Self-Pulsing, Breathing and Chaos in Optical Bistability and the Laser with Injected Signal
The subject of spontaneous pulsations in Optical Bistability has stimulated considerable interest following its prediction by Bonifacio and Lugiato1 in the framework of the plane-wave, ring cavity model and the realization by McCall2 of an electro-optical converter of cw coherent light into pulsed radiation. In a subsequent important development, Ikeda3 showed that, in the dispersive case, the Bonifacio-Lugiato instability leads to chaotic behavior (optical turbulence). This was later observed in a hybrid device by Gibbs, et al.4.
In this paper, we review some of the most relevant manifestations of self-pulsing in Optical Bistabilityility. First, we consider the absorptive case in which self-pulsing is a multi-mode phenomenon because the resonant mode remains stable while some of the sidebands can develop instability5. On the strength of the “dressed mode formalism” of optical bistability6, which represents an extension of Haken’s theory of generalized Ginzburg-Landau equations for phase-transition like phenomena in systems far from equilibrium7, we have arrived at an essentially analytical understanding of this phenomenon. When only the nearest two sidebands of the resonant mode become unstable, we have identified the entire domain of existence of the self-pulsing state in the plane of the control parameters. This region is divided into a soft- and a hard-excitation domain, where self-pulsing develops with a behavior that is reminiscent of second- or first-order phase transitions, respectively. In the hard-excitation region, one self-pulsing and two steady-state solutions are found to coexist, yielding hysteresis cycles that involve self-pulsing states along one of the branches. On approaching the boundary between the self-pulsing and the precipitation domains, the self-pulsing state becomes unstable. The instability is evidenced by a marked “breathing” behavior of the pulse envelope, whose transient nature has been linked to the unstable character of the limit cycle that bifurcates from the self-pulsing solution at the instability threshold.
When dispersive effects become important, self-pulsing behavior can arise even in the single-mode regime described by the well known mean field model8,9. For appropriate values of the control parameters, a large segment of the high transmission branch becomes unstable leading to periodic and irregular (chaotic) self-pulsing. On approaching the chaotic domain from either boundary, one finds a sequence of period doubling bifurcations of the type described by Grossman and Thomae10 and by Feigenbaum11 in the framework of the theory of discrete maps. As true also with most other dynamical models, the chaotic domain of this single-mode dispersive bi-stability contains “windows” of periodicity. We have characterized the behavior of bifurcating solutions and their critical slowing down in the neighborhood of each threshold with asymptotic power laws. These will be reviewed in detail. In the purely dispersive limit, we also show that our model reduces to the one analyzed recently by Ikeda and Akimoto12. If the absorbing medium is converted into an amplifier, the optically bistable system becomes a laser with an injected signal. We assume that the laser is above threshold and that the incident field frequency is detuned from the operating frequency of the laser. Obviously, when the injected field has a vanishingly small amplitude, the laser oscillates at its own frequency. On the other hand, when the incident field intensity is large enough, the entire system is expected to oscillate with the incident frequency (this phenomenon is well known as injection locking). We have analyzed the behavior of this system over the entire range of variation of the incident intensity below the locking threshold13. For a small incident field, the output power exhibits oscillations with a frequency equal to the mismatch between the incident and the laser frequencies, as originally proposed by Spencer and Lamb14. On increasing the incident intensity, the output becomes chaotic. A further increase in the injected signal leads to an inverted sequence of period doubling bifurcations with one- and two-periodic solutions exhibiting undamped breathing over limited domains. Finally, in the vicinity of the injection locking threshold, the system develops a chaotic spiking regime of an entirely different character from the irregular oscillations noted above. From our results, it is reasonable to speculate that the observations of chaos should be accessible with an all-optical system.
KeywordsResonant Mode Period Doubling Bifurcation Optical Bistability6 Bistable System Incident Intensity
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