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Network synchronization of lasers is critical for reaching high-power levelsand for effective optical computing. Yet, the role of network topology for thefrequency synchronization of lasers is not well understood. Here, we report oursignificant progress toward solving this critical problem for networks ofheterogeneous laser model oscillators with repulsive coupling. We discover ageneral approximate principle for predicting the onset of frequencysynchronization from the spectral knowledge of a complex matrix representing acombination of the signless Laplacian induced by repulsive coupling and amatrix associated with intrinsic frequency detuning. We show that the gapbetween the two smallest eigenvalues of the complex matrix generally controlsthe coupling threshold for frequency synchronization. In stark contrast withLaplacian networks, we demonstrate that local rings and all-to-all networksprevent frequency synchronization, whereas full bipartite networks have optimalsynchronization properties. Beyond laser models, we show that, with a fewexceptions, the spectral principle can be applied to repulsive Kuramotonetworks. Our results may provide guidelines for optimal designs of scalablelaser networks capable of achieving reliable synchronization.

Spectral Network Principle for Frequency Synchronization in Repulsive Laser Networks