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Topical Meeting on Optical Bistability (OB)

Instabilities in dispersive OB on the low transmission branch

Thomas Erneux, Paul Mandel

Published Jan 1, 1985• 0 citations
Publisher

Abstract

We study the mean field limit equations for optical bistability using two-level atoms as a model for the nonlinear medium. The resulting standard equations are /1/: The steady state equation is.
DOI
10.1364/obi.1985.wc2

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Physical Review A•Jan 1, 1986•14 citations
We study the semiclassical mean-field theory of single-mode dispersive optical bistability in the limit of a fully developed hysteresis cycle (i.e., in the limit of large values of the bistability parameter C) and when all decay rates have comparable magnitude. We locate (i) on the upper branch instabilities leading to time periodic regimes, (ii) on the lower branch instabilities leading to a switching before the limit point is reached. None of these instabilities are observed in the case of purely absorptive optical bistability.
In the first section we discuss the stationary solutions for a laser with an injected signal, the active counterpart of the optical bistability analyzed in Chapter 11. Next, in Section 13.2 we consider another very interesting system, namely a laser with a saturable absorber, a free-running laser in which the optical cavity contains a passive medium in addition to the active medium. The single-mode model for this system can be constructed by combining the single-mode models for the free-running laser and for optical bistability. The discussion is then focussed on its stationary states. In Section 13.3 we return to the single-mode model for optical bistability and, by introducing an appropriate “cubic limit” on the parameters in play, we derive a cubic model for dispersive optical bistability with two-level atoms. The stationary equation for such a model formally coincides with that derived and discussed for a Kerr medium in Section 11.2. In the last section we start from the equations for a quadratic nonlinear medium in the degenerate configuration derived in Chapter 6 and combine them with the appropriate boundary conditions for the two field envelopes in play, assuming that the quadratic medium is contained in a ring cavity as happens in an optical parametric oscillator or in second-harmonic generation in a cavity. Next, by introducing the low-transmission limit and the single-mode limit we derive a two-mode model that describes the dynamics of a degenerate optical parametric oscillator or of second-harmonic generation in a cavity. Finally, for the case of a parametric oscillator we illustrate extensively the stationary solutions of the equations. A laser with an injected signal From a technical point of view, this system is interesting because of the possibility of stabilizing the frequency and the phase of separate lasers by taking advantage of the known affinity of nonlinear oscillators to lock onto a periodic forcing element. In 1972 Spencer and Lamb [125] predicted the possibility of bistable behavior in a laser with an injected signal before the surge of interest in optical bistability in the passive configuration.
The purpose of this work is to study the behavior of atomic fluctuations and the spectrum of transmitted light by a nonlinear medium with absorption and dispersion. The authors make use of the system-size expansion and apply it to the quantum-mechanical Langevin's equations for the atomic operators and obtain the characteristic curves for bistable behavior, macroscopic atomic expectation values, and the spectrum of the atomic fluctuations. The calculated spectrum of transmitted light exhibits line narrowing near bistable thresholds, discontinuous formation of bands along the high-transmission bistable branch, and hysteresis effects, etc., for detuning $|\ensuremath{\delta}|$ less than a critical value ${\ensuremath{\delta}}_{c}$. For $|\ensuremath{\delta}|>{\ensuremath{\delta}}_{c}$, the discontinuous character of all the physical variables of the system vanishes; the system no longer exhibits bistable behavior. The system, however, still shows cooperative and single-atom behavior separated by a critical value of the intensity of the incident coherent pump at which a critical slowing down of the medium occurs, exhibiting a phase transition.
The stationary solution for optical bistability (OB) using a semiclassical exciton-biexciton model of CuCl is calculated exactly in the limit of large Fresnel number. The numerical solution of the second-order Maxwell equation is compared with corresponding results obtained with use of the slowly-varying envelope approximation (SVEA) as well as the mean-field approximation (MFA). It is found that results obtained using the SVEA are in close quantitative agreement with the numerical solution of the second-order Maxwell equation for each point in the parameter space considered, whereas the MFA correspondingly gives good qualitative results only. Furthermore, it is shown that the nonlinearity in the dielectric function for exciton-biexciton CuCl which causes OB is very nearly of the Kerr-medium type, and consequently, the two-photon biexciton resonance contributes only weakly to the OB characteristics.
We analyze instabilities in mixed absorptive and dispersive optical bistability in the rate-equation approximation and the mean-field and good-cavity limits. Our starting point is a set of multimode equations derived from the Maxwell-Bloch equations for ring-cavity boundary conditions. We obtain analytic expressions for the instability conditions. In a plane-wave analysis, we find that a portion of the lower transmission branch can be unstable in addition to the upper-branch instability found in purely absorptive bistability. Also, a new disconnected region of instability can exist on the upper branch. Our analysis becomes particularly simple for equal and opposite cavity detunings and we explore this case in detail. We extend our treatment to include a Gaussian transverse intensity profile and show that the instabilities remain in the presence of Gaussian averaging. We also show that many of the results obtained in the rate-equation approximation hold when the atomic linewidth and atomic decay rate are of the same order.