We study analytically a 1D array of N equally spaced identical single-mode Fabry-Perot lasers. Each laser is coupled to its nearest neighbors by an intensity-dependent loss. The steady state is stable if the coupling is smaller than a critical value. Above that value there appears a Hopf bifurcation to a self-pulsing regime. We construct, for even N, the small-amplitude periodic solution emerging from the Hopf bifurcation and prove that this self-pulsing solution is stable. We also prove a form of collective self-organization manifested by the fact that in the stable steady state regime, the decay of a small perturbation is characterized for each laser by a single relaxation oscillation associated to N damping rates, whereas the transient of the total intensity is characterized by a single frequency and a single decay rate, the largest among the individual lasers decay rates. The theoretical predictions are confirmed by numerical simulations.
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