Preferential instability in arrays of coupled lasers

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Abstract
We consider an array of N coupled class-B lasers in a ring geometry. We analyze the stability of the steady-state solutions for small values of the coupling strength and small damping. The problem is motivated by recent studies of laser-diode arrays, but analytical results on the possible instabilities remain limited to the case N=2. We consider N arbitrary and use the coupling strength as the bifurcation parameter. As this parameter increases from zero, we show that the first instability leads to a preferential mode of oscillations. For N even, we study this bifurcation to a time-periodic standing-wave solution and determine the direction of bifurcation. We discuss the bifurcation possibilities in terms of the parameter α, known as the linewidth-enhancement factor, in semiconductor lasers. Increasing α destabilizes phase locking between adjacent lasers but leads to a smooth bifurcation to periodic solutions. Inversely, decreasing α stabilizes the laser array, but the first bifurcation leads to a hard transition to time-dependent solutions. The predictions of our analysis are in agreement with the results of a numerical study of the laser equations.