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Instability and chaos of counterpropagating beams in a nonlinear medium.

Railing Chang

Published Jan 1, 1991• 0 citations
OpenAlex

Abstract

The dynamical behavior of counterpropagating light waves interacting with a nonlinear medium is studied numerically. It is found that the wave and the medium form a system that becomes unstable under certain conditions and exhibits self-oscillation. Because the interacting medium is modeled by an ensemble of two-level systems, we recognize the self-oscillation frequency as the Rabi-frequency of the constituent systems. The physical mechanisms responsible for this self-oscillation are the gain and the distributed feedback of the combined lightwave-medium system. When the environment changes, in particular when the intensity is increased, this system becomes more unstable and we find that the oscillation exhibits more complex behavior, including quasi-periodic motion and chaos. We analyze the output fields by using Fourier spectra, phase portraits, and autocorrelation functions. In the chaotic regime, the Lyapunov exponents and dimensions are also calculated. A physical interpretation of the quasiperiodic motion is given by an exact calculation of the absorption spectrum of our two-level medium. The negative absorption (gain) peaks are found at the frequencies of the quasi-periodic motions, thus implying that the gain of the combined light-medium system is responsible at least in part for the observed complex behavior. In addition, we investigate the stability of the propagating plane wave when a transverse wave is added to the system as a perturbation. Instabilities are analyzed by linearizing the nonlinear equations which model the lightwave-medium system. The results show that the instability is highly-correlated with the four-wave mixing phase conjugation.

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The dynamical behavior of counterpropagating light waves interacting with a nonlinear medium is studied numerically. It is found that the wave and the medium form a system that becomes unstable under certain conditions and exhibits self-oscillation. Because the interacting medium is modeled by an ensemble of two-level systems, we recognize the self-oscillation frequency as the Rabi-frequency of the constituent systems. The physical mechanisms responsible for this self-oscillation are the gain and the distributed feedback of the combined lightwave-medium system. When the environment changes, in particular when the intensity is increased, this system becomes more unstable and we find that the oscillation exhibits more complex behavior, including quasi-periodic motion and chaos. We analyze the output fields by using Fourier spectra, phase portraits, and autocorrelation functions. In the chaotic regime, the Lyapunov exponents and dimensions are also calculated. A physical interpretation of the quasiperiodic motion is given by an exact calculation of the absorption spectrum of our two-level medium. The negative absorption (gain) peaks are found at the frequencies of the quasi-periodic motions, thus implying that the gain of the combined light-medium system is responsible at least in part for the observed complex behavior. In addition, we investigate the stability of the propagating plane wave when a transverse wave is added to the system as a perturbation. Instabilities are analyzed by linearizing the nonlinear equations which model the lightwave-medium system. The results show that the instability is highly-correlated with the four-wave mixing phase conjugation.
The dynamical behavior of counterpropagating light beams interacting with a nonlinear medium is studied numerically. When the incident intensities are increased, this system becomes unstable and exhibits oscillations undergoing complex behavior, including quasiperiodic motion and chaos. We analyze the output fields by using Fourier spectra, phase portraits, and autocorrelation functions. In the chaotic regime, the Lyapunov exponents and dimensions are calculated. A physical interpretation of the quasiperiodic motion is also given by an exact calculation of the absorption spectrum of a two-level atom.
Two counterpropagating waves in a Kerr medium were shown to exhibit instabilities, which lead to self-oscillations, bifurcations, and chaos.1 These instabilities, as well as self-oscillation phenomena in bistable systems, can be explained by a gain-feedback mechanism.2 Fields at frequencies adjacent to the input pump-wave frequency can be amplified through four-wave-mixing interactions. If feedback is supplied, sidebands at these frequencies can be excited and beat with the pump wave to yield an oscillating output. In a counterpropagating-beam configuration, the feedback is distributed, and it originates from the grating that is formed by the interfering beams.
We have previously shown that counterpropagating waves in a Kerrlike medium exhibit spontaneous oscillations above a certain threshold intensity. The period of these oscillations was related to the medium length and to its response time. These oscillations were explained as results of parametric gain and feedback supplied by the grating formed through the nonlinear interaction. In this paper we extend the theory to a two-level medium. The complexity of the parametric interaction in this medium provides a reach spectrum of behaviors. The power exchange due to this interaction, between waves of different frequencies, is usually maximized when the frequency difference is the Rabi frequency. However, another maximum, which is smaller in magnitude, occurs in certain limits when the frequency difference is T1. Our derivation is based on performing a linear stability analysis to the steady-state solutions of the Maxwell-Bloch equations. We assumed a small signal absorption and large detuning from resonance. The solutions were studied in the limit of pure radiative broadening (T2 = T1) and rapid collisional dephasing (T2 ≪ T1). It is found that for most cases Rabi frequency self-oscillations are dominant. However, for T2 ≪ T1 and for large interaction length, we found slower oscillations, with a period of T1. We show that the results can be understood by the gain feedback model. We also discuss the effect of inhomogeneous broadening and present experimental parameters required for observation of T1 oscillations.
The physics of Raman-type self-oscillations in highly dispersive saturable two-level-medium devices is revisited by using the gain and reflectivity matrices. The probe gain when two identical pumps counterpropagate through a two-level system is demonstrated to be very different from the gain for a probe in a one-way device, and typically peaked near a frequency twice the one-way Rabi frequency, while multiwave mixing displays a generalized mirror behavior of the atomic cell. The multimode instability in a Fabry-P\'erot interferometer is also discussed and shown to be mainly a result of counterpropagating probe coupling while the coupling between copropagating probe and idler is negligible.
The steady-state polarizations of fields counterpropagating in nonlinear Kerr media can exhibit complicated behavior.1 We examine the temporal behavior for the case of an isotropic nonlinear Kerr medium and show that the polarizations are temporally unstable above a certain threshold intensity, even when the input polarizations are linear and parallel to each other. This polarization configuration is known to be spatially stable and is often chosen for the two strong pump waves in phase conjugation by degenerate four-wave mixing. We find that the threshold intensity for the polarization instability is approximately the intensity at which the phase conjugate reflectivity is equal to unity. Depending on the ratio of the two input intensities, this instability just above threshold can manifest itself as an abrupt rotation of the output polarizations to a new steady-state value or as a periodic oscillation of the output polarizations. At higher intensities the polarization can evolve chaotically in time, and depending on the ratio of the transit time through the medium to the response time of the medium the output intensity can fluctuate chaotically as well.