  We consider self-pulsing regimes in chains of Kerr nonlinear optical
microresonators. By means of a supermodal diagonalization procedure of the
conventional coupled-mode theory in time, we theoretically and numerically
study the bifurcation diagrams of a singly pumped three-cavity and a doubly
pumped four-cavity system: the latter allows us to predict thresholdless
frequency tripling of a GHz modulation. These self-pulsing regimes are proven
robust and will find applications in the generation and conversion of
microwaves on an optical carrier.
