Michel Nizette

Towards a classification of Euler-Kirchhoff filaments

Euler–Kirchhoff filaments are solutions of the static Kirchhoff equations for elastic rods with circular cross sections. These equations are known to be formally equivalent to the Euler equations for spinning tops. This equivalence is used to provide a classification of the different shapes a filament can assume. Explicit formulas for the different possible configurations and specific results for interesting particular cases are given. In particular, conditions for which the filament has points of self-intersection, self-tangency, vanishing curvature or when it is closed or localized in space are provided. The average properties of generic filaments are also studied. They are shown to be equivalent to helical filaments on long length scales.

On the Dynamics of Elastic Strips

Summary. {The dynamics of elastic strips, i.e., long thin rods with noncircular cross section, is analyzed by studying the solutions of the appropriate Kirchhoff equations. First, it is shown that if a naturally straight strip is deformed into a helix, the only equilibrium helical configurations are those with no internal twist and whose principal bending direction is either along the normal or the binormal. Second, the linear stability of a straight twisted strip under tension is analyzed, showing the possibility of both pitchfork and Hopf bifurcations depending on the external and geometric constraints. Third, nonlinear amplitude equations are derived describing the dynamics close to the different bifurcation regimes. Finally, special analytical solutions to these equations are used to describe the buckling of strips. In particular, finite-length solutions with a variety of boundary conditions are considered. }

Dynamics of vertical-cavity surface-emitting lasers with optical injection: a two-mode model approach

We present a comprehensive study of the bifurcation routes in a generic model of a two-polarization-mode semiconductor laser when subject to orthogonal optical injection, relevant to a single-transverse-mode vertical-cavity surface-emitting laser. For positive detunings, polarization switching (PS) occurs as the intensity of the noninjected mode gradually decreases and eventually shuts down, in some cases accompanied along the way by the onset and the subsequent vanishing of relaxation oscillations. We identify a Hopf bifurcation on a two-polarization-mode solution, which supports the onset of relaxation oscillation time-periodic dynamics in both injected (x-linearly polarized) and noninjected (y-linearly polarized) modes. For sufficiently large negative detunings, PS is a sharp transition occurring either at the injection-locking point or at a subcritical Hopf bifurcation and is often part of a hysteresis cycle.

Excitation of a two-mode limit cycle dynamics on the route to polarization switching in a VCSEL subject orthogonal to optical injection

We experimentally and numerically report on polarization switching (PS) mechanism which involves a two-mode limit cycle dynamics in a vertical-cavity surface-emitting laser (VCSEL) subject to orthogonal optical injection from a master laser (ML). The VCSEL (slave laser, SL) emits a horizontal linearly polarized (LP) fundamental mode, without optical injection. The VCSEL is injected by a vertically polarized light from ML. Dynamical characteristics of the VCSEL are investigated as a function of optical injection parameters, i.e., injection strength and frequency detuning between master and slave lasers. We experimentally resolve an injection parameter region for which, as the injection strength is increased for fixed detunings, a limit cycle dynamics in both non-injected and injected modes is abruptly excited. For larger injection strengths, the VCSEL switches from the two-mode to a single-mode limit cycle dynamics which involves only the injected mode. Using continuation methods, we numerically identify two torus bifurcation mechanisms, namely TR1 and TR2, which support such a switching scenario. We show that both TR1 and TR2 originate from a particular Hopf bifurcation which plays a key role in the polarization dynamics of the injected VCSEL. Furthermore, our results reveal that the newly observed switching dynamics are generic features of VCSEL two-mode systems.

Polarization dynamics in vertical-cavity surface-emitting lasers subject to optical injection or current modulation

1) Department of Applied Physics and Photonics (TW-TONA) of the Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels (Belgium) e-mail address: kpanajotov@tona.vub.ac.be 2) Institute of Solid State Physics, Tzarigradsko Chaussee blvd., 1784 Sofia, Bulgaria. 3) Université Libre de Bruxelles, Optique Nonlineáire Théorique, Campus Plaine, Code Postal Interne 231, 1050 Bruxelles, Belgium 4) Supélec and the Laboratoire Matériaux Optiques, Photonique et Systèmes (LMOPS), common laboratory between the University of Metz and Supélec, CNRS UMR-7132, 2 Rue Edouard Belin, F-57070 Metz (France) 5) Instituto de Física de Cantabria, CSIC-Universidad de Cantabria, E-39005 Santander, Spain

Bifurcation and nonlinear dynamics accompanying polarization switching in a VCSEL subject to orthogonal optical injection

In this paper, we investigate theoretically the interplay between polarization switching (PS) and nonlinear dynamics in a vertical-cavity surface emitting laser (VCSEL) subject to orthogonal optical injection from a master laser (ML). The free-running VCSEL is biased such that only the horizontal (x) linearly polarized fundamental (LP) mode is excited. The injected VCSEL is then modeled using an extension of the spin flip model (SFM). We perform a bifurcation analysis using continuation techniques that allow tracking bifurcation of both stable and unstable solutions.

Delay differential equations for passive-mode locking

This study analyzes a model for passive mode locking in which the approximations of small gain and loss per cavity round trip and weak saturation are avoided. The model is thus capable of describing mode locking in the parameter range of semiconductor lasers.