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Physical Review A

Long-time behavior of optical bistability

Yoshio Nishiyama

Published May 1, 1980• 8 citations
Publisher

Abstract

Long-time behavior of transmitted and reflected field amplitude of optical bistability by a cavity including an absorptive resonant medium from an initial to a final steady state on switching the incident field is investigated by solving numerically the semiclassical dynamical equations without the mean-field approximation. The transmitted field amplitude as a function of time is monotonic as a whole, and an oscillatory character is seen on it in a shorter period. The dynamical character such as the oscillation amplitude and the duration of a whole process depend strongly on two parameters included in the equations.

DOI
10.1103/physreva.21.1618

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Long-time behavior of transmitted and reflected field amplitude of optical bistability by a cavity including an absorptive resonant medium from an initial to a final steady state on switching the incident field is investigated by solving numerically the semiclassical dynamical equations without the mean-field approximation. The transmitted field amplitude as a function of time is monotonic as a whole, and an oscillatory character is seen on it in a shorter period. The dynamical character such as the oscillation amplitude and the duration of a whole process depend strongly on two parameters included in the equations.
We study the semiclassical mean-field theory of absorptive optical bistability in the limit of a fully developed hysteresis cycle, i.e., in the limit of large values of the bistability parameter CCC. We show by multiple time-scale perturbation analysis that it is possible to describe the transition between the two stable branches by simple equations. Near the low-transmission branch, the behavior of the system is governed by an equation for the atomic population, whereas, near the high-transmission branch, it is an equation for the field amplitude which determines the long-time evolution of the system. We then present an analytic study, completed by numerical results whenever necessary, of the time-dependent response of the system when the input field is swept across the bistable region. We study the influence of the initial conditions, the sweeping velocity, and the ratio of the atomic relaxation times on the dynamical hysteresis as well as its relation with the stationary hysteresis.
Within the framework of the theories of absorptive bistability recently proposed by various authors, the different sets of coupled differential equations describing the steady-state fields inside the Fabry-Perot cavity are solved analytically. Expressions are obtained for the bistability equation, which links the incident and transmitted fields, for different ranges of the mirror transmission coefficient. Comparison is made with the mean-field theory of Bonifacio and Lugiato. Conditions for stationary points in the bistability curve are derived.
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.
This paper presents a new model of optical bistability (OB). In view of the ideas of the mean field approximation, the slow envelope approximation and the adiabatic approximation, the OB system can be regarded as a box. Comparing with the theory of nonlinear oscillation, we find that the courses in different OBS's-includingdifferent substances and different cavities (E-P. C. or E. C. )-can be describedconsistently, in both steady state and temporal respect, with an appropriate nonlinear forced oscillation equation, in which the coefficients are measurable macroscopical parameters. Using the method of slowly changing phase and amplitude in oscillation theory, we develop the equation into a coupling and autonomous dynamic equation set of phase and amplitude. The general solutions of the equation set for OB steady state and transient behavior are obtained conveniently from this form. Several concrete results in literature are compared as the examples for the verification of the widely applicability o del.
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.
In this review, we will focus on optical bistability and the condition for its occurrence in cavity optomechanics. We study different optomechanical systems (basic optomechanical systems and hybrid optomechanical systems). we derived an equation that could predict the behavior of the system either bistable, transistor or monostable for each system considered depending on the value of its parameters. This equation is derived using analytical study for functional relation between the incident intensity field as a function of the optical cavity field intensity. Investigating the relation between any physical measurable quantity in the optomechanical system against the incident field intensity and its dependency on fundamental optical cavity field against the incident field intensity. Finally, studying the effects of integrating two types of atomic media into the optomechanical system, and tunability it could provide to the optomechanical system, each medium is separately studied. The atomic medium also tends to squeeze the domain of the bistability within the system relative to the basic optomechanical system