Low-frequency fluctuations in an external-cavity laser leading to extreme events

We experimentally investigate the dynamical regimes of a laser diode subject to external optical feedback in light of extreme-event (EE) analysis. We observe EEs in the low-frequency fluctuations (LFFs) regime. This number decreases to negligible values when the laser transitions towards fully developed coherence collapse as the injection current is increased. Moreover, we show that EEs observed in the LFF regime are linked to high-frequency pulsing events observed after a power dropout. Finally, we prove experimentally that the observation of EEs in the LFF regimes is robust to changes in operational parameters.

Figure 1

Experimental setup. Light from the laser diode (LD) is split by a beam splitter. One optical path provides feedback, which is adjusted for feedback strength $\eta$ and external cavity length $L$. The other path is used for detection of the optical intensity $I\left(t\right)$.

Figure 2

The standardized pdf's in the (a) LFF and (b) CC regimes. The vertical (solid red) lines at 6 (in units of $\sigma$) indicate $E{E}_{\mathrm{th}}$. The standardized pdf in the LFF regime ($J=9.8$ mA) has a long and exponential tail, which leads to a large EE frequency. In the CC regime ($J=17.4$ mA), the pdf has a significantly shorter tail. $L=55$ cm and $\eta =0.6$.

Figure 3

Examples of the time series $I\left(t\right)$ in each dynamical regime: (a),(c),(e) original time series, and (b),(d),(f) time series low-pass filtered at 100 MHz. The ECL starts in intermittency (a),(b) and then enters the stable LFF regime (c),(d) in which the intervals between dropouts decrease. The CC regime (e),(f) is recognized by an absence of dropouts. Intermittency is measured at 9.1, stable LFF at 9.8, and CC at 17.0 mA. $L=55$ cm and $\eta =0.75$.

Figure 4

Large-amplitude ringing in $I\left(t\right)$ occurs immediately after dropouts. $J=9.1$ mA, $L=55$ cm, and $\eta =0.6$. The red horizontal line shows $E{E}_{\mathrm{th}}$.

Figure 5

EE frequency as a function of $J$. EEs are most frequently observed in the LFF regime. As the ECL leaves the stable LFF regime, EE frequency starts to decrease. $L$ = 65 cm and $\eta$ = 0.75.

Figure 6

Normalized autocorrelation function of $I_{EE}\left(t\right)$, as defined in the text, in the LFF regime. $\tau$ is the delay time of an external cavity length. We observe a peak around the delay time $\tau$ ($t/\tau =1$) as well as around $7\tau$ corresponding to the average time separating consecutive dropouts. $J$ = 10.1 mA, $L$ = 65 cm, and $\eta$ = 0.75.

Figure 7

Statistics of waiting times between successive EEs in the LFF regime. The distribution has log-Poissonian characteristics. $J$ = 9.9 mA, $L$ = 65 cm, and $\eta$ = 0.75.

Figure 8

EE frequency for various values of $L$ and $\eta$ showing that the basic trends as a function of $J$ are essentially robust to changes in these parameters. The maximum EE frequency is observed in all cases to occur in the LFF regimes (blue solid lines). As labeled in Fig 5, the dashed lines (red) and dots indicate the intermittency and CC regimes, respectively.