is studied. Also, the reduced density matrices are stated. The present approach for the semiclassical treatment of resonant coherent interactions in three-level atoms represents a generalization of the Maxwell- Bloch equations for a two-level system. The Maxwell-Bloch equations reduced to the non-linear pendulum equation. The solution shows that the two pulses can propagate simultaneously without loss under some conditions about the pulses and the medium. The propagation of the two pulses through the medium without any constrains approves the self -induced transparency (SIT) phenomenon.
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 self-induced transparency solution of the Bloch-Maxwell equations is found in the case of inhomogeneously broadened resonant atomic lines and a host medium with a Kerr-type nonlinearity in the index of refraction. The characteristic feature of the solution is that the atomic functions are not factorized. The time dependence of the pulse phase is different from that in the case of a sharp atomic line. In addition to the known term quadratic in the pulse envelope, it contains a term increasing in time, attaining a maximum at the trailing edge of the pulse, and rapidly decreasing afterwards. The average carrier frequency is proportional to the square of the spectral width of the atomic line divided by the squared width of the pulse. A satisfactory solution is found by modulation of the center of the Lorentz distribution function of the purely resonant solutions. It is shown that in the presence of the inhomogeneous broadening the problem cannot be solved by a linear perturbation expansion with respect to the Kerr constant. This method leads to divergencies near the trailing edge of the pulse.