Spatio-temporal dynamics in semiconductor lasers
with delayed optical feedback
M. Münkel 
, F. Kaiser , O. Hess 
Institute for Applied Physics - Nonlinear Dynamics,
Technical University Darmstadt
Hochschulstrasse 4a,
64289 Darmstadt, Germany;
Institute for Technical Physics, DLR,
Pfaffenwaldring 38-40,
70569 Stuttgart, Germany
We investigate for the first time the influence of delayed optical feedback (DOF) on the dynamics of spatially extended laser systems. Focusing on multi-stripe semiconductor laser arrays, we show in which way DOF may trigger or inhibit spatio-temporal instabilities. For comparison with earlier results obtained on the basis of plane-wave models we initially discuss the single-stripe laser (SSL) with DOF. While the SSL without feedback always approaches cw-operation, we find that the presence of DOF leads to periodic, period-doubled, quasiperiodic and chaotic solutions. We identify the excitation of relaxation oscillations (RO) as a key mechanism causing the instabilities. Performing an eigenmode analysis by means of a singular value decomposition reveals that spatio-temporal effects are already present in this simple device. Then we turn to the twin-stripe laser (TSL) configuration which we may consider as one of the most simple semiconductor laser systems in which true spatio-temporal instabilities occur [1]. In the TSL, the spatially and temporally varying coupling between the two laser stripes due to diffraction and carrier diffusion has proven to be of crucial importance. Our results show that modified plane-wave ('coupled-mode') models intended to model the dynamics of coupled laser arrays on the assumption of discrete interstripe coupling are indeed insufficient in describing the spatio-temporal behavior of the TSL [2] and are to be considered even more inappropriate in the presence of DOF. Starting from a set of parameters, for which the TSL without DOF performs fast periodic 'strongly coupled oscillations (SCO)', DOF is found to induce spatio-temporal instabilities leading to spatio-temporal chaos. The singular value decomposition enables us to determine and identify the relevant spatio-temporal 'supermodes'. We identify the interplay between these modes representing antisymmetric SCO, symmetric RO and stripe coupling, respectively, as the cause for DOF-induced spatio-temporal complexity. Applying small feedback levels, we observe slow, lateral motions of the filaments corresponding to regular or irregular decoupling of the stripes. For higher feedback levels, we observe a competition between fast SCO (typical for the TSL) and slower RO (inherent in the isolated SSL). This characteristic intermittent behavior of the TSL under DOF-conditions is closely related to transitions between various synchronization states and symmetry properties. On the other hand, we show that delayed optical feedback can also produce coherent regimes from a state of spatio-temporal chaos. For appropriately chosen parameters, we managed to obtain periodic and even cw solutions. This last point might open the prospect for all-optical control-schemes highly relevant to a large number of applications. Finally, we present results for multi-stripe laser arrays which, on the one hand, display a considerably higher degree of spatio-temporal complexity but, on the other hand, show many analogies to the behaviour found in the TSL.