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Physical Review A

Fokker-Planck description of stochastic processes with colored noise

Paolo Grigolini A, Luigi A. Lugiato, R. Mannella, P. V. E. McClintock, M. Merri, M. A. Pernigo

Published Aug 1, 1988• 25 citations
Publisher

Abstract

A detailed theoretical discussion is presented of the effect of colored noise on nonlinear dynamical systems. The ideas arising from it are applied to a particular example of such a system: a model of dispersive optical bistability, considered in the contexts of both additive and multiplicative forcing. Analogue experiments and digital simulation techniques are used to explore the applicability and range of validity of theories proposed to date. The physical phenomenon of bimodality induced by the finite bandwidth (alone) of additive forcing is demonstrated in detail for the first time. The three existing theories capable of accounting for this phenomenon can each be categorized in terms of a single (constant) value of a characteristic parameter ${P}_{\mathrm{expt}}$; but it is shown that, in reality, ${P}_{\mathrm{expt}}$ varies weakly with the correlation time \ensuremath{\tau} of the noise. The variation of ${P}_{\mathrm{expt}}$ with \ensuremath{\tau} is investigated in the currently accessible range of 0.1\ensuremath{\le}\ensuremath{\tau}\ensuremath{\le}\ensuremath{\infty}. .AE

DOI
10.1103/physreva.38.1966

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We formulate 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 (1). This model is given by a set of three Langevin-type equations for the real and the imaginary part 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 describe also a Kerr medium or a two-level system in a suitable range of parameters, we are mainly interested in the case of miniaturized all–optical bistable devices which 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 onedimensional Fokker-Planck equation only in the limit of short correlation times.
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.
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.
With the unified colored noise approximation theory,the steepest descent method and the two-state approach theory,the effects of noise on the stochastic resonance and the mean first passage time of the dispersive optical bistability system are investigated.It is found that when there is a weak periodic signal included in the system,with both multiplicative colored noise and multiplicative white noise,the signal-to-noise ratio and the mean first passage time of the system are affected not only by the noise correlation time,but also by the intensity of both multiplicative colored noise and multiplicative white noise.It is shown that the stochastic resonance phenomenon depends on the initial conditions of the system.
The behavior of an optical bistable system in which the fluctuations play the dominant role has been studied on the basis of a Fokker-Planck description by Schenzle and Brand.1 However, all the analyses so far approximate the fluctuations as arising from a constant external noise source described by a δ-correlated, Gaussian random process.2,3 In this paper intensity-dependent shot noise fluctuations of a bistable system are experimentally studied for the first time, using a hybrid bistable optical device, and analyzed by a simple “ladder" model that fits the data well.