A comprehensive microscopic theory is reviewed which allows us to compute gain and refractive index properties of semiconductor heterostructures with varying degrees of strain and quantum confinement. The theory is applied to a model unstable resonator semiconductor laser. It is shown that the specific properties of the gain medium strongly influence the lateral mode stability.

A theoretical study is presented of multi-wave mixing dynamics in a single-mode semiconductor laser with monochromatic weak external injection. Three relevant regimes are overviewed, i.e., corresponding to locking, four-wave mixing, and multi-wave mixing. Moreover, much emphasis is put on four-wave mixing. For this regime, several analytical expressions are presented, some of which are new. A detailed theoretical explanation is given for the peculiar relaxation oscillation resonance behavior that was recently observed in experiments.

We report experimental measurements and numerical calculations of the effects of strong, nearly-degenerate optical injection in a nearly single-mode, Fabry-Perot laser diode. The noise spectra and the response to weak optical injection of the laser diode are well characterized by a model, based on the coupled electric-field and carrier-density equations, which includes the effects of gain saturation and the weak side modes. Beyond the perturbation regime, the laser exhibits new operating characteristics, including an output which involves rapid fluctuations among several longitudinal modes and spectral broadening of the principal mode. The laser dynamics strongly depends on the injected optical power and the frequency offset between the external input and the free-running oscillation. Numerical simulations are conducted using the full coupled-equation model for a strictly single-mode laser without noise.

In this paper we describe some of the effects of external optical feedback (OFB) on semiconductor lasers by simulation of the stochastic rate equations. Particular attention is paid to the laser's transition to optical chaos. In addition, we describe three techniques for avoiding this chaotic regime. The technique of high frequency injection, used in optical recording, can delay the onset of chaos till very high values of OFB. Experimental results are given and are in excellent agreement with the theory. A second technique called occasional proportional feedback can be used with some success to stabilize the chaotic output of semiconductor lasers. The final technique for controlling chaos consists of the optimization of various system and laser parameters so that the laser is least susceptible to OFB.

The delay differential equations modeling semiconductor lasers with optical feedback are numerically investigated. The low frequency fluctuations observed in the coherence collapsed regime are demonstrated to be crisis induced intermittency. The roles of antimodes and chaotic itinerancy are also determined.

The transient radiation oscillation process and generation instabilities in quantum-well laser systems are investigated in detail. A novel type of laser with quantum-well layers of various thickness is considered. The conditions for lasing switching with increasing the excitation current and for regular optical pulse generation have been determined. The calculations have been performed for the GaAs-AlGaAs system. The developed mathematical models and rate equations can be used to describe dynamics of laser action in quantum-well systems on other materials too.

Models of coupled lasers are investigated by exploring new asymptotic limits. The class B limit is based on the fact that the decay rate of the cavity is much larger than that of the inversion. By reformulating the laser equations as a weakly perturbed conservative system of equations, we may apply perturbation techniques appropriate for nonlinear oscillators. We illustrate the method by studying the bifurcation diagram of two coupled solid state lasers and determine conditions for a period doubling bifurcation. Semiconductor lasers are also class B lasers but, in addition, the normalized excess pump current is a small parameter. By taking this feature into account, we propose a new asymptotic analysis of the equations for an arbitrary number of coupled lasers. The leading order solution is then a linear combination of supermodes solutions.

Because of the obvious advantage in long time predictions it is useful to convert dynamical problems of flows into problems involving maps. For Hamiltonian flows this in effect is equivalent to identifying an area preserving map in the Poincare surface of section. The preservation of canonical structure of the Hamiltonian flow in the surface of section can lead to a description in terms of discrete canonical equations in the surface of section. This property is utilized here to convert the Hamiltonian flow problem of the dynamic evolution of self-focusing of an electromagnetic beam (width/phase front curvature dynamics) with beam propagation distance into an equivalent mapping problem for a wide range of initial conditions. The nonlinear Schrodinger equation is thereby converted to a map in a restricted sense.

