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Journal of Modern Optics

Degenerate two-photon propagation and the oscillating two-stream instability: The general solution for amplitude-modulated pulses

Heinz Steudel, David J. Kaup

Published Sep 1, 1996• 14 citations

Abstract

Coherent propagation of an optical pulse in a medium with a second-order resonance is investigated when we include both the dynamical Stark effect and a population-dependent refraction index. We show how to obtain the general solution when the incident pulse has no phase-modulation. The amplitude-modulation of the incident pulse can be arbitrary as can be the initial state of the medium. This general solution is reduced to obtaining the solution of an ordinary differential equation and quadratures. For an initially de-excited medium, the solution reduces to that given by Poluektov et al. {I. A. Poluektov, Yu. M. Popov and V. S. Roitberg, 1975, Kvant. Elektron. (Moscow), 2, 1147 [1975, Sov. J. Quant. Electron., 5, 620]}. We give some illustrative examples of solutions for various initial profiles. It is also shown that the oscillating two-stream instability (OTSI) from plasma physics is equivalent to the Maxwell-Bloch equations for degenerate two-photon propagation. We give a solution for the OTSI which contains five arbitrary real functions.

DOI: 10.1080/09500349608232856

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The resonant propagation of an optical pulse in a two-photon absorber with a coherent initial excitation of the medium may lead to a pulse steepening growing exponentially in space. In addition another example is given wherein the steepening is followed by flattening. These results are new although, principally, they could be obtained from the general solution of the respective Goursat problem of an amplitude-modulated initial wave (H Steudel and D J Kaup 1996 J. Mod. Opt. 43 1851). In addition, we also give a simple theorem whereby it is demonstrated that the spatial derivative of the phase of the electric field will always be bounded. Furthermore, it is also shown that there can be certain critical points at which the atomic state, as well as the electric field, can become stationary in time and the spatial direction. © 2012 IOP Publishing Ltd.

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The matrix exponent method which we applied recently for the solution of the Bloch equations [Sov. Phys. JETP 62, 1125 (1985)] allows us to obtain in the present paper the Bloch-equation solutions when the amplitude and the phase of the exciting field are arbitrary functions of time. The solution's validity conditions are derived. It is shown that one can also apply the matrix exponent method to the solution of the Maxwell-Bloch equations. The equation describing the evolution of the area of a pulse with modulated envelope and phase is obtained. This equation has an analytic solution which is a generalization of the well-known McCall-Hahn solution.

A comprehensive model of intense laser light propagation in a fully saturable medium is presented. This model employs the Maxwell-Bloch equations for a two-level atomic system. Light propagating through a medium creates intensity-dependent index of refraction gradients that reduce to a Kerr nonlinearity only for a weak field. When the laser detuning is positive, self-induced index gradients cause the beam to self-focus. Stationary filaments occur when self-focusing exactly balances diffraction. There exists a unique set of two-dimensional fundamental (and higher mode) solutions for which the filament transverse profile (not Gaussian in general) depends on the following quantities: density of saturable absorbers, laser detuning from resonance, laser power, and natural linewidth. (In contrast the Kerr solution does not have a unique filament diameter.) Emphasis will be placed on propagation of Gaussian incident beams; deviations from the stationary solution in either shape or power generally give rise to oscillating spatial patterns, including donutlike profiles, or beam breakup.