Selective Parametric Amplification of Degenerate Modes in Electrostatically Transduced Coupled Beam Resonators

TL;DR

The paper investigates how degenerate phase modes in a Double Ended Tuning Fork (DETF) resonator respond to parametric driving mediated by the coupling beam. By performing harmonic and parametric excitations on the degenerate $IP$ and $OP$ modes, the study uncovers a pronounced nonlinear and amplifying response in the out-phase mode, while the in-phase mode remains largely linear. A key result is the demonstration of phase-controlled parametric amplification with a maximum gain of about $13\ \mathrm{dB}$ in the $OP$ mode, achievable through a coupling-beam–driven stiffness modulation and phase difference between harmonic and parametric drives. These findings establish a pathway for mode-selective nonlinear control in coupled resonators, with potential applications in reconfigurable memory, logic, amplification, and high-sensitivity sensing in phononic systems, including prospects for mechanical-to-phononic information processing.

Abstract

Parametric excitation in coupled mechanical systems has enabled advances in sensing, computation, and phonon control. The function of distinct phase modes using parametric driving remains insufficiently explored. Here, we investigate the nonlinear and parametric response of degenerate phase modes in a Double Ended Tuning Fork (DETF) resonator. Our measurements reveal pronounced nonlinearity and parametric amplification in the out-phase mode, attributed to the dominant contribution of the coupling beam, while the in-phase mode remains predominantly linear. Uniquely, we demonstrate parametric excitation through the coupling spring, enabling selective amplification and de-amplification controlled via the relative phase between harmonic and parametric drives. A parametric gain of $\sim$13 dB is achieved in the out-phase mode, with phase-dependent modulation of amplification, indicating its suitability for signal processing, logic operations, and memory elements based on degenerate modes. These results establish a new approach to exploiting mode-specific nonlinear dynamics in coupled resonators for emerging applications in sensing and phononic control.

Figures (5)

• Figure 1: Device measurement technique: (a) Scanning Electron Microscope (SEM) image of the coupled beam resonators, known as DETF resonator, (b) the measurement setup for the nonlinear and parametric measurements using a lock-in amplifier, (c) the spring mass system of DETF resonator (d) the frequency response of the degenerate modes of the DETF, in-phase and out-phase modes.
• Figure 2: Device harmonic measurement: Frequency response of (a) in-phase mode and (b)out-phase mode with increasing actuation drive. The out-phase mode shows softening nonlinearity.
• Figure 3: Device parametric measurement: (a) in-phase shows no response, (b) out-phase mode shows nonlinear response from 1.1 V onwards, indicating the coupling beam effect
• Figure 4: Device parametric with harmonic measurement.(a) Frequency response of out-phase mode with varying phase angle between direct and parametric drive (b) Variation of gain with phase for out-phase mode (blue) and in-phase mode (red).
• Figure 5: (a). In-phase mode gain shows no effect of parametric pump drive at the angle of maxima and minima of the parametric oscillation, whereas, (b). Out-phase mode gain shows the effect of amplification at the maxima of the parametric oscillation and suppression at the minima of the oscillation