Synchronization properties of coupled semiconductor lasers subject to filtered optical feedback

We study numerically the synchronization properties of two unidirectionally coupled semiconductor lasers subject to filtered optical feedback. By adding a perturbation (a message) to the output of the master laser, we show that mutual information allows distinguishing between chaotic synchronization (at low to moderate coupling strengths) and injection locking (at large coupling strength). We find that a receiver subject to a feedback similar to that of the emitter (closed-loop receiver) shows better synchronization with the master laser when compared with a receiver without feedback (open-loop receiver). Closed-loop receivers also show better capability to recover weak messages. The filter in the feedback loop allows reducing the bandwidth of the chaotic carrier, improves the synchronization with respect to the conventional feedback case, and requires less coupling strength with a minor loss in complexity.

Figure 1

ML and SL in the (a) open- and (b) closed-loop configuration. In the open-loop configuration, we include the possibility of having an extra filter in front of the SL (see dotted box). ${\gamma }_{m,s}$ are the feedback strengths, $\tau$ is the feedback delay, ${\kappa }_r$ is the coupling strength, and $R\left(\omega \right)$ is the frequency response of the grating.

Figure 2

ML optical spectra for three ML filter widths. Parameters: ${\gamma }_m=25{\mathrm{ns}}^{-1}$, $I=1.5I_{\mathrm{th}}$.

Figure 3

Time series of the ML for different filter widths. (a) Without filter, (b) ${\Lambda }_m∕2\pi =16\mathrm{GHz}$ and (c) ${\Lambda }_m∕2\pi =8\mathrm{GHz}$. Parameters: ${\gamma }_m=25{\mathrm{ns}}^{-1}$, $I=1.5I_{\mathrm{th}}$.

Figure 4

(Color online) Autocorrelation time of the ML for different filter widths vs current. The inset is an enlargement of the autocorrelation time around $I∕I_{\mathrm{th}}=1.5$.

Figure 5

(Color online) Isochronous correlation of ML and SL in the open-loop scheme for increasing coupling strength, and three different ML filter widths, when the output of the ML is sent directly to the SL (full lines) and when it is further filtered before it is injected into the SL (dashed lines). Parameters: ${\gamma }_m=25{\mathrm{ns}}^{-1}$, ${\omega }_{m,p}=-8\mathrm{GHz}$, ${\gamma }_s=0$, $I=1.5I_{\mathrm{th}}$.

Figure 6

(Color online) Correlation at $t=\tau$ of ML and SL in the open-loop scheme for increasing coupling strength, and three different ML filter widths, when the output of the ML is sent directly to the SL (full lines) and when it is further filtered before it is injected into the SL (dashed lines). Parameters: see Fig. 5.

Figure 7

(Color online) Correlation at $t=\tau$ of ML and SL in the open-loop scheme for increasing injection current $I$ and three different ML filter widths. Parameters: ${\gamma }_m=25{\mathrm{ns}}^{-1}$, ${\omega }_{m,p}=0\mathrm{GHz}$, ${\gamma }_s=0$, ${\kappa }_r=25{\mathrm{ns}}^{-1}$.

Figure 8

(Color online) Synchronization of ML and SL in the closed-loop scheme for increasing coupling strength and three different filter widths. Parameters: ${\gamma }_m={\gamma }_s=25{\mathrm{ns}}^{-1}$, ${\Lambda }_s={\Lambda }_m$, ${\omega }_s={\omega }_m=-8\mathrm{GHz}$, $I=1.5I_{\mathrm{th}}$.

Figure 9

(Color online) Correlation at $t=0$ of ML and SL in the closed-loop scheme for increasing injection current $I$ and three different ML filter widths. Parameters: ${\omega }_s={\omega }_m=0\mathrm{GHz}$, ${\kappa }_r=\left(\mathrm{a}\right)$ 40 and (b) $50{\mathrm{ns}}^{-1}$. Other parameters as in Fig. 8.

Figure 10

(Color online) Synchronization of ML and SL in the closed-loop scheme for a mismatch in the width of the filters in ML and SL. Parameters: ${\gamma }_m={\gamma }_s=25{\mathrm{ns}}^{-1}$, ${\kappa }_r=60{\mathrm{ns}}^{-1}$, ${\omega }_s={\omega }_m$, $I=1.5I_{\mathrm{th}}$.

Figure 11

(Color online) Synchronization of ML and SL in the closed-loop scheme for a mismatch in the central frequency of the filters in ML and SL. Parameters: ${\gamma }_m={\gamma }_s=25{\mathrm{ns}}^{-1}$, ${\kappa }_r=60{\mathrm{ns}}^{-1}$, ${\Lambda }_s={\Lambda }_m$, $I=1.5I_{\mathrm{th}}$.

Figure 12

(Color online) Average mutual information in the closed-loop scheme vs coupling strength for three different filter widths at $I=1.5I_{\mathrm{th}}$. ●, $\Lambda ∕2\pi =8\mathrm{GHz}$, ▲, $\Lambda ∕2\pi =16\mathrm{GHz}$, ◼, $\Lambda =\infty$. Parameters: $ϵ=0.05$. Other parameters as in Fig. 8.

Figure 13

(Color online) Mutual information of ML (with and without message) and SL in the closed-loop scheme vs coupling strength for three different filter widths at $I=2I_{\mathrm{th}}$. ▲, $\Lambda ∕2\pi =16\mathrm{GHz}$, ◆, $\Lambda ∕2\pi =30\mathrm{GHz}$, ◼, $\Lambda =\infty$. Other parameters as in Fig. 12.

Figure 14

(a) Transmitted message. Message recovery in the closed loop for ${\kappa }_r=70$ (b) and (c) $45{\mathrm{ns}}^{-1}$. Parameters: $\Lambda ∕2\pi =16\mathrm{GHz}$. Other parameters as in Fig. 13.

Figure 15

(Color online) Mutual information of ML (with and without message) and SL in the open-loop scheme vs coupling strength for three different filter widths at $I=2I_{\mathrm{th}}$. ▲, $\Lambda ∕2\pi =16\mathrm{GHz}$, ◆, $\Lambda ∕2\pi =30\mathrm{GHz}$, ◼, $\Lambda =\infty$. Parameters: ${\gamma }_s=0$, $ϵ=0.15$. Other parameters as in Fig. 13.

Figure 16

(a) Transmitted message. Message recovery in the open loop for ${\kappa }_r=150$ (b) and (c) $60{\mathrm{ns}}^{-1}$. Parameters: $\Lambda ∕2\pi =16\mathrm{GHz}$. Other parameters as in Fig. 15.