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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Self-induced transparency for the supremely short pulses in conditions of partially spectral exceeding of the quantum transitions

С. В. Сазонов, A. F. Sobolevskii

Published Jan 14, 2000• 0 citations
Publisher

Abstract

Without using the slowly varying amplitude approximation a nonlinear propagation of supremely short pulses in the three-level quantum medium have been investigated. At this by the pulse spectral wide is exceeding only one of quantum transitions. The nonlinear wave equations of 'sin-Gordon + Korteweg-de Vries'-type and 'sin-Gordon + Korteweg- de Vries'-type, which describe pulse dynamics, was obtained. On the basis of approximate solution we come to a conclusion about minimum temporal duration of propagating pulse.

DOI
10.1117/12.375324

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Izvestiya VUZ Applied Nonlinear Dynamics•Nov 21, 1995•0 citations
Without using the slowly varying amplitude approximation a nonlinear propagation of supremely short pulses in the three-level quantum medium have been investigated. At this by the pulse spectral wide is exceeding only one of quantum transitions. The nonlinear wave equations of 'sin-Gordon + Korteweg-de Vries'-type and 'sin-Gordon + Korteweg- de Vries'-type, which describe pulse dynamics, was obtained. On the basis of approximate solution we come to a conclusion about minimum temporal duration of propagating pulse.
Propagation of broad-band ultimately short light pulses in a partially absorbing medium is analyzed in the framework of the three-level model. Nonlinear wave equations are obtained describing propagation of light pulses in media with quadratic (all three transitions are allowed) or cubic (one of the transitions is forbidden) nonlinearity in the range of optical transparency or with the sine-Gordon-type nonlinearity in the region of absorption. Using the averaged variational principle, the approximate solutions of equations in the form of unipolar soliton-like signals are found and conditions of their transverse stability are determined. A stable propagation of a broad-band pulse is shown to be possible under conditions when monochromatic signals exhibit self-focusing.
We study theoretically the phenomenon of self-induced transparency for ultrashort pulses (videopulses) propagating in a multilevel quantum medium under conditions in which the method of slowly varying amplitudes and phases breaks down and the pulse spectrum substantially overlaps the quantum transitions. A special class of transitions with one common level is considered. We find that the dynamics of the videopulses in such a medium is described by a double sine-Gordon equation. We establish the conditions under which steady-state traveling 0 π -, 2 π -, and 4 π -videosolitons are formed. It is found that 4 π -videosolitons can propagate both in equilibrium media and in some nonequilibrium media, while 0 π -videosolitons can propagate only in nonequilibrium media. We study amplification processes in highly nonequilibrium systems and show that, depending on the initial state of the medium, 2 π -and qπ -pulses (0< q <1) with increasing amplitudes may be formed. We conclude that the amplification of an ultrashort pulse occurs due to an increase in the photon number density and to an increase in the frequency of each photon. Finally, we study the possibility of an electromagnetic autosoliton being formed in a nonequilibrium dissipative medium.
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 propagation of ultrashort coherent light pulses through a resonant atomic medium, modelled by a two-quantum level of absorbers, is studied including the effects of level degeneracy. The governing equations, a sequence of n-tuple sine-Gordon (SG) equations, are solved by quadrature for the Q-transition, with an approximate method indicated for the (mathematically) more difficult P- and R- transitions. It is shown, in particular, that there is a surprisingly close relation between the usual SG equation and the double SG equation. A simple theorem is proved which shows that each travelling wave solution of the former immediately leads to a solution of the latter, which thus admits a restricted Backlund transformation. This direct solution of nonlinear dispersive equations, which apparently cannot be solved by the inverse method, has an interesting by-product: an approximate method for finding travelling wave solutions of relatively arbitrary Klein-Gordon equations.
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) .
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.