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Nonlinear Dynamics in Optical Systems

Fluctuational Transitions and Critical Phenomena in a Noise-Driven Optically Bistable Device

M. I. Dykman, G. P. Golubev, V. P. Golubchenko, D. G. Luchinsky, S. V. Tsuprikov, A. L. Velikovich

Published Jan 1, 1990• 0 citations
Publisher

Abstract

Fluctuational transitions between the stable states of a passive optically bistable device have been observed and investigated theoretically and experimentally in the absence of spatial effects (for a lumped-parameter system). The problem of calculation of transition probabilities in dynamical systems driven by weak Gaussian noise has been reduced to logarithmic accuracy to a variational problem which is solved for some types of colored noise. The procedure of the experimental determination of the transition probabilities from the data on transmitted intensity is proposed for the case of a broad-band noise in the intensity of incident radiation. The onset of an extremely narrow peak in the power spectrum of the system in the range of the kinetic phase transition is revealed.

DOI
10.1364/nldos.1990.ndd440

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Fluctuational transitions between the stable states of a passive optically bistable device have been observed and investigated theoretically and experimentally in the absence of spatial effects (for a lumped-parameter system). The problem of calculation of transition probabilities in dynamical systems driven by weak Gaussian noise has been reduced to logarithmic accuracy to a variational problem which is solved for some types of colored noise. The procedure of the experimental determination of the transition probabilities from the data on transmitted intensity is proposed for the case of a broad-band noise in the intensity of incident radiation. The onset of an extremely narrow peak in the power spectrum of the system in the range of the kinetic phase transition is revealed.
Fluctuational transitions between stable states of an all-optical lumped-parameter bistable system have been observed. The dependences of the transition probabilities, and of the statistical distribution of the transmitted-light intensity, on the intensities of incident light and noise have been investigated. Two types of stochastic modulation, that of the incident-light intensity and that of the device itself, have been used. They correspond to multiplicative and additive noise for the intracavity-phase gain and give rise to qualitatively different pictures of the fluctuations. In particular, for the latter type of noise the logarithm of the statistical distribution of the transmitted-light intensity is described by a multibranch function, and the peaks of the distribution are strongly asymmetric. The onset of a zero-frequency peak in the power spectrum of the transmitted light in the range of the kinetic phase transition has been observed. The experimental data are in good qualitative and quantitative agreement with the theory.
We formulate a running-wave, single-mode model for purely dispersive optical bistability that incorporates amplitude and frequency fluctuations in the incident-field, cavity-length fluctuations, and thermal noise in the radiation field and in the material. This model is given by a set of three Langevin-type equations for the real and the imaginary parts of the slowly varying electric field and for the material variable. In the white-noise limit, it is equivalent to a Fokker–Planck equation in three variables. Even if our model can also describe a Kerr medium or a two-level system in a suitable range of parameters, we are interested mainly in the case of miniaturized all-optical bistable devices that utilize semiconductor dispersive media. In this situation, we can adiabatically eliminate the field variables and therefore reduce the problem to a single stochastic differential equation in one variable, which contains several terms of additive and multiplicative noise. We find that noise in the imaginary part of the slowly varying electric field does not contribute to this equation. In the white-noise case, our stochastic equation is equivalent to a Fokker–Planck equation in one variable, whereas in the case of colored noise we obtain a one-dimensional Fokker–Planck equation only in the limit of short correlation times. Finally, we calculate the stationary solution of the resulting Fokker–Planck equation. We show that the presence of colored noise narrows the peaks of the probability distribution at steady state with respect to the white-noise case.
The role of external noise in the process of critical slowing down (CSD) has been analysed theoretically by Broggi and Lugiato/1/; these authors recently predicted a transient noise-induced optical bistability due to amplitude fluctuations in the light input. They found that under conditions of CSD the probability distribution of light field amplitudes becomes double-peaked in a sizeable time interval during the approach to the single-peaked steady state distribution. Moreover they found that in the presence of fluctuations the delay times τD observed in an experiment on CSD should have a broad distribution, which can be very asymmetric. With increasing noise levels the mean value of the delay time should be considerably shorter than in the deterministic case.
In a recent paper (1) we formulated a running wave, singlemode model for purely dispersive optical bistability, which incorporates amplitude and frequency fluctuations in the incident field, cavity length fluctuations, thermal noise in the radiation field and in the material. In the bad-cavity case of miniaturized bistable devices which utilize semiconductor dispersive media, our model reduces to a single stochastic differential equation for the material variable, which contains several terms of additive and multiplicative noise. In the white-noise case, it is equivalent to a Fokker-Planck equation in one variable, whereas in the case of colored noise it is equivalent to a multidimensional Fokker-Planck equation. On the basis of this model, we examined both problem of practical (P) and of fundamental (F) interest.
The first-order-like phase transition (FOLT) in the dispersive optical bistability is investigated when the fluctuation in the incident light field is considered as colored noise. A unified colored-noise approximation is applied to obtain the steady state distribution (SSD) when either the intensity or phase fluctuations of the incident field are included in the system. For intensity fluctuations only, the curve of SSD is changed from single extreme to two extremes, and then to three extremes. The colored nature of the noise can reduce the fluctuation in the system. However, for phase fluctuations only, the FOLT is mainly induced by the colored nature of the noise. The curve of SSD is changed from single extreme to three extremes directly. There is no FOLT existing for white noise.
A quantum description of the transmitted field is given. In the bistable situation, the stationary Glauber quasi probability distribution has two peaks thereby producing a first-order phase transition. For small incident field the linewidth of the transmitted light is proportional to the atomic density and becomes very narrow as the field approaches some critical value from below. Crossing this value the spectrum splits discontinuously into a triplet (dynamical Stark shift) where the central peak is as high as twice the side-bands.