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Quantum Electronics

Some models of propagation of extremely short electromagnetic pulses in a nonlinear medium

Andrei I. Maimistov

Published Apr 30, 2000• 118 citations
Publisher

Abstract

Some cases of model media considered in this paper allow analytical solutions to nonlinear wave equations to be found and the time dependence of the electric field strength to be determined in the explicit form for arbitrarily short electromagnetic pulses. Our analysis does not employ any assumptions concerning a harmonic carrier wave or the variation rate of the field in such pulses. The class of models considered includes two-level resonance and quasi-resonance systems. Nonresonance media are analysed in terms of models of anharmonic oscillators — the Duffing and Lorentz models. In most cases, only particular solutions describing the stationary propagation of a video pulse (a unipolar transient of the electric field or a pulse including a small number of oscillations of the electric field around zero) can be found. These solutions correspond to sufficiently strong electromagnetic fields when the dispersion inherent in the medium is suppressed by nonlinear processes.

DOI
10.1070/qe2000v030n04abeh001712

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Some cases of model media considered in this paper allow analytical solutions to nonlinear wave equations to be found and the time dependence of the electric field strength to be determined in the explicit form for arbitrarily short electromagnetic pulses. Our analysis does not employ any assumptions concerning a harmonic carrier wave or the variation rate of the field in such pulses. The class of models considered includes two-level resonance and quasi-resonance systems. Nonresonance media are analysed in terms of models of anharmonic oscillators — the Duffing and Lorentz models. In most cases, only particular solutions describing the stationary propagation of a video pulse (a unipolar transient of the electric field or a pulse including a small number of oscillations of the electric field around zero) can be found. These solutions correspond to sufficiently strong electromagnetic fields when the dispersion inherent in the medium is suppressed by nonlinear processes.
We consider the problem of the propagation of short-duration light pulses in a two-level medium in the regime of strong interaction, near the resonance, with no restriction on the strength of the coupling between field and medium. We prove that the local variations of the population difference created by the electric field actually induce a nonlinear effect by coupling the fundamental of the field Fourier component to its harmonics. As a result, the multiscale averaging limit of the Maxwell-Bloch system is a new system of coupled nonlinear Schrodinger equations for the slowly varying envelope of the electric field. This system has one integrable limit when the electric field is circularly polarized: it is the scalar nonlinear Schrodinger equation. Moreover, it possesses three constants of the motion and a Hamiltonian formulation. Then we prove that our system is nonintegrable as it does not pass the Painleve test. Finally, a stability analysis shows that it is modulationally unstable and thus it propagates steady pulses of coherent light.
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.
An analysis is made of the coherent interaction of a short pulse of polarised electromagnetic radiation with a medium in which the energy levels are degenerate in respect of the angular momentum projection. In the case of the j a = 1 → j b =0 transition it is found, without recourse to the approximation of slowly varying envelopes, that the system of the Maxwell–Bloch equations has an exact solution in the form of a steady-state solitary wave of duration equal to half the period of emission as a result of the investigated atomic transition. A new solution is obtained, in addition to a simple generalisation corresponding to a polarised video pulse. When the populations of the excited states with different projections of the angular momentum are different, the spherical components of the vector representing the electric field intensity do not behave proportionately: if one of them corresponds to a unipolar spike of the electric field, the other component is a variable-sign solitary wave.
A nonlinear medium with nonreciprocal propagation of electromagnetic field pulses is proposed and studied. The medium consists of a long waveguide surrounded by two-level media implanted in a straight spiral. Respective Maxwell-Bloch system describing evolution of the pulses in the waveguide with nonlinear feedback is derived. The influence of the neighbors' dipole fields taken into account in the tight bounding approximation manifests itself in the appearance of the nonlinear differential terms. Under the condition of unidirectional propagation of field pulses, the Maxwell-Bloch system of equations is reduced to an integrable system. Analysis of obtained exact solutions have shown that these differential terms are responsible for the nonreciprocity of the propagation of field pulses in the waveguide. Using the soliton solution as an example, it is shown that the spiral arrangement of atoms yields critical dependence of the evolution of field pulses on the direction of their propagation or on the chirality of the medium.
Frequency-dependent and nonlinear susceptibilities are common in nonlinear optics, but they are scarcely used in coherent resonant pulse propagation because of their complexity. Early studies1 include the frequency quadratic susceptibilities, but none was proportional to the spectral pulse strength other than in the time-dependent case.2 We consider the propagation of short duration pulses through a medium described by an ensemble of resonant two-level atoms embedded in a nonresonant but nonlinear host medium. The nonlinear susceptibility is given by monomial terms of the spectral pulse strength.