Shear-induced bifurcations and chaos in models of three coupled lasers

Copyright © 2011 Society for Industrial and Applied MathematicsWe study nonlinear dynamics in a linear array of three coupled laser oscillators with rotational $\mathbb{S}^1$ and reflectional $\mathbb{Z}_2$ symmetry. The focus is on a coupled-laser model with dependence on three parameters: laser coupling strength, $\kappa$, laser frequency detuning, $\Delta$, and degree of coupling between the amplitude and phase of the laser, $\alpha$, also known as shear or nonisochronicity. Numerical bifurcation analysis is used in conjunction with Lyapunov exponent calculations to study the different aspects of the system dynamics. First, the shape and extent of regions with stable phase locking in the $(\kappa,\Delta)$ plane change drastically with $\alpha$. We explain these changes in terms of codimension-two and -three bifurcations of (relative) equilibria. Furthermore, we identify locking-unlocking transitions due to global homoclinic and heteroclinic bifurcations and the associated infinite cascades of local bifurcations. Second, vast regions of deterministic chaos emerge in the $(\kappa,\Delta)$ plane for nonzero $\alpha$. We give an intuitive explanation of this effect in terms of $\alpha$-induced stretch-and-fold action that creates horseshoes and discuss chaotic attractors with different topologies. Similar analysis of a more accurate composite-cavity mode model reveals good agreement with the coupled-laser model on the level of local and global bifurcations as well as chaotic dynamics, provided that coupling between lasers is not too strong. The results give new insight into modeling approaches and methodologies for studying nonlinear behavior of laser arrays

Parameter-dependent behaviour of periodic channels in a locus of boundary crisis

This is the final version of the article. Available from EDP Sciences via the DOI in this recordA boundary crisis occurs when a chaotic attractor outgrows its basin of attraction and suddenly disappears. As previously reported, the locus of a boundary crisis is organised by homo- or heteroclinic tangencies between the stable and unstable manifolds of saddle periodic orbits. In two parameters, such tangencies lead to curves, but the locus of boundary crisis along those curves exhibits gaps or channels, in which other non-chaotic attractors persist. These attractors are stable periodic orbits which themselves can undergo a cascade of period-doubling bifurcations culminating in multi-component chaotic attractors. The canonical diffeomorphic two-dimensional Hénon map exhibits such periodic channels, which are structured in a particular ordered way: each channel is bounded on one side by a saddle-node bifurcation and on the other by a period-doubling cascade to chaos; furthermore, all channels seem to have the same orientation, with the saddle-node bifurcation always on the same side. We investigate the locus of boundary crisis in the Ikeda map, which models the dynamics of energy levels in a laser ring cavity. We find that the Ikeda map features periodic channels with a richer and more general organisation than for the Hénon map. Using numerical continuation, we investigate how the periodic channels depend on a third parameter and characterise how they split into multiple channels with different properties

Locking bandwidth of two laterally coupled semiconductor lasers subject to optical injection

We report here for the first time (to our knowledge), a new and universal mechanism by which a two-element laser array is locked to external optical injection and admits stably injection-locked states within a nontrivial trapezoidal region. The rate equations for the system are studied both analytically and numerically. We derive a simple mathematical expression for the locking conditions, which reveals that two parallel saddle-node bifurcation branches, not reported for conventional single lasers subject to optical injection, delimit the injection locking range and its width. Important parameters are the linewidth enhancement factor, the laser separation, and the frequency offset between the two laterally-coupled lasers; the influence of these parameters on locking conditions is explored comprehensively. Our analytic approximations are validated numerically by using a path continuation technique as well as direct numerical integration of the rate equations. More importantly, our results are not restricted by waveguiding structures and uncover a generic locking behavior in the lateral arrays in the presence of injection

Nonlinear dynamics of solitary and optically injected two-element laser arrays with four different waveguide structures: A numerical study

We study the nonlinear dynamics of solitary and optically injected two-element laser arrays with a range of waveguide structures. The analysis is performed with a detailed direct numerical simulation, where high-resolution dynamic maps are generated to identify regions of dynamic instability in the parameter space of interest. Our combined one- and two-parameter bifurcation analysis uncovers globally diverse dynamical regimes (steady-state, oscillation, and chaos) in the solitary laser arrays, which are greatly influenced by static design waveguiding structures, the amplitude-phase coupling factor of the electric field, i.e. the linewidth-enhancement factor, as well as the control parameter, e.g. the pump rate. When external optical injection is introduced to one element of the arrays, we show that the whole system can be either injection-locked simultaneously or display rich, different dynamics outside the locking region. The effect of optical injection is to significantly modify the nature and the regions of nonlinear dynamics from those found in the solitary case. We also show similarities and differences (asymmetry) between the oscillation amplitude of the two elements of the array in specific well-defined regions, which hold for all the waveguiding structures considered. Our findings pave the way to a better understanding of dynamic instability in large arrays of lasers

A Journey Through the Dynamical World of Coupled Laser Oscillators

The focus of this thesis is the dynamical behaviour of linear arrays of laser oscillators with nearest-neighbour coupling. In particular, we study how laser dynamics are influenced by laser-coupling strength, $\kappa$, the natural frequencies of the uncoupled lasers, $\tilde{\Omega}_j$, and the coupling between the magnitude and phase of each lasers electric field, $\alpha$. Equivariant bifurcation analysis, combined with Lyapunov exponent calculations, is used to study different aspects of the laser dynamics. Firstly, codimension-one and -two bifurcations of relative equilibria determine the laser coupling conditions required to achieve stable phase locking. Furthermore, we find that global bifurcations and their associated infinite cascades of local bifurcations are responsible for interesting locking-unlocking transitions. Secondly, for large $\alpha$, vast regions of the parameter space are found to support chaotic dynamics. We explain this phenomenon through simulations of $\alpha$-induced stretching-and-folding of the phase space that is responsible for the creation of horseshoes. A comparison between the results of a simple {\it coupled-laser model} and a more accurate {\it composite-cavity mode model} reveals a good agreement, which further supports the use of the simpler model to study coupling-induced instabilities in laser arrays. Finally, synchronisation properties of the laser array are studied. Laser coupling conditions are derived that guarantee the existence of synchronised solutions where all the lasers emit light with the same frequency and intensity. Analytical stability conditions are obtained for two special cases of such laser synchronisation: (i) where all the lasers oscillate in-phase with each other and (ii) where each laser oscillates in anti-phase with its direct neighbours. Transitions from complete synchronisation (where all the lasers synchronise) to optical turbulence (where no lasers synchronise and each laser is chaotic in time) are studied and explained through symmetry breaking bifurcations. Lastly, the effect of increasing the number of lasers in the array is discussed in relation to persistent optical turbulence