Because of the obvious advantage in long time predictions it is useful to convert dynamical problems of flows into problems involving maps. For Hamiltonian flows this in effect is equivalent to identifying an area preserving map in the Poincare surface of section. The preservation of canonical structure of the Hamiltonian flow in the surface of section can lead to a description in terms of discrete canonical equations in the surface of section. This property is utilized here to convert the Hamiltonian flow problem of the dynamic evolution of the nonlinear Schrodinger Equation which is thereby converted to a map in a restricted sense. The evolution of perturbed soliton with initial inhomogeneous chirp factor is governed by this equation and the corresponding map is analyzed.

The dynamics of stimulated Brillouin scattering (SBS) in single mode optical fibers is considered in the presence of arbitrary amounts of external feedback from the fiber ends. At high feedback levels the Stokes wave oscillates periodically with an intensity-dependent frequency. The irregular structure at very low feedback levels is found to be stochastic, not deterministically chaotic through analysis of the recorded time series.

A new and novel training algorithm, based upon the matrix pseudoinverse least-squares method, is introduced for training hidden layer, forward-feed neural networks with high accuracy and speed for nonlinear and chaotic time series prediction. Model-generated chaotic time series, including that of the Lorenz system, are used to measure performance and robustness. Our new training algorithm has rendered application of forward-feed, hidden-layer neural networks for adaptive chaotic time series analysis, as well as other signal processing, practical and near real time using standard desktop computation facilities. We have applied our method, in conjunction with other standard methods, to the analysis of stimulated Brillouin scattering under cw pump conditions involving a single Stokes and pump signal in a single- mode optical fiber as the nonlinear medium. We use Stokes signal data generated from a standard model and correlate the training performance of our algorithm with statistical and dynamical characteristics of the system determined by other means.

A high strain resolution fiber optic interferometer is utilized to characterize the nonlinear magnetostrictive dynamics of amorphous ferromagnetic ribbons. Strain response of the magnetostrictive oscillator is found to exhibit several interesting routes to chaos as a function of applied dc and ac magnetic fields. In particular, we study parametric effects near a period doubling bifurcation. Small signal gain predicted near a super-critical period doubling bifurcation is verified in this system and has been used to demonstrate a fiber optic magnetic field sensor with gain. For appropriate parameter settings, the magnetostrictive oscillator also displayed a hysteretic sub-critical bifurcation. Stochastic resonance (SR) has been observed near such a sub-critical bifurcation. Suggestions are made regarding the exploitation of the SR effect for making novel magnetic field sensors.

Periodic pulling, the incomplete entrainment of a driven, nonlinear oscillator, is observed experimentally in a relaxation oscillator for the first time. A neon-bulb relaxation oscillator is driven by a chopped, neon-resonant, laser beam. For driving frequencies just outside the range for entrainment, the signatures of periodic pulling are verified. The experimental results are qualitatively reproduced in numerical solutions of the driven van der Pol equation.

The behavior of the time-dependent portion of the current i(t) as a function of the total mean current <I> flowing through a glow-discharge tube filled with noble gases (He, Xe, Kr) is analyzed. The experiments were carried out in dc-operated low-pressure spectroscopic lamps immersed in an electrically insulated heath bath. A dc regulated high- voltage supply was used to sustain the discharge. The data points, i(t) sampled at regular time intervals, were used to reconstruct the dynamics of the system. Evidence was found of intermixed irregular and regular regimes. The discharge in the He-filled tube exhibits a sequence of bifurcations between regular states as the current <I> was increased reaching at the higher currents an irregular (stochastic or chaotic) regime. In Xe and Kr only the stochastic regime was observed with no trace of the regular ones.

