Observation of Locked Phase Dynamics and Enhanced Frequency Stability in Synchronized Micromechanical Oscillators

Even though synchronization in autonomous systems has been observed for over three centuries, reports of systematic experimental studies on synchronized oscillators are limited. Here, we report on observations of internal synchronization in coupled silicon micromechanical oscillators associated with a reduction in the relative phase random walk that is modulated by the magnitude of the reactive coupling force between the oscillators. Additionally, for the first time, a significant improvement in the frequency stability of synchronized micromechanical oscillators is reported. The concept presented here is scalable and could be suitably engineered to establish the basis for a new class of highly precise miniaturized clocks and frequency references.

Figure 1

Coupled silicon DETF micromechanical resonators. (a) An optical micrograph of the device fabricated in a commercial foundry process using a standard silicon-on-insulator MEMS process. (b) The linear frequency response is plotted at several $V_{\mathrm{dc}}$ voltages and at a fixed $v_{\mathrm{ac}}$ of 16 mV. In these responses, a reduction in the resonance frequency is a function of dc voltage results due to a reduction in the stiffness of the resonator. This characteristic can be used to reduce the frequency mismatch between the resonators. (c) The nonlinear response is plotted at several $v_{\mathrm{ac}}$ voltages and at a fixed 25 V $V_{\mathrm{dc}}$ voltage. At higher excitation levels, the resonant response skews towards the right, eventually exhibiting the Duffing bifurcation.

Figure 2

Response of the coupled oscillators. (a) Two marked peaks are observed in the frequency spectrum corresponding to the different output frequencies of the unsynchronized oscillators, while in the case of synchronization, a single peak is observed, indicating an entrainment condition. (b) Various frequency locking ranges are shown when the oscillators are operated at $V_F$ values ranging from 32 to 160 mV and at (b) 15 V and (c) 20 V $V_{{\mathrm{dc}}_2}$ voltages. Here, $f_1$ is the output frequency of oscillator 1, while $\Delta f$ represents the frequency difference of the uncoupled oscillators.

Figure 3

Relative phase response ($\Delta \varphi$) of synchronized and unsynchronized oscillators. (a) Several phase locked states are observed when the oscillators are operated at different values of $V_F$ due to the variation in $\Delta f$ and the initial phases. In these plots, the fluctuations in $\Delta \varphi$ (in degrees) show the limited interaction of the inherent noise with the phase trajectories of each oscillator. A clear reduction of fluctuations in $\Delta \varphi$ is observed with higher $V_F$ voltages. (b) In this plot, curve $A$ corresponds to an entrainment state while curve $B$ demonstrates the phase slip phenomenon, indicating a state of nearly synchronized oscillators. As the frequency mismatch increases, the oscillators oscillate with the different frequencies, resulting in uniform boundless growth in $\Delta \varphi$ (curves $C$, $D$, and $E$).

Figure 4

Allan deviation [${\sigma }_y(\tau )$] of the implemented MEMS oscillators. (a) A nonoverlapping estimation of ${\sigma }_y(\tau )$ of the uncoupled oscillator 2 is shown here in units of parts per billion (ppb) with respect to various integration times. A trade-off between $V_F$ and ${\sigma }_y(\tau )$ can be observed in these plots due to the dependency of the output signal power and the $a\mathrm{\text{−}}f$ correlation on $V_F$. (b) ${\sigma }_y(\tau )$ of the oscillator for uncoupled and synchronized cases are compared here. In both the cases, the oscillator is operated at a $V_F$ of 50 mV. A clear enhancement in the frequency stability is visible when the oscillator is entrained, compared to the uncoupled case.