A two-parameter study of the locking region of a semiconductor laser subject to phase-conjugate feedback

We present a detailed bifurcation analysis of a single-mode semiconductor laser subject to phase-conjugate feedback, a system described by a delay differential equation. Codimension-one bifurcation curves of equilibria and periodic orbits and curves of certain connecting orbits are presented near the laser's locking region in the two-dimensional parameter plane of feedback strength and pump current. We identify several codimension-two bifurcations, including a double-Hopf point, Belyakov points, and a T-point bifurcation, and we show how they organize the dynamics. This study is the first example of a two-parameter bifurcation study, including bifurcations of periodic and connecting orbits, of a delay system. It was made possible by new numerical continuation tools, implemented in the package DDE-BIFTOOL, and showcases their usefulness for the study of delay systems arising in applications

MULTISCROLL ATTRACTORS IN SEMICONDUCTOR LASER BY OPTICAL FEEDBACK AND DIRECT CURRENT MODULATION

Chaotic behavior with multiscroll attractors and equilibrium points of semiconductor laser dynamics subjected to optical delay feedback and sinusodial injection current modulation observed numerically. The complicated dynamical behavior performed based on numerical simulation of modified Lang-Kobayashi model with direct current modulation term. The results reveal different dynamical regimes involving steady state, periodic, quas-periofdic, mixed modes, chaotic state with high power and 1-D 10 scrll attractors. These dynamics analyzed by sequences of observation analysis, FFT, phase portrait in two and three dimensions that qualify sensitivity of the system to initial conditions and give measure of the rate at which the trajectories separate one from the other(fixed point attractor). These prove important use in Chaos synchronization and networks

SPILKING GENERATION AND SYNCHRONIZATION IN SEMICONDUCTOR LASER BY MEANS OF OPTICAL FEEDBACK

Nonlinear dynamics of a semiconductor laser subjected optical feedback observed numerically. The investigation performed based on numerical simulation of Lang-Kobayashi time delay rate equations over wide range of optical feedback strength. The results show that under small, moderate and high optical feedback strength semiconductor laser output power goes different dynamical regimes involving steady state, periodic, mixed modes and chaotic spiking. These dynamics analyzed by time series and their FFt power spectrum with phase space trajectory. The bifurcation diagram is drawn as a function of optical feedback strength. Chaos synchronization in unidirectional coupling scheme numerically presented

Deterministic continutation of stochastic metastable equilibria via Lyapunov equations and ellipsoids

Numerical continuation methods for deterministic dynamical systems have been one of the most successful tools in applied dynamical systems theory. Continuation techniques have been employed in all branches of the natural sciences as well as in engineering to analyze ordinary, partial and delay differential equations. Here we show that the deterministic continuation algorithm for equilibrium points can be extended to track information about metastable equilibrium points of stochastic differential equations (SDEs). We stress that we do not develop a new technical tool but that we combine results and methods from probability theory, dynamical systems, numerical analysis, optimization and control theory into an algorithm that augments classical equilibrium continuation methods. In particular, we use ellipsoids defining regions of high concentration of sample paths. It is shown that these ellipsoids and the distances between them can be efficiently calculated using iterative methods that take advantage of the numerical continuation framework. We apply our method to a bistable neural competition model and a classical predator-prey system. Furthermore, we show how global assumptions on the flow can be incorporated - if they are available - by relating numerical continuation, Kramers' formula and Rayleigh iteration.Comment: 29 pages, 7 figures [Fig.7 reduced in quality due to arXiv size restrictions]; v2 - added Section 9 on Kramers' formula, additional computations, corrected typos, improved explanation

Bifurcation analysis of a spatially extended laser with optical feedback

Vertical cavity surface-emitting lasers (VCSELs) are a new type of semiconductor laser characterized by the spatial extent of their disk-shaped output apertures. As a result, a VCSEL supports several optical modes (patterns of light) transverse to the direction of light propagation. When any laser is coupled to other optical elements, there is unavoidable optical feedback via reflecting surfaces, which influences the stability of the laser output. For a VCSEL, the question is how the transverse optical modes interact dynamically in the presence of optical feedback and how this affects stability of the system. In this paper, we start from a PDE description of the VCSEL. We proceed by using an expansion in suitable eigenfunctions to resolve the spatial dependence. In the presence of optical feedback we obtain a model in the form of a system of delay differential equations (DDEs). As we show with the example of a VCSEL that supports two transverse modes, the spatially expanded DDE model is small enough to allow for a multiparameter bifurcation analysis with numerical continuation tools. Specifically, we present stability regions of steady states and periodic solutions in dependence on the feedback strength and a homotopy parameter that models the amount of self- versus crossfeedback between the two modes. Bifurcations of more complicated spatiotemporal mode dynamics are then discussed. © 2009 Society for Industrial and Applied Mathematics

Dynamics of delay-coupled semiconductor laser systems

Nonlinear laser dynamics has received considerable attention because of possible applications, but also fundamental physical and mathematical aspects are of great interest. This thesis is concerned with the dynamical behavior of semiconductor lasers subject to external delayed perturbations. In particular the time delay in the coupling to external elements is of importance, because it substantially complicates the dynamical behavior. This time delay arises from finite signal propagation times and, hence, is large compared to the laser internal time scales so that it cannot be neglected. Specifically, the thesis investigates two different delay-coupled semiconductor laser systems: (I) a semiconductor laser subject to delayed filtered optical feedback, where a part of the laser emission is filtered by a Fabry-Perot filter and then feed back into the laser, and (II) two semiconductor lasers that are mutually delay-coupled via their optical fields. With concepts and tools from dynamical systems theory a comprehensive study of the underlying bifurcation structure of two systems is presented. Knowledge of this underlying structure is the key to understanding complicated laser dynamics. The results from the bifurcation analysis are interpreted in terms of the dynamics of the real laser system and compared with experiments.Krauskopf, B. [Promotor]Lenstra, D. [Promotor