Global bifurcations and bistability at the locking boundaries of a semiconductor laser with phase-conjugate feedback

Delay-induced homoclinic bifurcations in modified gradient bistable systems and their relevance to optimization

Nonlinear dynamical systems with time delay are abundant in applications but are notoriously difficult to analyze and predict because delay-induced effects strongly depend on the form of the nonlinearities involved and on the exact way the delay enters the system. We consider a special class of nonlinear systems with delay obtained by taking a gradient dynamical system with a two-well "potential" function and replacing the argument of the right-hand side function with its delayed version. This choice of the system is motivated by the relative ease of its graphical interpretation and by its relevance to a recent approach to use delay in finding the global minimum of a multi-well function. Here, the simplest type of such systems is explored for which we hypothesize and verify the possibility to qualitatively predict the delay-induced effects, such as a chain of homoclinic bifurcations one by one eliminating local attractors and enabling the phase trajectory to spontaneously visit vicinities of all local minima. The key phenomenon here is delay-induced reorganization of manifolds, which cease to serve as barriers between the local minima after homoclinic bifurcations. Despite the general scenario being quite universal in two-well potentials, the homoclinic bifurcation comes in various versions depending on the fine features of the potential. Our results are a pre-requisite for understanding general highly nonlinear multistable systems with delay. They also reveal the mechanisms behind the possible role of delay in optimization.

Optimized properties of chaos from a laser diode

We perform an experimental parametric study of the chaos generated by a laser diode subjected to phase-conjugate feedback. In addition to the typical figure of merit, i.e., chaos bandwidth, the corresponding spectral flatness and permutation entropy at delay is analyzed. Our experimental observations reveal that the chaos can be generated with a bandwidth of ≈29 GHz, a spectral flatness up to 0.75, and a permutation entropy at delay of up to 0.99. These optimized performances are maintained over a large range of parameters and have not been achieved in the conventional optical feedback configuration. Interestingly, reducing the pump current reduces the chaos bandwidth while keeping the spectral flatness and the permutation entropy at delay the same as observed for increased pump current. Our experimental findings are consistent with the presented numerical simulations produced using the Lang-Kobayashi model.

Mapping of external cavity modes for a laser diode subject to phase-conjugate feedback

We numerically investigate the dynamics of a semiconductor laser subject to phase-conjugate optical feedback. We explore the effects of the laser model and feedback parameters for the generation of time-periodic oscillations of the output power at harmonics of the external cavity frequency, i.e., dynamical solutions that have been named external cavity modes. We point out that both the experimentally tunable and other parameters have an influence on the frequency of such dynamics. Since the delay has to exist, it is not the relevant parameter as we show that the feedback rate fixes the frequency of the periodic self-pulsations. The interaction length of the crystal and the ratio between carrier and photon lifetimes tend to filter out high frequencies as they increase. Finally, the linewidth enhancement factor unlocks high frequencies as it increases. We conclude by providing a situation which leads to periodic solutions with higher frequencies using a set of realistic values of parameters.

High-order external cavity modes and restabilization of a laser diode subject to a phase-conjugate feedback

We experimentally report the sequence of bifurcations destabilizing and restabilizing a laser diode with phase-conjugate feedback when the feedback rate is increased. Specifically, we successively observe the initial steady state, undamped relaxation oscillations, quasi-periodicity, chaos, and oscillating solutions at harmonics up to 13 times the external cavity frequency but also the restabilization to a steady state. The experimental results are qualitatively well reproduced by a model that accounts for the time the light takes to penetrate the phase-conjugate mirror. The theory points out that the system restabilizes through a Hopf bifurcation whose frequency is a harmonic of the external cavity frequency.

