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Physical Review A

Instability of counterpropagating beams in a two-level-atom medium

Israel Bar-Joseph, Yaron R. Silberberg

Published Aug 1, 1987• 40 citations
Publisher

Abstract

Self-oscillations are predicted for a simple nonlinear optical system consisting of two counterpropagating waves in a two-level medium. Oscillations at frequencies of ${T}_{1}^{\mathrm{\ensuremath{-}}1}$ and the Rabi frequency are expected for certain limits of system parameters. The physical origin of these instabilities is traced to parametric amplification in the medium. The possibility of experimental verification is discussed.

DOI
10.1103/physreva.36.1731

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Self-oscillations are predicted for a simple nonlinear optical system consisting of two counterpropagating waves in a two-level medium. Oscillations at frequencies of T1\ensuremath−1{T}_{1}^{\mathrm{\ensuremath{-}}1}T1\ensuremath−1​ and the Rabi frequency are expected for certain limits of system parameters. The physical origin of these instabilities is traced to parametric amplification in the medium. The possibility of experimental verification is discussed.
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 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.
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.
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.
Nonlinear optical systems can transform a continuous wave input beam, if it is intense enough, into an oscillating light output. The system which has been most thoroughly studied is an optical resonator, usually of the ring type, containing either a Kerr medium1,2 or a two-level medium. Unfortunately, the mathematical approaches of these studies do not lend themselves to a tractable intuitive explanation of the self-pulsation phenomena, and do not identify the physical origin of instabilities. In this paper we consider a different approach, showing that optical instabilities stem from one source, namely, gain generated by four-wave mixing processes. This gain, together with feedback supplied by the resonator, leads to oscillation of sidebands of the input field. The beating between these sidebands and the input field causes the oscillating output intensity. The salient feature of our approach lies in its intuitive appeal, namely, it is possible to predict the behavior of the system for a wide range of parameters. Moreover, our model enables understanding of the system's evolvement beyond the first bifurcation, something usually not possible with the other approaches.