Zero-lag synchronization and bubbling in delay-coupled lasers

We show experimentally that two semiconductor lasers mutually coupled via a passive relay fiber loop exhibit chaos synchronization at zero lag, and study how this synchronized regime is lost as the lasers’ pump currents are increased. We characterize the synchronization properties of the system with high temporal resolution in two different chaotic regimes, namely, low-frequency fluctuations and coherence collapse, identifying significant differences between them. In particular, a marked decrease in synchronization quality develops as the lasers enter the coherence collapse regime. Our high-resolution measurements allow us to establish that synchronization loss is associated with bubbling events, the frequency of which increases with increasing pump current.

Figure 1

Schematic experimental setup (top panel). PC: polarization controller, PD: fast photodetector, OC: optical coupler, OI: optical isolator. Lower panel: (a) Experimental time series of synchronized fast intensity dynamics in the coherence collapse regime. A short desynchronization event is highlighted. The pump current corresponds to $1.25I_{\text{thr}}$, with $I_{\text{thr}}$ being the solitary lasing threshold. (b) Corresponding normalized intensity difference (synchronization error).

Figure 2

Cross-correlation at zero lag (black circles), fraction of the sliding cross-correlation above the correlation threshold of $C_{\mathrm{thr}}=0.5$ (red squares), and mean integrated synchronization error $\chi$ (blue diamonds), respectively, vs the applied pump current and the pump current normalized with respect to $I_{\mathrm{thr}}$.

Figure 3

Synchronization of LFF dynamics. Output intensity time series of the two lasers (a, b) and corresponding sliding cross-correlation (c, d) for a long time interval (a, c) and a magnification in time (b, d). The intensity time series were vertically shifted for better visibility. The applied pump current was $I_p=12$ mA, which is closely above the solitary thresholds of both lasers.

Figure 4

Synchronization in the coherence collapse regime. Output intensity time series of the two lasers (a, b) and corresponding sliding cross-correlation (c, d) for a long time interval (a, c) and a magnification in time (b, d). The intensity time series were vertically shifted for better visibility. The pump current was $I_p=16$ mA $\approx 1.3I_{\text{thr}}$.

Figure 5

Normalized distributions of the sliding cross-correlation coefficients for six different pump currents. The histograms have 401 bins.

Figure 6

Histograms of the interevent intervals (left) and the bubbling duration (right) for pump currents $I_p=14–17$ mA. A bubbling event is considered to occur when the SLCC decreases below a value $C_{\mathrm{thr}}=0.5$. We also introduce a threshold for a minimum event duration $\Delta T_{\mathrm{thr}}=0.5$ ns, and a minimum interevent interval IEI${}_{\mathrm{thr}}=0.5$ ns below which two isolated desychronization events are considered part of a single, longer one.