Synchronization of High-Dimensional Dynamical Systems

Abstract

There are many examples of high-dimensional systems in nature. Often these systems behave in synchrony even though they possess a large number of degrees of freedom. Far fewer of these types of systems exist in the laboratory, and even fewer techniques exist with which to analyze them. As experimental capability increases, and more high-dimensional laboratory systems are fabricated, universal tools must be developed to observe and analyze the dynamics of these systems. This thesis will present experiments and analysis of two high-dimensional systems, coupled fiber ring lasers and a liquid crystal spatial light modulator with optoelectronic feedback. Two identically constructed mutually coupled erbium doped fiber ring lasers were studied and were found to synchronize at very low coupling strengths. Synchronization error was characterized as a function of coupling strength. Optical frequency-locking and hopping as a result of the mutual coupling was also observed. Methods for detecting the leader and follower laser as well as role switching, a form of spontaneous symmetry-breaking, were developed. These include a spatiotemporal representation of the intensities within each ring laser and the use of Karhunen-Loeve decomposition. A delay-differential equation model was developed and the numerical simulations were in agreement with the experiment. Chaotic communication was achieved in this system with bit rates of 125 MHz, limited by the detection speed. A liquid crystal spatial light modulator (SLM) was also studied. When used as a dynamic holographic grating, this device allowed the fabrication of a variety of reshaped laser beams, including multiple Gaussian beams, optical billiards, and propagating Bessel beams. When configured in an optoelectronic feedback loop, the SLM displays spatiotemporal chaos and using the auxiliary system method, we have achieved generalized synchronization of this system. The space-time patterns as well as the transients to synchronization have been characterized as a function of the bias voltage across the liquid crystal. The analysis techniques used in this thesis can be applied to other high-dimensional systems.