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Digest on Nonlinear Optics: Materials, Phenomena and Devices

Scaling Behavior Of Optical Bistability In Nonlinear Medium

Yao‐Huang Kao, Ching-Sheu Wang

Published Jan 1, 1990• 0 citations
Publisher

Abstract

Optical bistability in nonlinear media is potentially useful because of its fast transition speed for all-optical logic operation [1-3]. The dynamics of these mirrorless systems have been described by a anharmonic Duffing oscillator [1-4]. After normalization the equation of motion is described by x + k x + x - f(x) = F sinwt, where k is the damping factor, F is the driving amplitude, w is the driving frequency and f(x) is a nonlinear function of x. Such a model predicts the existence of bistability under appropriate conditions [3]. It would be particularly important for this system to be tuned to some situation in order to efficiently generate these nonlinear phenomena. In this summary we would provide a global picture of possible dynamic behaviours in Eq.(1), which can be used as a guide line in real experiment.

DOI
10.1109/nlo.1990.695902

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Optical bistability in nonlinear media is potentially useful because of its fast transition speed for all-optical logic operation [1-3]. The dynamics of these mirrorless systems have been described by a anharmonic Duffing oscillator [1-4]. After normalization the equation of motion is described by x + k x + x - f(x) = F sinwt, where k is the damping factor, F is the driving amplitude, w is the driving frequency and f(x) is a nonlinear function of x. Such a model predicts the existence of bistability under appropriate conditions [3]. It would be particularly important for this system to be tuned to some situation in order to efficiently generate these nonlinear phenomena. In this summary we would provide a global picture of possible dynamic behaviours in Eq.(1), which can be used as a guide line in real experiment.
Introduction The transient behavior of nonlinear optical resonators which display bistability and its reaction to fluctuations of system parameters are of extreme importance in all possible applications of these devices. Moreover, however, studies on these processes may give insight into some more general aspects of the behavior of nonlinear dynamical systems. In fact it has been emphasized in the early stages of the development of the field that optically bistable systems, in a suitable model, may exhibit a first-order phase transition (Bonifacio and Lugiato, 1978). Since the threshold behavior of a laser can be described as an example of a second-order phase transition (Haken, 1964,1975; see also Haken, 1977), optical bistability in (passive) nonlinear resonators complements nicely lasers as nonlinear dissipative systems in optics. Both types of optical systems are easily accessible to measurements, and the time scales coming into play are convenient for detailed experimental studies of the dynamics. It is worth mentioning that there is a close connection between laser operation and optical bistability in nonlinear (passive) resonators (see Lugiato, 1984). The standard single-mode laser model with distributed losses (Haken, 1970; Sargent, Scully and Lamb, 1974) can be generalized to include the effect of an external field that continuously drives the atoms within the optical resonator. The standard model of absorptive bistability, the Bonifacio–Lugiato model, is a special case of this generalized model. It represents the situation of a pumping mechanism being absent.
Intrinsic (mirrorless) optical bistability (MOB) has been predicted independently by a number of authors1. Recent experiments have demonstrated at least one major aspect of this phenomenon, i.e., MOB due to increasing absorption2. Other categories of MOB which have been predicted from theoretical models are caused by: interatomic correlation in a many-body quantum model, the anharmonic contribution in a classical Duffing oscillator, the local field correction in the macroscopic Maxwell-Bloch formulation, and coherent dipole-dipole interaction in a microscopic semiclassical model of a finite number of damped, two-level atoms driven by an external field3. All of these models share a single common characteristic which causes MOB in some region of the parameter space of each model. The characteristic is simply: an interaction-dependent nonlinearity which causes a renormalization of the resonance frequency. In each case, the sign and magnitude of the incident field detuning from the unrenormalized resonance is a crucial parameter.
Ferroelectric oxides with perovskite structure such as PbLaTiO3 (PLT), BaTiO3, PbTiO3, SrBi2Ta2O9 and LiNbO3 are very attractive class of materials which possess numerous useful properties such as high dielectric constant, large spontaneous polarization, and remarkable optical nonlinearity. Potential applications of these materials include real-time holography, correlation filtering and various novelty filter applications (Sutherland 1996 and Eaton 1991). They are also popular materials for the fabrication of nonvolatile memories (Ramesh 2001). Over the past few years, ferroelectric oxides have been widely investigated for various nonlinear optical applications (Shi 2006; Xuan 1998; Zhang 1999; Zhao 1996) especially for optical switches. Optical switches are devices invented to perform multiplexing at very fast speeds and with less delay than the customary switches works with electronic signals. A ferroelectric optical switch is expected to allow the processing of millions of signals at a speed of terahertz. To obtain optical bistability phenomenon, two ingredients are necessary, a nonlinear process and a feedback mechanism (Gibbs, 1977, 1979, and 1985). In all optical systems, the feedback can be “distributed”, “extrinsic” or “intrinsic”. In multilayer systems with alternating nonlinear materials, the feedback is “distributed”; it arises from the interaction of the propagating wave with many cross-sections of a nonlinear medium. In a Fabry-Perot (FP) resonator, the feedback is “extrinsic”; it arises as a result of reflection from the mirrors placed at its interface. In a single nonlinear layer, the feedback can be “intrinsic” or “mirrorless”; it arises in each elementary oscillator due to the strong local nonlinear response of an individual atom or molecule. In the usual or the standard analysis to study the optical bistability (Marburger 1978, Gupta 1987, Biran 1990, Danckaert 1989, Shen 1984, and Haelterman 1989) in nonlinear optics, the governing equation for optical propagation within the nonlinear medium is a nonlinear wave equation in the electric field derived from Maxwell’s equations. The usual constitutive relation between the nonlinear polarization and the electric field is then obtained by expanding the nonlinear polarization as a Taylor series in the electric field. The usefulness of this constitutive relation is that the polarization is a natural source term in the Maxwell’s
The theory of dynamic optical bistability and self-pulsations in condensed media is elaborated. The nonlinearity is due to a process in which two excitons are bound into a biexciton by virtue of their Coulomb interaction. A computer experiment is used to study the switching times between the optical bistability branches. The scenario for the transition to the dynamical chaos mode is found. The action of external periodical force upon the system is investigated and the range of amplitude values and frequencies of external harmonic pump were found at which the chaotic regime becomes nonlinear periodic by transformation of strange attractor into a limit cycle. Finally, we discuss the possibility of detecting these phenomena in experiments.
The theory of dynamic optical bistability and self-pulsations in condensed media is elaborated. The nonlinearity is due to a process in which two excitons are bound into a biexciton by virtue of their Coulomb interaction. A computer experiment is used to study the switching times between the optical bistability branches. The scenario for the transition to the dynamical chaos mode is found. The action of external periodical force upon the system is investigated and the range of amplitude values and frequencies of external harmonic pump were found at which the chaotic regime becomes nonlinear periodic by transformation of strange attractor into a limit cycle. Finally, we discuss the possibility of detecting these phenomena in experiments.