Laser diode nonlinear dynamics from a filtered phase-conjugate optical feedback

The rate equations for a laser diode subject to a filtered phase-conjugate optical feedback are studied both analytically and numerically. We determine the Hopf bifurcation conditions, which we explore by using asymptotic methods. Numerical simulations of the laser rate equations indicate that different pulsating intensity regimes observed for a wide filter progressively disappear as the filter width increases. We explain this phenomenon by studying the coalescence of Hopf bifurcation points as the filter width increases. Specifically, we observe a restabilization of the steady-state solution for moderate width of the filter. Above a critical width, an isolated bubble of time-periodic intensity solutions bounded by two successive Hopf bifurcation points appears in the bifurcation diagram. In the limit of a narrow filter, we then demonstrate that only two Hopf bifurcations from a stable steady state are possible. These two Hopf bifurcations are the Hopf bifurcations of a laser subject to an injected signal and for zero detuning.

Numerical study of extreme events in a laser diode with phase-conjugate optical feedback

Extreme intensity pulses sharing statistical properties similar to rogue waves have been recently observed in a laser diode with phase-conjugate feedback [A. Karsaklian Dal Bosco, D. Wolfersberger, and M. Sciamanna, Opt. Lett. 38, 703 (2013)], but remain unexplained. We demonstrate here that a rate equation model of a laser diode that includes an instantaneous phase-conjugate feedback field reproduces qualitatively well the statistical features of these extreme events as identified in the experiment, i.e., the deviation of the intensity statistics to a Gaussian-shape statistics and the statistics of the time separating extreme events. The numerical simulations confirm the importance of the feedback strength in increasing the number of such extreme events and allow us to explain how extreme events emerge from a sequence of bifurcations on self-pulsating solutions, the so-called external cavity modes.

Global bifurcation investigation of an optimal velocity traffic model with driver reaction time

We investigate an optimal velocity model which includes the reflex time of drivers. After an analytical study of the stability and local bifurcations of the steady-state solution, we apply numerical continuation techniques to investigate the global behavior of the system. Specifically, we find branches of oscillating solutions connecting Hopf bifurcation points, which may be super- or subcritical, depending on parameters. This analysis reveals several regions of multistability.

External cavity modes of semiconductor lasers with phase-conjugate feedback

External cavity modes (ECMs) of a semiconductor laser with phase-conjugate feedback are defined as time-periodic pulsating intensity solutions exhibiting a frequency close to an integer multiple of the external cavity frequency. As the feedback rate progressively increases from zero, they sequentially appear as stable attractors in the bifurcation diagram. We construct a simple analytical approximation of these pulsating intensity solutions and determine their frequencies. We show that branches of ECMs are isolated. Finally, the validity of our approximation is tested by comparing numerical bifurcation diagrams obtained by simulation and continuation techniques with our analytical results.

Delay dynamics of semiconductor lasers with short external cavities: bifurcation scenarios and mechanisms

We present a comprehensive study of the emission dynamics of semiconductor lasers induced by delayed optical feedback from a short external cavity. Our analysis includes experiments, numerical modeling, and bifurcation analysis by means of computing unstable manifolds. This provides a unique overview and a detailed insight into the dynamics of this technologically important system and into the mechanisms leading to delayed feedback instabilities. By varying the external cavity phase, we find a cyclic scenario leading from stable intensity emission via periodic behavior to regular and irregular pulse packages, and finally back to stable emission. We reveal the underlying interplay of localized dynamics and global bifurcations.

Stability and rupture of bifurcation bridges in semiconductor lasers subject to optical feedback

The bifurcation diagram of a single-mode semiconductor laser subject to a delayed optical feedback is examined by using numerical continuation methods. For this, we show how to cope with the special symmetry properties of the equations. As the feedback strength is increased, branches of modes and antimodes appear, and we have found that pairs of modes and antimodes are connected by closed branches of periodic solutions (bifurcation bridges). Such connections seem generically present as new pairs of modes and antimodes appear. We subsequently investigate the behavior of the first connection as a function of the linewidth enhancement factor and the feedback phase. Our results extend and confirm existing results and hypotheses reported in the literature. For large values of the linewidth enhancement factor (alpha=5-6), bridges break through homoclinic orbits. Changing the feedback phase unfolds the bifurcation diagram of the modes and antimodes, allowing different types of connections between modes.