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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Deterministic Chaos And Noise In Optical Bistability

R. Neumann

Published Jul 10, 1987• 0 citations
Publisher

Abstract

The time-dependent solutions of the mean-field Maxwell-Bloch equations for optical bistability are studied numerically for the deterministic equations and the stochastic equations with additional noise sources. From the solution of the deterministic equations a discrete, one-dimensional return map is constructed which proves the existence of deterministic chaos and shows that the Feigenbaum scenarium is valid. Inclusion of noise sources leads to destabilization of higher periodic solutions.

DOI
10.1117/12.937119

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The time-dependent solutions of the mean-field Maxwell-Bloch equations for optical bistability are studied numerically for the deterministic equations and the stochastic equations with additional noise sources. From the solution of the deterministic equations a discrete, one-dimensional return map is constructed which proves the existence of deterministic chaos and shows that the Feigenbaum scenarium is valid. Inclusion of noise sources leads to destabilization of higher periodic solutions.
The theory of optical bistability can be developed at various levels of sophistication.(1) In this overview I have chosen an approach which I hope will make the simplicity and generality of the phenomenon clear, and be suited to the emphasis of this meeting. I will describe a deterministic, semiclassical theory of optical bistability, based on coupled Maxwell-Bloch equations. Statistical and quantum statistical theories will not be included. I also limit my attention to questions concerning steady states and their stability, leaving the theory of higher bifurcations to other speakers.
Based on the instability criterion of semi-classical Maxwell -Bloch equations of single-mode optical systems, the global distribution of instability region is depicted in △-θ parameter space (△ detuning of the atoms, θ mistuning of the cavity). In the instability region predicted, various bifurcations and chaos are disclosed. We find that chaos may appear in both upper and lower branches of one bistable curve Moreover, a new time-dependent solution coexisting with the stable stationary solution is found.
The problem of dispersive optical bistability has so far been treated only in the mean-field approximation. A rigorous justification of the mean-field theory can only be obtained from exact solutions of the steadystate Maxwell-Bloch equations which retain the spatial dependence of the field. In this paper exact analytic solutions are presented for these equations. The authors demonstrate that the mean-field equation connecting the input and the output fields follows naturally from these solutions in the limits $T\ensuremath{\rightarrow}0$, ${\ensuremath{\delta}}_{F}\ensuremath{\rightarrow}0$, and $\ensuremath{\alpha}L\ensuremath{\rightarrow}0$ for the mirror transmission coefficient, the detuning of the field from the cavity resonance, and the linear absorption, respectively, with $\frac{\ensuremath{\alpha}L}{2T}$ and $\frac{{\ensuremath{\delta}}_{F}L}{2cT}$ remaining finite. The results are illustrated with the help of graphs showing the output versus input intensity for different values of the relevant parameters. The effect of these parameters on the phase shift of the output field is also displayed.
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.
The cooperative behavior of a system of atoms contained in a ring cavity is investigated under the condition that the fundamental atomic transition is a two photon transition1. Maxwell-Block equations are used to obtain the input vs. output relation by adiabatically eliminating all the atomic variables. The fundamental equation that results is a complex two dimensional map--which in the mean field limit reduces to the standard bistability equations. The characteristics of the two dimensional map are numerically investigated. Such a map is shown to lead to chaotic behavior following the Feigenbaum2 scenario. The sequence of events following first regime of chaos is a set of period halving bifurcations. The characteristics of the power spectrum in the region of chaos are presented. The effect of noise on the period doubling bifurcations in the present model, is also investigated and the connection with the theoretical predictions3 is established. The changes in the dynamical characteristics of the system when atomic inversion relaxes on the same time scale as the cavity round trip time will also be discussed in detail.