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Journal of the Optical Society of America B

Linear analysis of single-feedback-mirror spatiotemporal instabilities

Martine Le Berre, Elisabeth Ressayre, Andree A. Tallet, J.‐J. Zondy

Published Jul 1, 1990• 15 citations
Publisher

Abstract

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.

DOI
10.1364/josab.7.001346

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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 linear stability analysis of the plane wave stationary solution of the dispersive single-feedback mirror device displays a 2τR oscillation only if the phases of the stationary forward and backward field amplitudes are 2π shifted, δφ = 2kπ. The condition provides an instability boundary in the space of the atom field detuning and the input intensity, the shape of which agrees with experimental results. The 2τR oscillation is the signature of an instability which develops in a phase conjugate resonator, constituted of the atomic cell and a totally reflecting ordinary mirror. Thus, a stationary symmetric profile may undergo a 2τR Hopf bifurcation together with a spontaneous breaking of the cylindrical symmetry. The a-c profiles are constituted of spots located on a ring, as observed in the Tucson experiment [G. Giusfredi et al., J. Opt. Soc. Am. B 5, 1181 (1988)]. A mean phase δφp is defined, which results from an average of the grating, weighted with the mode TEM0, ±p. Therefore the system itself determines the number of spots in order that δφp fulfills the requirement δφ= 2kπ.
The instability threshold characteristics are investigated for two different optical devices, the ring cavity and the feedback mirror, with passive weakly absorbing two-level atoms cell. The linear stability analysis of the ring cavity predicts emission of several conical shells, depending on the optical arrangement and the buffer gas pressure. In the case of pure vapor, each conical shell has its own frequency. All the cones display the same frequency in the case of large buffer gas pressure. The feedback mirror device displays very different instability characteristics, with only one frequency emitted along a unique cone, in agreement with the experiment. The self-oscillation is shown to be controlled by the reflectivity properties of the cell, which behaves at threshold as a mixture of ordinary mirror and phase conjugate mirror
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.
The phase-conjugate reflectivity produced by degenerate four-wave mixing in a Kerr medium in the Bragg scattering regime is shown to be unstable when the intensity of the external pump beams is greater than a threshold value that can be predicted analytically. Below this threshold, a single branch of the reflectivity curve is stable, eliminating any possibility for bistability. A numerical simulation indicates that the unstable regime is characterized by fast oscillations with a period related to the transit time in the nonlinear medium. A finite relaxation time of the nonlinearity eliminates the high-frequency oscillations but can increase the amplitude of the low-frequency components. The insertion of a phase-conjugate mirror into an optical cavity leads to an instability whose threshold can be evaluated analytically. The temporal evolution of the field intensity in the cavity is shown to change from a stable regime to periodic oscillations and then to quasi-periodic oscillations as the pump intensity is increased.
A transverse optical bistability is defined here as a bistability that would not occur with a plane-wave input; it is characterized by pronounced changes in transverse profiles, but not in total transmission.1-5 In order to use transverse bistabilities to study instabilities of the type first studied by Ikeda6 for a ring cavity, one needs a medium response time τM less than the round-trip time tR between the medium and the feedback mirror. Here instabilities are studied experimentally using a short length of sodium vapor and a far-field mirror (Fig. 1).7,8 This system has advantages over previous9,10 all-optical passive systems for studying instabilities: its input is cw, permitting the study of slowly evolving waveforms; it has little sensitivity to laser frequency shifts, because it has no cavity; its optical nonlinearity arises from a vapor of one-electron atoms, giving hope for a detailed quantitative comparison of experiments and calculations.