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Quantum Electronics

Stochastic aspects of transient processes in optical bistable systems

V. V. Pomortsev

Published May 31, 1995• 0 citations
Publisher

Abstract

A stochastic theory of relaxation in optical bistable systems is developed. The Fokker—Planck equation for the amplitude distribution function of an electromagnetic field in a passive medium is derived from fundamental principles in the adiabatic approximation. An analysis of the lowest eigenstates is used to construct a theory of relaxation of metastable and unstable states.

DOI
10.1070/qe1995v025n05abeh000390

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A stochastic theory of relaxation in optical bistable systems is developed. The Fokker—Planck equation for the amplitude distribution function of an electromagnetic field in a passive medium is derived from fundamental principles in the adiabatic approximation. An analysis of the lowest eigenstates is used to construct a theory of relaxation of metastable and unstable states.
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A bistable absorber in the good-cavity limit and the Fokker-Planck equation describing the time evolution of the quasi-probability distribution for the transmitted field amplitude are considered. It is shown that the system evolves in two stages. The first consists of a local relaxation around the minimum (or minima) of a suitably defined potential. The second stage, which occurs over a much longer time scale, results from tunneling from one minimum to the other across the local maximum of the same potential. In this paper, only the first stage is analyzed. The time evolution of the expectation value of the transmitted field is described analytically. The accuracy of this approach is tested by numerical solutions of the Fokker-Planck equation.
On the basis of the absorptive bistability model, including the fluctuations, we rederive in the good-cavity limit the Fokker-Planck equation describing the time evolution of the quasiprobability distribution for the transmitted field amplitude. It is shown that on a semiclassical level different variants of the Fokker-Planck equation are possible. The equation of Bonifacio, Gronchi, and Lugiato is solved in terms of a path integral which involves a generalized Wiener measure. In order to calculate the averages of transmitted field moments as functions of time, a Monte Carlo integration scheme is used. This scheme consists in constructing random trajectories which result from the Langevin equation statistically equivalent to the derived Fokker-Planck equation. Numerical results in the bistable domain are presented for large values of the fluctuation parameter.
Systems exhibiting dispersive optical bistability are considered as examples of nonequilibrium steady states lacking detailed balance. Methods for describing such states are developed. The Fokker-Planck equation for the steady-state distribution function of the transmitted electromagnetic field is derived. In the limit of small fluctuations the Fokker-Planck equation is reduced to the form of a Hamilton-Jacobi equation for a "nonequilibrium thermodynamic potential." This equation is solved in various approximations. The nonequilibrium thermodynamic potential acts like a free-energy for the first-order-type transition far from thermodynamic equilibrium lacking detailed balance. The potential entirely determines the steady-state distribution, the coexistence curve in the bistable domain, a Lyapunoff function of the deterministic equations of motion, and allows us to cast the deterministic equations of motion into the standard form of nonequilibrium thermodynamics.
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.