Mismatch and synchronization: Influence of asymmetries in systems of two delay-coupled lasers

We study the synchronization properties of the delay dynamics of two identical semiconductor lasers coupled through a semitransparent mirror. Via an analytical and numerical approach, we investigate the influence of asymmetries, in particular mismatches of self- and cross-coupling strength and differences in self- and cross-coupling delay. We show that the former mismatch affects the stability of the zero-lag state but not the dynamics within the synchronization manifold, while the latter mismatch does not affect the quality of synchronization but alters the dynamics significantly. Our results are extended to different unidirectional coupling schemes. This is highly relevant for communication schemes utilizing chaotic dynamics. Finally, the influence of nonlinear gain saturation on the dynamics and stability of synchronization is discussed.

Figure 1

Symmetric setup of bidirectionally coupled lasers with feedback. The distances are given in time for a one-way trip. $L$ and $K$ are the strengths of the feedback and the coupling, respectively.

Figure 2

Zero-lag cross correlation $C(0)$ of the two coupled lasers’ intensities vs the coupling strength $K$ and feedback strength $L$. Here, the coupling and the feedback have the same delay. Parameters: $\mu =0.26$, $p=1.0$, $\alpha =4.0$, $T=200$, $\tau =1000$, and noise $\beta ={10}^{-5}$.

Figure 3

Dynamics of the symmetrized solution $(E_S,n_S)$ of two lasers coupled through a semitransparent mirror with different reflectivity-transmission mismatches, plotted in $(\omega ,n_S)$ phase space. $\omega =[{\varphi }_S(t)-{\varphi }_S(t-\tau )]/\tau$ is the delay-averaged frequency. Here the symmetrized variable ${\varphi }_S=\frac{1}{2}({\varphi }_1+{\varphi }_2)$ describes the phase in the synchronization manifold, $n_S=\frac{1}{2}(n_1+n_2)$ is the corresponding carrier density, and $E_S$ is the field amplitude. The dark blue (black) part of the time trace is well synchronized, the light blue (light gray) part is not. Transversely stable modes are (green) triangles and unstable modes are (red) circles. The insets show the corresponding intensity difference. Parameters: $\tau =500$, $T=200$, $\alpha =4$, $p=0.1$, $\beta ={10}^{-5}$, and (a) $K=L=0.06$, (b) $K=0.045$, $L=0.075$, and (c) $K=0.03$, $L=0.09$.

Figure 4

Evolution of the peaks of the autocorrelation and cross-correlation function $C(\Delta t)$, respectively, with the mismatch $L-K$. Both correlations are calculated for the lasers’ intensities: (a) autocorrelation and (b) cross-correlation. Black thick line: cross-correlation at zero lag $C(0)$. Blue circles: peak at one delay time $C(\tau )$. Red diamonds: $C(2\tau )$. Green squares: $C(3\tau )$. Magenta triangles: $C(4\tau )$. Parameters: $\tau =1000$, $\mu =0$, $p=1$, $T=200$, $\beta ={10}^{-5}$.

Figure 5

Scheme for a closed-loop setup as a drive-response configuration. The feedback delay is $\tau$.

Figure 6

Scheme of a bidirectional coupling setup with a delay mismatch of the self-feedback. The distances are given in time for a one-way trip. Both self-feedback strengths have the value $L$ and the coupling strengths both equal $K$.

Figure 7

Dynamics and correlation of two delay-coupled lasers with a delay mismatch of $\Delta \tau =100$ in the coherence collapse regime. Upper panel: time series of laser output intensities $I_1(t)$ [green (dark gray) line] and $I_2(t)$ (black line). The time traces exhibit a time lag of $\Delta \tau$. For demonstration purposes, the noise was disregarded in this simulation. Lower panel: cross-correlation function of laser intensities. The time shift of the maximum correlation peak equals the delay mismatch: $\Delta t=-\Delta \tau$. Simulated with noise $\beta ={10}^{-5}$. Parameters for both simulations are $\tau =2000$, $\alpha =4.0$, $p=1.0$, $\mu =0.26$, $T=200$; all coupling strengths were identical $L=K=0.1$.

