The topic of this symposium is Optical Chaos. It is essential for those working in this field to know how to recognize, identify and characterize chaotic behavior and how to distinguish it from noise. General principles, definitions of terminology, and some brief review and commentary on the current status of techniques for measuring chaos are presented.

Instabilities in various classes of lasers are now well recognized phenomena, and in recent years these phenomena have become increasingly well understood. Tor most purposes gas lasers have been found to the easiest systems to study, and many of the instability effects found in gas lasers have been at least qualitatively modelled. In our meeting presentation we review some of the basic concepts of laser instability modelling with emphasis on results pertaining to gas lasers. The more qualitative aspects of the presentation are briefly summarized in these notes.

A short introduction to instabilities in optically pumped systems is given. We have put emphasis on homogeneously broadened media. Main results on line competition and stability analyses of single-mode operation are reviewed and compared to experiments. The conditions for Lorenz chaos in cw optically pumped far-infrared (FIR) lasers are formulated. They are more stringent than those derived in two level systems because of the pump laser induced coherence.

We have used optically pumped NH3-far-infrared lasers to investigate self-pulsing in-stabilities and transition to chaotic emission, theoretically predicted to occur for homogeneously broadened lasers high above threshold when the 3 characteristic times of the laser: decay time of population, polarization, and field, are of comparable magnitude. We have observed pulsing instabilities and chaotic emission from these systems for the first time and find qualitative agreement with the laser dynamics predicted from the simplest laser model (Lorenz equations).

A new kind of spontaneous mode-splitting is predicted for resonant (near resonant) pumped molecular lasers, resulting from pump field induced Rabi splitting of the emission line-shape. General conditions are presented for the occurrence of mode-splitting in homogeneously broadened systems. Results of detailed analysis pertaining to specific mid- and far-infrared lasing transitions in NH3 show that such effects should be observable under typical experimental conditions as oscillatory instabilities in the emission. For the case of off-resonant (Raman) emission, pump-intensity dependent unidirectional cavity mode frequency shifting is predicted. This is confirmed for a CO2 laser pumped 12.08 μm laser.

We discuss a simple rate equation laser model, containing intensity-dependent parameters. Our calculations suggest that the model is consistent with the emergence of self-pulsing in the good cavity limit even at very low pump levels. The dynamical behavior of two- and four-level systems shows rather good qualitative agreement with the result of previous experimental observations.

We show the existence of self-pulsing at two characteristic frequencies in a single mode CO2 laser for low pump rates. The dynamical behavior of the system as a function of the pump rate, cavity losses and detuning between atomic and cavity frequencies has been investigated. These results can be explained by a recently developed theoretical model which incorporates intensity dependent parameters.

New experimental results are presented which show the behavior of a homogeneously broadened ring dye laser both below and above the threshold for the onset of the two-frequency instability. It is shown that the spectrum of the output of the laser is extremely sensitive to the detuning of the initial lasing frequency from the center of the gain curve. Spectral splittings as large as 340 A are reported. The close connection between this instability and the production of femtosecond mode-locked laser pulses is discussed.

A simplified dynamical model describing switching of fundamental and first-order modes in laser diodes is derived. Application of Hopf bifurcation criteria to the model is facilitated via a partition of the dynamical system matrix.

Some recent developments in the field of optical bistability are reviewed. Emphasis is placed upon those new areas which are currently most active and which demonstrate the greatest interest for the future.

The physics behind magnetically-induced polarization switching for atoms undergoing a Jlower=1 - Jupper =0 transition in a polarization degenerate cavity is discussed. We present preliminary theoretical results for self-pulsing between a+ and a- modes.

We report on systematic studies of the static and dynamic behavior of intensity and polarization of the light transmitted by a sodium-filled Fabry-Perot cavity. The device displays a rich variety of phenomena, including optical bistability, tristability, pitchfork bifurcation, as well as intrinsic oscillations. The behavior can be traced back to the simultaneous presence of Zeeman pumping and hyperfine pumping in the sodium ground state, combined with velocity selective hole burning effects.

Periodic waveforms appearing within the chaotic domain of an acoustooptic bistable device are shown. The stability of the well-known frequency-locked waveforms is first discussed. New periodic waveforms are also shown which are the time-continuous counterparts of the discrete cycles predicted by the 1-D model. The results are analyzed within the framework of the linearized stability analysis of the differential-difference equation describing the system.

The mechanisms for the main sources of optical nonlinearities in semiconductors are briefly reviewed. Oscillatory and chaotic instabilities in passive and active semiconductor etalons are discussed.

