Master stability function for frequency synchronization in laser networks

Network synchronization of lasers is of critical importance for reaching high-power levels as well as for effective optical computing. Yet, the role of network topology for frequency synchronization of lasers is not well understood. In this talk, we report our significant progress towards solving this critical problem for networks of heterogeneous lasers with repulsive coupling. We derive a master stability function for predicting the stability of frequency synchronization from the spectral knowledge of a complex matrix representing a combination of the signless Laplacian induced by the repulsive coupling and a matrix associated with intrinsic frequency detuning. We show that the gap between the two smallest eigenvalues of the complex matrix determines the coupling threshold for frequency synchronization. The application of our master stability function shows that, in stark contrast with Laplacian networks, dense long-range interactions do not necessarily lower the synchronization threshold. We reveal a non-trivial interplay between the network topology and geometric frustrations and demonstrate that full bi-partite networks have the optimal synchronization properties determined by the lowest synchronization threshold. Our results may provide guidelines for optimal designs of scalable laser networks capable of achieving reliable synchronization.