Dynamics of an array of mutually coupled semiconductor lasers

We consider the dynamics of a linear array of coupled semiconductor lasers. Particular attention is paid to the synchronous states, which are caused by the permutation of two outer lasers. A system of three coupled lasers is studied in more details. We report different types of multistability of synchronous and asynchronous states including chaotic ones. We identify parameter values, for which a synchronous chaos can occur. Moreover, we show that transition to the synchronization occurs via blowup of the synchronous transversely unstable invariant set within the synchronization manifold. Finally, we present numerical analysis of larger arrays of coupled lasers and note some common qualitative features of the synchronization regions, which are independent of the number of lasers.

Figure 1

Schematic representation of the array of linearly coupled lasers. PD1 and PD2 are the photodiodes, which detect the output from the outer lasers.

Figure 2

Two-dimensional bifurcation diagram for the stability of stationary states within the manifold $M_s$. The gray shaded regions show the parameter values, for which there exists a stable stationary state. Solid lines denote Hopf bifurcations, dotted lines, the fold bifurcations, and broken lines the pitchfork bifurcations. Fixed parameters are $\alpha =2$, $J=1$, $\epsilon =0.03$.

Figure 3

Numerically obtained parameter region for the stability of the synchronization manifold $M_s$.

Figure 4

One-dimensional bifurcation diagram of the synchronous system attractor (a) and corresponding ranges of synchronization (b). $\phi =\pi ∕2$ is fixed.

Figure 5

Bifurcation diagrams for synchronous (a) and asynchronous (b) attractors. Zoom of the interval $0.022<\eta <0.035$. The region of multistability is indicated.

Figure 6

Behavior of the Lyapunov exponents. (a) Transverse Lyapunov exponents and (b) the largest longitudinal Lyapunov exponent. The parameters are the same as in Fig. 4. Bifurcation points $A–E$ are explained in the text.

Figure 7

Lyapunov exponents. Zoom of the interval $0.022<\eta <0.035$.

Figure 8

(Color) The phase portraits [projection onto the $(a_1=\mid E_1\mid ,N_1)$ plane] of the synchronous attractor before the blowup (a) at $\eta =0.0282$ and after (b) at $\eta =0.0287$. Those parts of the attractors which possess negative local Lyapunov exponents are marked by the black points. Period-3 unstable periodic orbit is shown by broken line (red).

Figure 9

(Color online) (a) The bifurcation diagram for the period-3 unstable periodic orbit. LP—fold bifurcation $(\eta =0.02762)$, PD—period-doubling bifurcation $(\eta =0.02783)$. $T$ denotes the actual period of the orbit. (b) The three-band chaotic saddle (CS), which is created through the period-doubling cascade of period-3 orbit (broken line, red online) at $\eta =0.0280$.

Figure 10

Distance between the synchronous attractor $\mathcal{A}$ and the period-3 orbit, which is embedded in the chaotic saddle. At $\eta =0.0284$, the distance rapidly decreases indicating that the period-3 orbit becomes a part of the attractor with increasing $\eta$.

Figure 11

The regions in the $\phi$-$\eta$ parameter space, for which an orbit started from randomly chosen initial condition is attracted to the synchronization manifold $M_s$. This corresponds to the asymptotic synchronization of the outer lasers. (a) Five coupled lasers, (b) ten coupled lasers.