Figure 8

Cross-correlation $C(\Delta t)$ at shift $\Delta t=\Delta \tau$ of the output intensities of both lasers (upper red line), and the peak at $\Delta t=2\tau$ of the autocorrelation function of one laser (lower blue line), versus the delay mismatch parameter $\Delta \tau$. The peaks coincide: higher regularity (peak in the autocorrelation) means stronger cross-correlation (peak in the correlation coefficient). Parameters are $\tau =1000$, $\alpha =4.0$, $p=1.0$, $\mu =0.26$, $T=200$, $L=K=0.1$, $\beta ={10}^{-5}$.

Figure 9

Cross-correlation $C(\Delta t)$ at shift $\Delta t=\Delta \tau$ of the field intensities of two coupled lasers in a configuration, according to Fig. 6, vs $L$ and $K$. In comparison with Fig. 2, the system is more sensitive to a mismatch of the coupling $L-K$. $\Delta \tau =20$. Other parameters are $\tau =1000$, $\mu =0.26$, $\alpha =4.0$, $p=1.0$, $T=200$, $\beta ={10}^{-5}$.

Figure 10

Dynamics of the symmetrized solution $(E_S,n_S)$ in the $(\omega ,n_S)$ phase space for different values of the delay mismatch parameter $\Delta \tau$ after $t=5\times {10}^4$. For a definition of $\omega$ and $n_S$, see Fig. 3. The ECMs for a laser subject to two delayed feedbacks with different delay times are aligned inside the area of the mode ellipse for the case where the delays are equal. Green (light gray) circles are transversely stable modes; red (black) triangles are transversely unstable modes and antimodes. The itineracy is drawn in blue (dark gray) and marked with an arrow if stabilized. The mean delay time $\tau$ is fixed at $\tau =1000.$ The other parameters are $\alpha =4.0$, $p=1.0$ (except lower right), $\mu =0.26$, $K=0.1$. Note that the lower right plot has a different $y$ scale than the others.

Figure 11

Cross-correlation coefficient (upper red lines), and auto-correlation at $\Delta t=2\tau$ (lower blue lines), vs the delay mismatch parameter $\Delta \tau$ for rational ratios of the delay times ${\tau }_1$ and ${\tau }_2$. Magnifications of plot in Fig. 8. (a) $r_{\tau }=\frac{{\tau }_1}{{\tau }_2}=1/1$, (b) $r_{\tau }=2/1$, (c) $r_{\tau }=3/1$, (d) $\Delta \tau \approx \tau$. Parameters as in Fig. 8.

Figure 12

Bifurcation diagram of the intensity extrema of the two lasers vs the delay mismatch parameter $\Delta \tau$. Upper green points are maxima; lower red points are minima. Varying the delay mismatch results in radical changes in the dynamics, which is evident, e.g., by the changes in the variance of the intensity outputs. Magnifications are shown for a small delay mismatch $\Delta \tau <15$ (lower left), a delay times ratio of $r_{\tau }\approx 2/1$ (lower middle), and one very short delay ${\tau }_1\gg {\tau }_2$ (lower right). Other parameters are $\tau =1000$, $\alpha =4.0$, $\mu =0.26$, $p=1.0$, $T=200$, $\beta ={10}^{-5}$.

Figure 13

Effect of the nonlinear gain saturation on the alignment and transverse stability of the external cavity modes in the ($\omega ,n_S$) phase space; $p=1.0$. The larger the pump current value is, the more pronounced is the effect. Red (black) circles: transversely unstable antimodes. Green (light gray) circles: transversely stable modes. The dynamics is shown in blue (dark gray). The other parameters are $\tau =1000$, $\Delta \tau =0$, $\alpha =4.0$, $T=200$, $K=0.1$, $\beta ={10}^{-5}$. Inset: bifurcation diagram with varying $\mu$. Upper green points: maxima of intensities; lower red points: minima of intensities.

Figure 14

Cross-correlation of the field intensities vs $K$ and $L$ with and without delay mismatch. (a) Symmetric delays: $\Delta \tau =0$, zero-lag synchronization; (b) with delay mismatch $\Delta \tau =20$, cross-correlation at $\Delta t=\Delta \tau$. The nonlinear gain saturation is ignored for these simulations. Parameters: $\mu =0$, $p=1.0$, $\alpha =4.0$, $T=200$, $\tau =1000$, and noise $\beta ={10}^{-5}$.