Semiconductor lasers with axially non-uniform structures are known to exhibit bistable charactreristics which can be switched by the application either of electrical or optical pulses. The paper describes the physical mechanism of the appearance of bistable characteristics in InGaAsP/InP DH lasers and gives accounts why this laser tends to bistable operation in comparison with GaAs lasers. Temperature dependence and control of hysteresis width by non-uniform pumping are also discussed.

A model describing the dynamics of an irradiated SF6 molecule was previously shown to yield chaotic trajectories. We have found that neutrally stable oscillatory states are also possible within windows of the parameter space. When dissipation is added to the model, we have found bistability, both between fixed points and limit cycles, and between limit cycles and other limit cycles. By dynamically varying the laser detuning, we have proven that hysteresis cycles result.

We investigate optical bistability in CdS single crystal platelets and related materials either due to electronic or thermal effects, depending on the length of the excitation pulses. An induced absorber in a hybrid ring cavity with cw-input shows self oscillations.

The existence of Hopf bifurcations in twin-stripe laser diodes is demonstrated in a generalisation of an analysis developed for gain-guided laser diodes. Both electronic and optical injection control of the bifurcation phenomenon are found to be effective.

The fabrication procedure and properties of optical nonlinear guided-wave structures are reported. The structures consist of a thin film of Corning 7059 glass deposited by RF planar magnetron sputtering on a surface of CdSxSe1-x doped glass. The coupling of light into the waveguides characterized by a power dependent propagation constant was studied. The experiments were performed at two wavelengths λ= 0.4880 μm and λ= 0.5145 μm with a laser working in cw and mode-locked regimes of operation. It is shown that integrated optical structures fabricated exhibit strong nonlinear effect with subnanosecond response time.

Novel deterministic instability, featuring subharmonic locking and chaos, is demonstrated in a semiconductor laser strongly coupled to an external cavity. A quasiperiodic transition to chaos is found to exist in locking boundaries as well as within locking zones.

We propose a new scheme for the classification of the optical instabilities that emerge within the framework of the Maxwell-Bloch model. Our scheme is based on an appropriate ex-tension of the notions of amplitude and phase instability advanced in earlier publications and generalized in this work to include the free running laser, optical bistability and the laser with an injected signal. We address our considerations to a unidirectional ring cavity model, in the plane wave approximation and with a homogeneously broadened, two-level active medium.

This is a short review of the investigations done at Istituto Nazionale di Ottica on the onset of deterministic chaos in single mode, homogeneous line lasers. While carrying these researches, we discovered that bistabilty (or more generally multistability) is common to most nonlinear dynamical systems, that is, usually there are many coexisting attractors for the same control parameters. In such bistable (or multistable) situations, addition of a small amount of noise yields long memory effects displaying low frequency spectra of 1/f type. A theory of these effects is outlined.

Characteristic features associated with the presence of non-white pump noise in single and two-mode dye lasers are discussed. A slow initial decay of the steady state intensity correlation function is characterized in terms of an effective eigenvalue which is calculated exactly. For the single mode case, non-white pump noise produces a large reduction of its value and suppresses the existence of a minimum. For the two-mode case the crossed-correlation effective eigenvalue becomes non zero due to pump noise when the two pump parameters are different. Transient statistics for the single mode case are characterized by the passage time distribution and the anomalous fluctuations. The mean first passage time is essentially independent of pump noise while the variance is enlarged by pump noise but partially reduced by its nonwhite character. Similar phenomenon is observed in the anomalous fluctuations.

In the deeply modulated laser diode, chaotic behaviors are observed. There are two types of routes to chasos depending on the modulation frequency. When the modulation frequency is at least 1.7 times higher than the relaxation frequncy, period-two instabilties are easily induced the period-two and the period-four instabilties are observed on the way to chaos.

Recent research on instabilites in the four-wave mixing process is summarized. The four-wave mixing process can cause a laser beam to become unstable to the growth of new frequency components as the beam propagates through a nonlinear medium. Similar instabilities can occur in more complicated optical systems such as lasers and bistable optical devices. The nature of the instability is very different for nonlinearities resulting from one- and two-photon resonances in the susceptibility describing four-wave mixing. The four wave mixing process can lead to chaotic behavior in a self-pumped phase conjugate mirror.

Intracavity nearly-degenerate four-wave mixing is used to study instabilities occurring in Fabry-Perot laser cavities operating above or below threshold. Two-level theoretical calculations and experimental results obtained with semiconductor lasers are reported.

