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Physical Review A

Noise-induced switching of photonic logic elements

P. Filipowicz, J. C. Garrison, Pierre Meystre, Ewan M. Wright

Published Feb 1, 1987• 6 citations
Publisher

Abstract

We study the stability of optical bistable elements against holding-field noise by considering the specific case of a purely dispersive Kerr medium. For small devices, i.e., those for which the round-trip time is small compared to the material-response time, we show that operation of an optical switch with a holding-field intensity a few percent away from the switching value will not be subject to problems due to the noise in the holding laser. This is due to the extremely rapid increase of the average time between noise-induced switching events as the distance, in intensity, from the switching point increases. White noise or Ornstein-Uhlenbeck noise models are not sufficiently smooth for a correct description of this problem, and so we use a more elaborate colored-noise model to evaluate the diffusion constant in the Fokker-Planck equation for the nonlinear phase shift. The time between switching events is then obtained by solving a first-passage-time problem.

DOI
10.1103/physreva.35.1172

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We study the stability of optical bistable elements against holding-field noise by considering the specific case of a purely dispersive Kerr medium. For small devices, i.e., those for which the round-trip time is small compared to the material-response time, we show that operation of an optical switch with a holding-field intensity a few percent away from the switching value will not be subject to problems due to the noise in the holding laser. This is due to the extremely rapid increase of the average time between noise-induced switching events as the distance, in intensity, from the switching point increases. White noise or Ornstein-Uhlenbeck noise models are not sufficiently smooth for a correct description of this problem, and so we use a more elaborate colored-noise model to evaluate the diffusion constant in the Fokker-Planck equation for the nonlinear phase shift. The time between switching events is then obtained by solving a first-passage-time problem.
When trying to operate optically bistable systems as logical switches, one is confronted with the question of stability against noise in amplitude and frequency of the holding laser beam. It is generally accepted that this problem is governed by scaling laws which are, to a large extent, independent of the detailed description of the nonlinear medium. Therefore, we consider the case of a purely dispersive medium whose nonlinear susceptibility satisfies a Debye relaxation equation. The cavity is taken to be of moderate finesse and large bandwidth (short round-trip time).
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.