This paper reports nonlinear dynamics that arise through the interaction of oscillating modes of class-B multimode lasers, in which polarization dynamics can be adiabatically eliminated. They include self-organized collective relaxation oscillations in transient regimes and in the vicinity of stationary states of free-running lasers, antiphase dynamics in unstable regimes of modulated lasers, and mode-partition-noise-induced chaotic burst generation in compound- cavity laser schemes.

The two-photon laser represents an entirely new class of quantum optical oscillator that promises to display a wealth of new and exciting nonlinear behavior. For example, the prediction that the turn-on behavior of the laser is indicative of a first-order phase transition was verified in recent studies of the first continuous-wave two-photon laser. In addition, it was found experimentally that the field generated by the laser displayed dynamical instabilities under some conditions. We briefly review the properties of two-photon lasers that are pertinent to its dynamical characteristics and show that many of the interesting properties can be understood from a simple rate-equation model. Also, we describe our efforts at Duke to address the origin of instabilities in two-photon lasers.

The behavior of a cascade laser in which two fields are simultaneously amplified is theoretically investigated. Each field is coupled with one of the transitions of a ladder three- level atomic or molecular system in conditions of homogeneous broadening. Steady-state as well as dynamic solutions are considered. Cooperative emission of both fields is found in steady state regime. Single-field solutions can loose stability through two different types of Hopf bifurcations. One of them can appear below the Lorenz-Haken second laser threshold, and the `bad cavity' condition is not necessarily required. In the unstable regimes transitions to chaos via the quasiperiodicity road are found. The influence of cavity detuning and the passage from cascade to two-photon laser are considered.

This paper highlights some of the new contributions nonlinear dynamics has made in the areas of control and tracking. In particular, emphasis is placed on the model independent approach to control and tracking: The connections between the classical control and the control based on time series embedding methods are made. In experiments of control, our approach does not necessarily imply new equipment is needed in the loop. Rather, it is the control settings which are constructed off-line so that location of the control point and gain are determined without trial and error. Using the model independent approach also allows one to locate and control many other accessible unstable phenomena without having to construct a global nonlinear model. Tracking gives a constructive approach to control inaccessible states, as well as maps out the global regions of phase space.

Further modifications of the method proposed by Ott, Grebogi and Yorke to control chaos [Phys. Rev. Lett. 64, 1196 (1990)] have been achieved allowing us to stabilize and characterize unstable states (stationary or periodic) in their whole domain of existence. We demonstrate the possibility of stabilizing unstable periodic orbits in an experiment by applying a continuous feedback method. It has been checked experimentally on a CO₂ laser with a modulated parameter. This kind of method is very attractive opening the way to the control of chaos in very fast systems.

In most encounters, chaos is considered a nuisance, if not a down right detriment to system performance, especially in laser devices. However, the presence of chaos in a system can act as a rich source of complex frequencies if one only had a way of accessing them. In this work we present a discussion of the recent work of Ott, Grebogi and Yorke on controlling chaos as applied to a semiconductor diode laser subject to optical feedback via an external mirror. In the regime in which the laser is chaotic, stabilization can be achieved by sampling the output intensity and feeding back minuscule amounts of a correcting signal on the pumping current at the appropriate time interval. We present the results of our numerical investigations.

A theoretical investigation has been undertaken to study the amplification of small-signal by an acousto-optic bistable system operating in the Bragg regime. Numerical solutions of the difference-differential equation show strong coupling between small-signal and the optical bistable system. Near the bifurcation point an amplification of 23 is obtained. computed time plots for different parameters and the beat characteristic curve are given.

Using the concept that pattern recognition can be conceived as pattern formation, and using the mathematical methods of synergetics for the treatment of the spontaneous formation of spatial, temporal or spatio-temporal patterns, we study several laser configurations in which both, competition and pre-processing, allow one to deal with the problem of pattern recognition.

