Bubbling in delay-coupled lasers

We theoretically study chaos synchronization of two lasers which are delay coupled via an active or a passive relay. While the lasers are synchronized, their dynamics is identical to a single laser with delayed feedback for a passive relay and identical to two delay-coupled lasers for an active relay. Depending on the coupling parameters the system exhibits bubbling, i.e., noise-induced desynchronization, or on-off intermittency. We associate the desynchronization dynamics in the coherence collapse and low-frequency fluctuation regimes with the transverse instability of some of the compound cavity’s antimodes. Finally, we demonstrate how, by using an active relay, bubbling can be suppressed.

Figure 1

Schematic setup.

Figure 2

(Color online) Maximum transversal Lyapunov exponent ${\lambda }_{\perp }$ (red dashed) and maximum parallel Lyapunov exponent ${\lambda }_{\parallel }$ (blue solid) as a function of the feedback strength $K$ for (a) passive relay and (b) active relay $\left(p_{\text{relay}}=4.0\right)$. At the two blow-out bifurcations $B1$ and $B2$ the maximum transversal Lyapunov exponent of the chaotic attractor changes sign. Other parameters: $T=200$, $p=1.0$, $\tau =1000$, $\alpha =4$, $\phi =0$

Figure 3

Carrier density of the symmetric variable $n_{\mathbf{S}}=\frac{1}{2}\left(n_1+n_2\right)$ and intensity difference $\left|I_1-I_2\right|/⟨I_1+I_2⟩$ (normalized by the mean intensity) representing the deviation from the synchronized state vs time. (a) Bubbling in the coherence collapse regime $\left(p=1.0\right)$. (b) Bubbling in the low-frequency fluctuation regime during power dropouts $\left(p=0.1\right)$. Other parameters: $T=200$, $K=0.12$, $\tau =1000$, $\alpha =4$.

Figure 4

(Color online) Projection of the dynamics of the symmetrized solution $n_{\mathbf{S}}$, $E_{\mathbf{S}}=A_{\mathbf{S}}\text{exp}\left(i{\varphi }_{\mathbf{S}}\right)$ (black trajectory) onto the $\left(\omega ,n\right)$ plane for (a) passive relay $\left(p=1.0\right)$ and (b) active relay ($p=1.0$, $p_{\text{relay}}=4.0$). Transversally stable (blue circles) and transversally unstable (red squares) ECMs are also shown. (a) The competition between chaotic itinerancy and antimodes leads to bubbling during global antimode dynamics. Yellow diamonds mark the onset of desynchronization. Solid and dashed parts of the trajectory correspond to synchronized and desynchronized periods, respectively. The inset in (a) shows the ECM ellipse and bubbling dynamics in a larger range. (b) The system evolves around the transversally stable compound laser modes and bubbling is suppressed. Parameters are as in Fig. 3.