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Physical Review A

Influence of phase diffusion on spontaneous oscillations in driven optical systems

M. Brambilla, L. A. Lugiato, G. Strini, Lorenzo M. Narducci

Published Aug 1, 1986• 6 citations
Publisher

Abstract

We analyze how phase noise degrades or, sometimes, even creates temporal structures in the laser with an injected signal and in optical bistability. In the case of a laser with an injected signal, the influence of the phase fluctuations of the driving field is considerably larger than what one might expect on the basis of intuitive arguments.

DOI
10.1103/physreva.34.1237

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Topical Meeting on Optical Bistability (OB)•Jan 1, 1983•0 citations
We analyze how phase noise degrades or, sometimes, even creates temporal structures in the laser with an injected signal and in optical bistability. In the case of a laser with an injected signal, the influence of the phase fluctuations of the driving field is considerably larger than what one might expect on the basis of intuitive arguments.
We derive and analyze the master equation for absorptive and dispersive optical bistability in the high-Q cavity limit where the driving laser has both amplitude and phase fluctuations which are larger than the small amplitude and phase fluctuations caused by incoherent spontaneous emission in the cavity. We treat the amplitude and phase fluctuations of the driving laser as independent time dependent Gaussian processes. The causes of jitter and amplitude fluctuations will vary from laser to laser and the fluctuations need not be Gaussian. However, we use Gaussian processes for mathematical tractability and because we expect them to be qualitatively correct. Recently,1 we have shown in the case of absorptive optical bistability when the driving laser has phase fluctuations the fundamental process which determines the statistical properties of the internal field is the competition between locking of the internal field phase to the phase of the driving laser and the diffusion of the phase of the internal field as a consequence of the driving laser fluctuations. The driving laser fluctuations have little effect if the phase locking rate is larger than the phase diffusion rate caused by the driving laser’s fluctuations. However, when the diffusion rate becomes comparable to the locking rate we find the amplitude of the 2 internal field is reduced and the dimensionless parameter2 measuring the importance of transitions from one branch of the OB curve to the other due to fluctuations takes on a value which leads to an appreciable transition rate due to fluctuations.
A generalized Fokker-Planck equation is derived for a system undergoing optical bistability when the dominant source of fluctuations is due to fluctuations in the incident field rather than spontaneous emission. The fluctuations are treated as a time-dependent Gaussian process whose properties are determined by the individual laser's characteristics. Both amplitude and phase fluctuations are included on the incident laser field. The effect of fluctuations of the incident laser on optical bistability are different for amplitude and phase fluctuations and are different for absorptive and dispersive optical bistability. The combination of incident laser phase fluctuations and dispersive optical bistability leads to large amplitude fluctuations near the turning points of the optical bistability curve and causes the system to make a transition from one branch of the optical bistability curve to the other before reaching the mean-field turning points. We take the high-QQQ cavity limit of our generalized Fokker-Planck equation, solve the linearized equations, and provide criteria for the magnitude of fluctuations in optical bistability in terms of laser and cavity parameters. The magnitude of incident laser fluctuations are typically orders of magnitude larger than fluctuations due to spontaneous emission and are therefore important for practical applications of optical bistability devices.
The fluctuations in the phase of the pump in a degenerate optical parametric oscillator are modeled by a classical phase-diffusion process, and their influence on squeezing is studied. To this end, appropriate quadrature variables, relative to the instantaneous phase of the pump, are introduced. The squeezing is calculated both inside and outside the cavity. In both cases it is found to be degraded by the phase fluctuations, which may be understood in terms of the time lag in the response of the system to an instantaneous change in the pump phase. This effect might account for a small part of the residual noise present in current experiments. Correlation functions for the field of the oscillator for finite pump linewidth are derived and discussed.
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 al4.
In this paper we discuss the influence of laser fluctuations on the hysteresis cycle in Optical Bistability. For the simplest case of an Absorptive Optical Bistability in a good cavity driven by an external laser light with non-white Gaussian amplitude fluctuations we derive a Fokker-Planck equation in the limit of large laser linewidth and good amplitude stabilization. This Fokker-Planck equation has a nonconstant diffusion term which shifts the most probable values of the steady state probability distribution from its deterministic positions. From the form of the steady state solution of the Fokker-Planck equation with non-white amplitude fluctuations we discuss the relative stability of the bistable branches. As in the case of quantum fluctuations the random amplitude of the laser field leads to a small range of values of the injected laser intensity in which the two bistable peaks have a comparable area. The transmitted field averaged with respect to the stationary distribution function coincides with one of the two deterministic branches except on the narrow transition region where we have large fluctuations.