The Incipit of Complexity in Self-Coupled Lasers (from Deterministic Behaviour to Periodic Oscillations and to Chaos)

At the simplest level of implementation for an optical system, chaos and complexity are generated by the self-coupled laser configuration, when the field back-reflected from a remote target is allowed to re-enter the laser cavity. The delay of the go-and-return path is the parameter governing complexity, and together with the second-order cavity resonance and coupling of gain to amplitude and phase, gives rise to a number of characteristic phenomena at increasing coupling, such as perturbed deterministic behavior, onset of period-one and period-two bifurcations, and finally route to chaos. We present detailed results of the regimes found in a semiconductor laser subject to a minute (a few percent) back-reflection, as a function of strenght of coupling K, propagation delay, and alfa-factor (or linewidth broadening). The portrait of regimes in the K-phi plane, where phi= 2ks is the phase of the field returning from target distance s and k= 2 pi/lambda is the wavevector, starts with unperturbed and period-1 bands to proceed, as distance s is increased, to zones of periodic regime sandwiched with increasing chaos bands, up to the point that all the K-phi space is filled with chaos. Using the new results, we will revisit the Tchak and Chaprivly (T-C) diagram of self-coupling, that is commonly used to diagnose disturbance effects in laser diodes intended for optical fiber communications. We show that the T-C diagram unveils intricated overlap of regimes. We then conclude with a focus on the several applications of the high level dynamics to measurement of optical paths, cryptography for communication of sensible data, and to high speed random number generation for computer applications.