Thermal and parametric noise in dispersive optical bistability I General formalism

Abstract

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.