Third Order Dispersion in Time-Delayed Systems: Applications to the Passive Mode-locking of VECSELs

Julien Javaloyes
Dept. de Física & IAC3 - Universitat de les Illes Balears. Cra. de Valldemossa, km 7.5 E-07122 Palma, SPAIN.

Time-Delayed dynamical systems (DDSs) materialize in situations where distant, point-wise, nonlinear nodes exchange information that propagates at a finite speed. They describe a large number of phenomena in nature and they exhibit a wealth of dynamical regimes such as localized structures, fronts and chimera states. A fertile perspective lies in their interpretation as spatially extended diffusive systems which holds in the limit of long delays. However, DDSs are considered devoid of dispersive effects, which are known to play a leading role in pattern formation and wave dynamics. In particular, second order dispersion in nonlinear extended media governs the Benjamin-Feir (modulational) instability and also controls the appearance of cavity solitons in injected Kerr fibers. Third order dispersion is the lowest order non-trivial parity symmetry breaking effect, which leads to convective instabilities and drifts.

In this contribution, we review our recent results regarding how second and third order dispersion may appear naturally in DDSs by using a more general class of Delayed Systems, the so-called Delay Algebraic Delay Differential Equations. This class of DDS appears for instance in the modeling of Vertical External-Cavity Surface-Emitting Lasers (VECSELs) and we illustrate our general result studying the effect of third order dispersion onto the optical pulses found in the output of a passively mode-locked VECSEL and link our results with the Gires-Tournois interferometer. We show that third order dispersion leads to the creation of satellites on one edge of the pulse which induces a new form of pulse instability. Our results are in good agreement with the experiment. Finally, we connect these results with the possibility of obtaining Light bullets, that is to say, pulses of light that are simultaneously confined in the transverse and the propagation directions, in mode-locked VECSELs.