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Physical Review A

Spectrum of transmitted light in optical bistability: Effects of phase fluctuations of the driving laser

Aurora Hernández-Machado, Maxi San Miguel

Published Apr 1, 1986• 10 citations
Publisher

Abstract

Time-dependent correlation functions and the spectrum of the transmitted light are calculated for absorptive optical bistability taking into account phase fluctuations of the driving laser. These fluctuations are modeled by an extended phase-diffusion model which introduces non-Markovian effects. The spectrum is obtained as a superposition of Lorentzians. It shows qualitative differences with respect to the usual calculation in which phase fluctuations of the driving laser are neglected.

DOI
10.1103/physreva.33.2481

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Time-dependent correlation functions and the spectrum of the transmitted light are calculated for absorptive optical bistability taking into account phase fluctuations of the driving laser. These fluctuations are modeled by an extended phase-diffusion model which introduces non-Markovian effects. The spectrum is obtained as a superposition of Lorentzians. It shows qualitative differences with respect to the usual calculation in which phase fluctuations of the driving laser are neglected.
We derive and analyze the master equation for absorptive and dispersive optical bistability in the high-Q cavity limit where the driving laser has both amplitude and phase fluctuations which are larger than the small amplitude and phase fluctuations caused by incoherent spontaneous emission in the cavity. We treat the amplitude and phase fluctuations of the driving laser as independent time dependent Gaussian processes. The causes of jitter and amplitude fluctuations will vary from laser to laser and the fluctuations need not be Gaussian. However, we use Gaussian processes for mathematical tractability and because we expect them to be qualitatively correct. Recently,1 we have shown in the case of absorptive optical bistability when the driving laser has phase fluctuations the fundamental process which determines the statistical properties of the internal field is the competition between locking of the internal field phase to the phase of the driving laser and the diffusion of the phase of the internal field as a consequence of the driving laser fluctuations. The driving laser fluctuations have little effect if the phase locking rate is larger than the phase diffusion rate caused by the driving laser’s fluctuations. However, when the diffusion rate becomes comparable to the locking rate we find the amplitude of the 2 internal field is reduced and the dimensionless parameter2 measuring the importance of transitions from one branch of the OB curve to the other due to fluctuations takes on a value which leads to an appreciable transition rate due to fluctuations.
The main purpose of this paper is to study the behavior of the atomic fluctuations and the spectrum of the light transmitted by an absorptive bistable device. To this end we develop an approximation scheme based on the so-called system-size expansion and apply it to the quantum-mechanical Langevin equations for the atomic fluctuation operators. The Bonifacio-Lugiato mean-field equations for bistability are derived from the lowest-order approximation to the system-size expansion, while the atomic correlation functions result from the next-higher-order expansion. The calculated spectrum of the transmitted light exhibits line narrowing near the bistable thresholds, discontinuous formation of sidebands along the high-transmission branch of the device, and hysteresis effects.
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.
We derive, solve, and analyze the linearized master equation and evaluate the spectrum for absorptive optical bistability in the high-QQQ cavity when the incident laser has jitter or frequency fluctuation. The fluctuations are treated as a time-dependent Gaussian process whose properties are determined by the individual laser's characteristics. With our results we can explain the qualitative results of machine simulations and provide criteria for the magnitude of fluctuations in optical bistability in terms of laser and cavity parameters.
A generalized Fokker-Planck equation is derived for a system undergoing optical bistability when the dominant source of fluctuations is due to fluctuations in the incident field rather than spontaneous emission. The fluctuations are treated as a time-dependent Gaussian process whose properties are determined by the individual laser's characteristics. Both amplitude and phase fluctuations are included on the incident laser field. The effect of fluctuations of the incident laser on optical bistability are different for amplitude and phase fluctuations and are different for absorptive and dispersive optical bistability. The combination of incident laser phase fluctuations and dispersive optical bistability leads to large amplitude fluctuations near the turning points of the optical bistability curve and causes the system to make a transition from one branch of the optical bistability curve to the other before reaching the mean-field turning points. We take the high-QQQ cavity limit of our generalized Fokker-Planck equation, solve the linearized equations, and provide criteria for the magnitude of fluctuations in optical bistability in terms of laser and cavity parameters. The magnitude of incident laser fluctuations are typically orders of magnitude larger than fluctuations due to spontaneous emission and are therefore important for practical applications of optical bistability devices.
We consider the dynamical behavior of the (nonlinear) amplitude fluctuations in the transmitted radiation of the optical bistability in the transition region. A Fokker-Planck model with nonlinear diffusion coefficient (multiplicative Gaussian white noise) is used to describe the system in the "good-quality" cavity limit. Choosing a parameter set corresponding to a numerically small but tractable tunneling rate between the stable and metastable states we study, as functions of the external coherent field and system size, the spectrum and the response to a small additional injected coherent signal. The numerical calculations are facilitated by the construction of complex-valued continued-fraction representations for the spectrum of amplitude fluctuations and for the response function. This representation allows us to discuss the memory effects caused by deviations from a simple Gaussian-Markov behavior around the transition region. In addition, such a representation enables the construction of "a posteriori" error bounds on the power spectrum.