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Nonlinear Dynamics in Optical Systems

Spatiotemporal Instabilities in a Two-Level Medium Interacting with Counterpropagating Waves

Martine Le Berre, Elisabeth Ressayre, Andree A. Tallet

Published Jan 1, 1990• 0 citations
Publisher

Abstract

The threshold of the static and periodic instabilities of the feedback mirror experiment is shown to be well reproduced by taking into account the non zero homogeneous response time of the medium, together with the transverse structure of the field. The phase conjugation operation is discussed.
DOI
10.1364/nldos.1990.ob214

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•Jan 1, 1991•0 citations
Spatiotemporal instabilities of a single-feedback-mirror device are investigated with the help of the linear stability analysis of the stationary regime for a saturable medium that has a complex nonlinear susceptibility; the outcome is compared with experimental results. The first striking result of the analysis is the occurrence of an oscillation with a period nearly twice the round-trip time, regardless of its value with respect to the atomic linewidth for given physical situations, when only longitudinal grating effects are considered. This result cannot be interpreted as a pure phase-conjugate resonator operation, which is expected to occur only when transverse grating effects are included. When such effects are included, the second striking feature is the prediction of the spontaneous breaking of the revolution symmetry; symmetry is broken by rotating light spots, with an instability threshold that depends on the number of spots; this prediction agrees with experiment.
The threshold characteristics of the static and periodic instability of the single feedback mirror device are investigated for a homogeneously broadened two-level medium as a function of the input intensity, the detuning, the buffer-gas pressure, the atomic concentration, and the round-trip time d, and compared with experimental results. For a large range of parameters that agrees with the experiment, the self-oscillation process is shown to be mainly induced by the longitudinal grating, which ensures a large instability domain, and controlled by the buffer-gas pressure, which must be low enough in order that the threshold period of the oscillation is constant on a doughnutlike transverse profile and nearly equal to 2d. This oscillation, called the 2d oscillation, is neither a result of pure phase-conjugate-mirror operation of the atomic medium nor a result of a Rabi sideband gain process. It is shown to take place through a mixture of phase-conjugate-mirror and ordinary mirror operations, which are both allowed by the specific optical arrangement of the device.
An analysis is made of double phase conjugation in media with a local response. A threshold is found for the generation of a phase-conjugated wave by the double process in a medium with a local response and with additional amplification. An analysis is made of the further possibilities of generation of phase-conjugated radiation in the case of a four-wave interaction characterized by a feedback loop established when double phase conjugation takes place in a nonlinear medium. The threshold intensity of the input radiation ensures possible generation of a phase-conjugated wave as a result of a four-wave interaction characterized by a feedback loop and merging of counterpropagating beams in a nonlinear medium.
We explore various models for the pattern forming instability in a laser-driven cloud of cold two-level atoms with a plane feedback mirror. Focus is on the combined treatment of nonlinear propagation in a diffractively thick medium and the boundary condition given by feedback. The combined presence of purely transverse transmission gratings and reflection gratings on wavelength scale is addressed. Different truncation levels of the Fourier expansion of the dielectric susceptibility in terms of these gratings are discussed and compared to literature. A formalism to calculate the exact solution for the homogenous state in presence of absorption is presented. The relationship between the counterpropagating beam instability and the feedback instability is discussed. Feedback reduces the threshold by a factor of two under optimal conditions. Envelope curves which bound all possible threshold curves for varying mirror distances are calculated. The results are comparing well to experimental results regarding the observed length scales and threshold conditions. It is clarified where the assumption of a diffractively thin medium is justified.
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.