Synchronization in a mechanical resonator array coupled quadratically to a common electromagnetic field mode

Abstract

Optomechanical systems are based on the nonlinear coupling between the electromagnetic (EM) field in a resonator and one or more bulk mechanical resonators such that the frequency of the EM field resonator depends on the displacement coordinates of each of the mechanical resonators. In this paper we consider the case of multiple mechanical resonators interacting with a common field for which the frequency of the EM resonance is tuned to depend quadratically (to lowest order) on the displacement of the resonators. By using the method of amplitude equations around a critical point, it is shown that groups of near-identical bulk mechanical resonators with low driving fail to synchronize unless their natural frequencies are identical, in which case the resulting system can exhibit multistability.

Synchronization in coupled mechanical resonator arrays

“…Everywhere to the left of the bold red dashed line (which once more marks the saddle-node bifurcations of limit cycles) there exists one stable limit cycle and one unstable limit cycle. This figure was published in [4]. of R i0 H ir0 = 1.…”

“…Perhaps the most elegant explanation for this phenomenon was described by Kuramoto [38], who showed that for every weakly coupled nearly identical phase-only limit cycle oscillator, the long-term dynamics are given by [33] Using the order parameter, Kuramoto attempted to explain why the sudden onset of synchronization occurred above a certain value of K. In terms of the order parameter, his model is reduced to 4) making the dependence of the individual oscillators onr andψ explicit, so rather than each oscillator depending on itself and each of the N − 1 others, each oscillator depends on itself andr andψ. Using this form, Kuramoto noticed that the effective strength of the coupling is Kr, so a steady state value ofr = 0 just gave independent motion of the oscillators at their natural frequencies, whilstr ≈ 1 gave the near-maximum value for the effective strength, attracting more oscillators to the synchronized pack, thereby further strengthening the effective coupling and so on.…”

“…The area of coupled oscillators that is explored in this thesis is opto-mechanics, which is the study of the interaction of light with mechanical systems [2,4]. The concepts associated with opto-mechanics are not new-Maxwell predicted that electromagnetic radiation could exert forces on objects, and this phenomenon was first observed over a century ago [6][7][8].…”

“…These schemes lead to a wealth of new and interesting dynamical behaviors, experimental advantages, and applications, such as parametric instablities [31], extremely low optical loss [4], and interferometric gravitational wave observatories, where the free spectral range may match relevant mechanical frequencies [31], among others. In Chapter 2 of this thesis, an opto-mechanical system with a single optical mode that is coupled to many mechanical elements is analyzed [4]. While this model is superficially similar to a simpler model which has already been investigated [2], the results show that the dynamics are quite different and they exhibit rich dynamical behavior.…”

“…While this model is superficially similar to a simpler model which has already been investigated [2], the results show that the dynamics are quite different and they exhibit rich dynamical behavior. In its realization as an opto-mechanical system, it could also offer the advantage of low optical loss [4]. Chapters 3 and 4 of this thesis are concerned with the coupling of two opto-mechnical systems, each of which mediates the coupling of multiple mechanical elements.…”