Parametric Oscillation, Frequency Mixing, and Injection Locking of Strongly Coupled Nanomechanical Resonator Modes

We study locking phenomena of two strongly coupled, high quality factor nanomechanical resonator modes to a common parametric drive at a single drive frequency in different parametric driving regimes. By controlled dielectric gradient forces we tune the resonance frequencies of the flexural in-plane and out-of-plane oscillation of the high stress silicon nitride string through their mutual avoided crossing. For the case of the strong common parametric drive signal-idler generation via nondegenerate parametric two-mode oscillation is observed. Broadband frequency tuning of the very narrow linewidth signal and idler resonances is demonstrated. When the resonance frequencies of the signal and idler get closer to each other, partial injection locking, injection pulling, and complete injection locking to half of the drive frequency occurs depending on the pump strength. Furthermore, satellite resonances, symmetrically offset from the signal and idler by their beat note, are observed, which can be attributed to degenerate four-wave mixing in the highly nonlinear mechanical oscillations.

Figure 1

(a) Color-coded frequency spectra versus dc voltage for a strong parametric drive power $P_{\mathit{d}}=14.6\mathrm{dBm}$ at $f_{\mathit{d}}=13.18163\mathrm{MHz}$. Parametric oscillation resonances ($f_1$, $f_2$), satellites from degenerate four-wave mixing ($f_3$, $f_4$), and complete injection locking ($f_5$) to half of the drive frequency are observed. (b) The extracted peak frequencies of the above spectra (red dots) and calculated satellite resonances from Eq. (2) (blue daggers) exhibit excellent agreement. The gray dots display the avoided crossing of the two modes under a weak resonant drive with a mode splitting of 0.97 kHz. Some gray dots are missing for the in-plane mode due to the bad detection efficiency of this mode.

Figure 2

(a) Color-coded frequency spectra versus dc voltage for a weak parametric drive power $P_{\mathit{d}}=-1.5\mathrm{dBm}$ at $f_{\mathit{d}}=13.18454\mathrm{MHz}$. Parametric oscillation resonances ($f_1$, $f_2$), satellites from degenerate four-wave mixing ($f_3$, $f_4$), a higher order satellite ($f_{4^{\prime }}$), and a chaotic regime are observed. (b) The extracted peak frequencies of the above spectra (red dots) and calculated satellite resonances from Eq. (2) (blue daggers) exhibit excellent agreement. The gray dots display the avoided crossing of the two modes under a weak resonant drive with a mode splitting of 6.16 kHz. Some gray dots are missing for the in-plane mode due to the bad detection efficiency of this mode.

Figure 3

(a) Color-coded frequency spectra versus dc voltage for an intermediate parametric drive power $P_{\mathit{d}}=13\mathrm{dBm}$ at $f_{\mathit{d}}=13.18502\mathrm{MHz}$. Parametric oscillation resonances ($f_1$, $f_2$), satellites from degenerate four-wave mixing ($f_3$, $f_4$), and a partial injection locking peak ($f_5$) are observed. Additionally, equidistant injection pulling peaks appear ($f_6$ to $f_{11}$). (b) The extracted peak frequencies of the above spectra (red dots) and calculated satellite resonances from Eq. (2) (blue daggers, yellow crosses) exhibit excellent agreement. The gray dots display the avoided crossing of the two modes under a weak resonant drive with mode splitting of 1.87 kHz. Some gray dots are missing for the in-plane mode due to bad detection efficiency of this mode.