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Annalen der Physik

Noise Influence on a Dispersive Optical Bistable System with Trilinear Interaction

Kurt Germey, R.‐L. Herger, R. Herklotz, M. Mareyen, F.‐J. Schütte, R. Tiebel, K. Worlitzer

Published Jan 1, 1985• 0 citations
Publisher

Abstract

Abstract The influence of fluctuations in the driving fields on an optical bistable multimode system inside a ring resonator with a second order nonlinear material is considered. Steady state solutions of the resulting Fokker‐Planck equations are found for different approximations in which one or two of the modes can be eliminated adiabatically. The results of the stochastics are compared with the deterministic behaviour.

DOI
10.1002/andp.19854970103

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Abstract The influence of fluctuations in the driving fields on an optical bistable multimode system inside a ring resonator with a second order nonlinear material is considered. Steady state solutions of the resulting Fokker‐Planck equations are found for different approximations in which one or two of the modes can be eliminated adiabatically. The results of the stochastics are compared with the deterministic behaviour.
Abstract An externally driven light mode in a ring cavity may show multistable behaviour if linear losses and dispersive nonlinear coupling compete with each other. The paper considers the trilinear (sum frequency) system with two externally driven modes which shows bistability simultaneously in these two modes. It is described in a fully quantum manner. The master equation and the Fokker‐Planck equation for the Glanmber‐Sudarshan quasi‐probalbility are deduced with appropriate consideration of fluctuations. A closed steady solution of the Fokker‐Planck equation is given for one mode only under strong restrictive assumptions.
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.
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.
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.
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.