Locking of electrostatically coupled thermo-optically driven MEMS limit cycle oscillators

A model of coupled limit cycle oscillators is developed and analyzed to discover and map out the system's locking behavior. This sixth order model is motivated by the physical problem of a pair of closely spaced doubly clamped, thin silicon beams, coupled to each other through electrostatic fringing fields. The beams are assumed to be detuned with respect to each other. The beams are optically thin and are situated above a thick silicon substrate. When illuminated with continuous laser light a cavity interferometer is formed. Coupling of this optical interference with thermal stresses creates an inherent feedback loop that can drive the beams into limit cycle oscillation. Numerical analysis is used to study the range of coupling strengths and detunings over which 1:1, and other integer frequency ratio locking can be obtained. Results show that 1:1 locking can occur over a broad range of detuning even at relatively low levels of coupling. For coupling strengths just above the threshold for locking, both locked and drift states can exist, depending on the initial conditions. Locking at 2:1, 3:1, 3:2 and 5:2 frequency ratios are observed for detunings that are close but not exactly equal to these integer ratios.