Correlations in electrically coupled chaotic lasers

We show how two electrically coupled semiconductor lasers having optical feedback can present simultaneous antiphase correlated fast power fluctuations, and strong in-phase synchronized spikes of chaotic power drops. This quite counterintuitive phenomenon is demonstrated experimentally and confirmed by numerical solutions of a deterministic dynamical system of rate equations. The occurrence of negative and positive cross correlation between parts of a complex system according to time scales, as proved in our simple arrangement, is relevant for the understanding and characterization of collective properties in complex networks.

Figure 1

Setup for the experiments on the power correlations and chaos synchronization of two electrically coupled lasers.

Figure 2

Output power of the coupled lasers. Laser 1 (top line) that has optical feedback shows LFF power drops, while laser 2 (lower line) had no optical feedback. (a) Experimental light intensity and (b) ${\left|E_j\left(t\right)\right|}^2$ from numerical integration of Eqs. (1, 2, 3) (the vertical scales were displaced for better visualization).

Figure 3

Output power of the lasers coupled in parallel and having optical feedback. The top line is laser 1 and the bottom is laser 2. (Power scales were displaced for visualization.) Synchronism is clearly observed. (a) Experimental time series including a few power drops. (b) Numerical integration for the coupled laser equations.

Figure 4

Signal showing antiphase fine structure fluctuations superimposed on a pair of drops of the in-phase synchronized LFF when ${\kappa }_1={\kappa }_2$. (a) Experiment and (b) theory.

Figure 5

Experimental LFF time interval histograms: (a) Time interval between successive drops in one of the lasers. (b) Theory for (a). (c) Time interval between drops of the two lasers. (d) Theory for (c).

Figure 6

Theoretical numerical cross-correlation dependence on the feedback coefficient ratio.

Figure 7

Detail of Fig. 6. (a) Cross correlation between the lasers calculated at zero delay time $\left(\tau =0\right)$ as a function of ${\kappa }_2/{\kappa }_1$. (b) Cross correlation between the lasers calculated when laser 2 has no feedback $\left({\kappa }_2=0\right)$.

Figure 8

Theoretical numerical cross-correlation dependence on the coupling coefficient $\eta$. Same parameters from Table 1 but $N_{02}=0.998\times {10}^8$ and $J_0=2.02J_{\text{th}1}$.

Figure 9

Setup for the experiments on the power correlations and chaos synchronization of two electrically coupled lasers.

Figure 10

Laser power variations during the LFF drop when only laser 1 had feedback. (a) Experimental. (b) Theory with $J_0/2=1.01J_{{\text{th}}_1}$ and $J_{{\text{th}}_1}=7.87503\times {10}^{16}{\text{s}}^{-1}$. In both figures the black curves correspond to laser 1 and the red (online) curves to laser 2.

Figure 11

Pump current variations during one LFF drop of laser 1 while laser 2 had no feedback. (a) Experimental. (b) Theory. In both figures a $29\text{MHz}$ low-pass filter was used. The black curves correspond to laser 1 and the red (online) curves to laser 2.

Figure 12

Laser power variations during a pair of synchronized LFF drops. (a) Experimental. (b) Theory with $J_0/2=1.01J_{{\text{th}}_1}$ and $J_{{\text{th}}_1}=7.87503\times {10}^{16}{\text{s}}^{-1}$. In both figures the black curve corresponds to laser 1 and the red (online) curve to laser 2.

Figure 13

Pump current variations during a pair of synchronized LFF drops. (a) Experimental. (b) Theory with $J_0/2=1.01J_{{\text{th}}_1}$ and $J_{{\text{th}}_1}=7.87503\times {10}^{16}{\text{s}}^{-1}$. In both figures a $29\text{MHz}$ low-pass filter was used. The black curves correspond to laser 1 and the red (online) curves to laser 2.

Figure 14

Output power of the lasers coupled in parallel and having optical feedback. The pump current and feedback strengths were set to partial synchronism of LFF. The top line is laser 2 and the bottom is laser 1. (Power scales were displaced for visualization). (a) Experimental and (b) numerical integration from theory. Whenever a coincidence drop is missed there is a power jump up in one of the lasers.

Figure 15

Numerically calculated optical phase for the two coupled lasers. (a) Segment of the laser intensity for the sake of comparison. (b) Plot of the two phases evolving in time. (c) Phase difference showing the large time scale synchronism. (d) Optical frequency excursion of the two lasers. (e) Details of the intensity fluctuations close to a synchronized drop and (f) the instantaneous frequency of the two lasers.

Figure 16

Experimental correlations of the lasers coupled in parallel and having optical feedback. (a) Autocorrelations of one laser. (b) Cross correlation showing negative values due to antiphase fluctuation contributions, superimposed to large time scale positive contributions due to in-phase fluctuations.

Figure 17

Numerical correlation functions: (a) Autocorrelation of laser 1. (b) Cross correlation showing the downgoing antiphase fine structure superimposed on the in-phase synchronized LFF. Filters were used to enhance the proportions of antiphase (negative contribution) and in-phase (positive contribution) signals visible.

Figure 18

Numerical cross correlation obtained when two simple signals are composed by harmonics in phase and antiphase, with different frequencies and having variable amplitudes. (a) 3D map showing the dominant antiphase term contribution evolving to the in-phase cross correlation when $B$ in Eq. (7) varies from $-A$ to $A$. (b) Details of (a) showing the undulations associated to antiphase overriding the wide positive correlation curves.

Figure 19

Details of the numerical cross correlation obtained when two simple signals are composed by harmonics in phase and antiphase with different frequencies and having variable amplitudes. (a) Variation of the zero time $\left(\tau =0\right)$ cross correlation showing the excursion from fully antiphase, $-1$, to almost complete in phase near $+1$. The value $+1$ is not reached because a small antiphase contribution is always present. (b) Correlation profile when $B=A$.