Non-integral Maxwell equations are derived for an interfering beam scheme, by taking into account a tensor property inherent in a third-order nonlinear susceptibility, χ (3), and provide new insights into four-wave mixing theory, in particular, polarization bistability and instability.

Noise-induced transitions in a delay-differential-equation model of an optical bistable device are studied. In the limit of long delay times, two different mechanisms for the transition process are identified, consistent with the predictions of one-dimensional map models. For small delay times different mechanisms operate. The effects of the statistics of the external noise on the transition process are also investigated.

The interaction of light with a-cyclodextrin-iodine inclusion compounds was studied with aim of obtaining optically bistable elements. A simple model, developed to explain the dichroic properties of these compounds, predicts important photodichroism. This is reported along with preliminary experiments. Finally, other means of inducing dichroism are surveyed.

The analysis of a nonlinear, single-mode fibre ring resonator is presented. The transmission characteristics of this device features steady and self-pulsing states, leading to chaos.

A Stokes vector formalism is used to describe the nonlinear polarization dynamics of counterpropagating beams. Employing the full SO(3) covariance of the Stokes parameters, a large class of material symmetries can be easily discussed.

Nonlinear Fabry-Perot interferometers, in the form of nonlinear semiconductor etalons, have been proposed and demonstrated some years ago as pulse shaping elements 1,2. Such etalons can shorten the pulse duration drastically if their finesse is high enough. On the other hand, the nonlinear Fabry-Perot interferometer is also, under certain conditions, a bistable element 3,4, and the fast switching associated with this property may also be used to reshape optical pulses. However, the switching time and the pulse shortening ability of such nonlinear interferometers are limited by the finite lifetime of the cavity and transient effects5. A shorter cavity requires a stronger nonlinearity, most often associated with a significant absorption, and a long response time.

Nonlinear pictorial feedback systems may show either deterministic chaos in space and time or cooperative behavior. Specifically interesting is the evolution of stable patterns which show autoassociativity, i.e. which restore themselves after perturbation. We quantify the dynamics by crosscorrelation, entropy and energy.

We present here a theoretical discussion of the effect of nonlocality and saturation in transverse optical bistability. It is found that the switching conditions, switching power and characteristics are considerably different from the situation where these effects are not accounted for. We also include an analytical study of the transverse self-phase modulation effect and relate the key parameters explicitly to the transverse bistability switching condition and bistability.

We use a fully quantum, many-body treatment to show mirrorless optical bistability in terms of the spacial properties of interacting two-level systems. The bistability effects are related to the coherent dipole-dipole interactions which introduce an inversion-dependent renormalization of the frequency (or relaxation). Conditions of stability and instability are discussed for a thin sample of two-level atoms with a width smaller than a wavelength. For a long sample of two-level atoms, we find that the two phases of high transmissivity and low transmissivity may coexist spacially in the material.

The infrared CO2 laser with an intracavity saturable gas absorber presents the bistable, unstable and chaotic regimes. The bistable and unstable regimes have been experimentally investigated through the different static and dynamic responses. The static behaviour is described by a phase diagram. In the dynamic regims, where a control parameter is swept in time or is modified through a step function, the amplitude of the bistability region,the delay time in the response of the system and the fluctuations in the response have been measured. The experimental results have been described through a rate-equation approach.

The experimental investigations of the behavior of an internally modulated CO2 laser reveals new chaotic features. In addition to the period-doubling transition to chaos, it has been shown to present (i) generalized bistability between different attractors when the laser cavity is sufficiently detuned from the molecular resonance and (ii) crises when its most usual strange attractor collides with an unstable state. This laser is also a good prototype to study experimentally the influence of noise on bifurcations. A simple model of this system proves useful to understand the dependence of the bifurcation diagram on different parameters.

First, the possibility of semiconductor laser induced free carrier reflection in a few typical semiconductors was investigated. When used in telecommunication through the atmosphere this should allow modulation up to 100 MHz. Secondly an all optical semiconductor switch was used to Q-switch a waveguide CO2 laser.

An improved adiabatic elimination procedure, based on center manifold theory, is used to determine a two-dimensional flow which describes a large class of lasers characterized by slow relaxation of the population inversion. A detailed analysis of this model shows a) a quasi-conservative motion in a Toda potential, b) Haken second threshold for the expected parameter values, c) undamped oscillations in agreement with the Lorenz equations and d) a more accurate description of the relaxation processes than the usual rate equations. The relevance of the model to experiments is also briefly discussed.