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D Pieroux

Publications and source records attributed to D Pieroux.

9 recordsLinked to original sources
PubMed2003

Bifurcation diagram of a complex delay-differential equation with cubic nonlinearity.

We reduce the Lang-Kobayashi equations for a semiconductor laser with external optical feedback to a single complex delay-differential equation in the long delay-time limit. The reduced equation has a time-delayed linear term and a cubic instantaneous nonlinearity. There are only two parameters, the real linewidth enhancement factor and the complex feedback strength. The equation displays a very rich dynamics and can sustain steady, periodic, quasiperiodic, and chaotic regimes. We study the steady solutions analytically and analyze the periodic solutions by using a numerical continuation method. This leads to a bifurcation diagram of the steady and periodic solutions, stable and unstable. We illustrate the chaotic regimes by a direct numerical integration and show that low frequency fluctuations still occur.

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PubMed2003

Dynamical properties of lasers coupled face to face.

We derive a reduced model to describe two identical lasers coupled face to face. Two limits are introduced in the Maxwell-Bloch equations: adiabatic elimination of the material polarization and large distance between the two lasers. The resulting model describes coupled homogeneously broadened lasers, including semiconductor lasers. It consists of two coupled delay differential equations with delayed linear cross-coupling and an instantaneous self-coupling nonlinearity. The study is analytical and numerical. We focus on the properties of steady and periodic amplitudes of the electric fields. In steady state, there are symmetric, antisymmetric, and asymmetric solutions with respect to a permutation of the two fields. A similar classification holds for the periodic states. The stability of these solutions is determined partly analytically and partly numerically. A homoclinic point is associated with the asymmetric periodic solutions.

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PubMed2002

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.

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PubMed2001

Bridges of periodic solutions and tori in semiconductor lasers subject to delay.

For semiconductor lasers subject to a delayed optical feedback, branches of steady states sequentially appear as the feedback rate is increased. But branches of time-periodic solutions are connecting pairs of steady states and provide bridges between stable and unstable modes. All bridges experience a change of stability through a torus bifurcation point. Close to the bifurcation point, the torus remains localized near a specific fixed point in phase space. As the feedback rate increases, the torus envelope suddenly unfolds and its trajectory visits two or more unstable fixed points, anticipating the rich dynamics observed at larger feedback rates.

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PubMed2001

Interacting pairs of periodic solutions lead to tori in lasers subject to delayed feedback.

Models of class-B lasers subject to either an optoelectronic or an optical feedback are investigated analytically and numerically. We derive slow time amplitude equations from the laser delay differential equations and find multiple bifurcating and isolated branches of periodic solutions. We then show that secondary bifurcations to tori result from the interaction of pairs of Hopf bifurcations. The branches emerging from these bifurcations are followed numerically using a continuation method developed for delay differential equations.

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