We demonstrate that the optical parametric oscillator is an ideal system for studying spatial pattern formation and quantum effects in spatial structures. In the first part of the article we analyze a semiclassical model which includes diffraction in the paraxial approximation, and we show the formation of rolls, zig-zag patterns, dislocations, filamentation, and optical chaos. In the second part of the paper we describe the spatial structure of the squeezed vacuum state emitted by the optical parametric oscillator below the threshold for signal generation.

Hexagonal structures have been observed and predicted in a number of nonlinear optical systems. After a brief general review of the field, the case of a slice of Kerr medium with single feedback mirror is analyzed in some detail. Its analogy to liquid-crystal light valve systems with feedback loop is noted. Recent experimental results are mentioned, and analytical results based on symmetry analysis are presented, along with simulations, for the practically important case of Gaussian beam excitation.

A video image circulating in a loop with a local nonlinearity and a convolution operator can be considered as a high dimensional nonlinear dynamical system, or as a specific neural net. The evolving image displays spatiotemporal deterministic chaos, oscillations, and stable patterns. The specific behavior of the system is strongly determined by the coupling of pixels, i.e., by the synaptic pattern and by the nonlinearity. The system can be trained to display a certain given image as a fixed point. Thus, it is able to associatively restore perturbed input images, independently of a spatial shift.

The paper reviews our recent research on the dynamics of a complex phase-conjugate neural network model with a Hopfield-like energy function. The model is shown to have a close analogy to optical fields generated in a phase-conjugate resonator. A physical interpretation is given to the model and its dynamics. The results of experiments and computer simulations are presented that demonstrate the behaviors of the complex neural fields predicted by the theory.

Two-dimensional patterns of a passive optical ring cavity with delayed feedback are presented in a case where multiconical emission is predicted by a linear stability analysis. Multiconical emission can indeed be obtained when a diaphragm is inserted inside the cavity. The first cone exhibits hexagonal pattern, only on the defocusing side, while a metastable pentagon is found on the focusing side; multi-cross-rolls are observed on the outer cones. A modified version of the mean field model predicting multiconical emission is discussed.

We present the results of theoretical investigations and computer simulations of switching waves and diffractive autosolitons in wide-aperture lasers with additional nonlinear elements, nonlinear interferometers, and arrays of optically coupled passive nonlinear cavities excited by laser radiation.

The nonlinear distributed optical system with rotation in 2-D feedback contour is considered. A particular case of rotational multi-petal waves is described by bifurcational periodic solutions in the form of power series expansion of a small parameter. Conditions of its stability are obtained. The finite-dimensionality of the studied system is defined in terms of the number of determining modes.

The light-induced hydrodynamic instabilities in initially homogeneously oriented nematic liquid crystals (NLC) are detected and studied for the first time at the excitation of convective flows and orientational dynamic structures with complex (periodic) topology, producing the self-induced diffraction and scattering of the light accompanied with optical bistability. These phenomena are caused solely by intrinsic multiplicative feedback arising in a spatially distributed system (highly nonlinear absorbing NLC) for an all-optical experiment without any external quasi-stationary field and initial temperature gradient.

The role of stochastic effects on the nonlinear dynamics of stimulated Brillouin scattering in single mode fiber with and without external feedback is presented. Distinction between chaos and stochastic behavior is established using newly developed measurement procedures. Also presented are first experimental observation of spatio-temporal patterns, defect formations, and preliminary evidence of defect induced turbulence in stimulated Brillouin scattering generated in multi-mode optical fiber. Results of a theoretical analysis of these phenomena are given.

Edge-emitting semiconductor lasers in different external cavity geometries were chosen to study laser chaos. The objective of this work is to obtain a general understanding of the onset of chaotic behavior in semiconductor lasers from easily interpreted experiments.

We have demonstrated experimentally a proportional feedback algorithm for the synchronization of chaotic time signals generated from a pair of independent diode resonator circuits. Synchronization was easily obtained and occurred for relative feedback levels between three and eight percent of the driving voltage. Once established, the synchronization persisted throughout the whole range of the resonator bifurcation diagram without varying the gain of the feedback.