Acoustically-Coupled MEMS Transducer Pairs with Loss and Gain

TL;DR

This work quantifies acoustically mediated coupling between MEMS ultrasound transducers immersed in water and explores non-Hermitian dynamics arising from gain and loss. It derives a rotating-wave model for a two-DOF PMUT system, fabricates pairs with varying diameter and pitch, and extracts coupling by analyzing the amplitude ratio $A_2/A_1$ (which follows $A_2/A_1 = \Gamma/[2(\delta_2-i\gamma_2/2)]$). The study then implements a gain-feedback loop to induce self-oscillations and identifies exceptional-point (EP) and Hopf-bifurcation thresholds, fitted via $-\gamma \pm \mathrm{Im}\{\sqrt{\Gamma(\Gamma+iG)}\}=0$ for Hopf/EP and $4\Gamma^2=-G^{*2}$ for the EP case, with in-water approximations $\gamma_1\approx\gamma_2$ and $\delta_1\approx\delta_2$. Results show a distance-dependent coupling that decays roughly as $1/$pitch, with closed-loop measurements yielding larger coupling magnitudes than open-loop but consistent phase behavior, and Hopf-type and EP-type measurements giving similar coupling values. The findings enable improved modeling and design of 2D MEMS transducer arrays and illustrate experimentally accessible EP and Hopf phenomena in acoustically coupled MUTs.

Abstract

This work treats the dynamics of pairs of microelectromechanical ultrasound transducers (MUTs) that are immersed in water and acoustically coupled through the fluid medium. A series of these transducer pairs with varying diameters (and thus resonance frequency) and pitch separation (and thus coupling strength) are fabricated and measured. The work presented here models and quantifies the open-loop coupling between the MEMS transducer pairs and its dependence on pitch. Furthermore, a gain feedback loop is systematically applied to one of the device pair and the dynamics of the acoustically-coupled gain-loss system is investigated, and the formation of an exceptional-point or of an Hopf bifurcation is equally used to quantify the coupling coefficient. This work provides an experimental study of acoustic coupling in MUT transducers, as well as an exploration of the formation of exceptional points in acoustically-coupled MEMS transducers.

Figures (4)

• Figure 1: (a) Schematic representation of the measurement setup, showing the possibility to have either gain feedback loop or direct drive depending on which of the two switches is closed. Also shown is a schematic of the in-water measurement layout showing the PMUT sample (black and grey), the PMMA water containers (dark blue) and the PDMS acoustic absorber (light blue). (b) Optical microscope picture of the experimental samples showing the driven and the un-driven MEMS devices, also visible are the alternating diameters per row (100 and 60 $\mu$m diameters).
• Figure 2: Coupling between two PMUTs. (a) Resonance peaks (top) measured in air (left) and in water (right), and the ratio of the PMUT2 (red) to PMUT1 (blue) amplitudes (lower panels) showing that the ratio of the two indeed results in a Lorentzian peak, as well as demonstrating that the experimental noise is amplified when considering the ratio, especially for water measurements. (b) Dependence of the coupling strength on distance (top panels), in log-log scale, for the 100 $\mu$m (red), 80 $\mu$m (yellow), 60 $\mu$m (green), and 40 $\mu$m (blue), demonstrating a near 1/distance dependence both in air (left) and water (right). Fitted coupling phase (lower panels) showing a slowly changing phase for all diameters in water, and a slowly changing phase for the 40 $\mu$m devices in air, but a rapidly changing one for the otehr diameters in air.
• Figure 3: Self-oscillations and exceptional points. (a) Effect of the feedback gain on resonance peaks for the 100 $\mu$m devices for 295 $\mu$m pitch (left) and 155 $\mu$m pitch (right) (top row), demonstrating the increase in quality factor, amplitude, and onset of self-oscillation for feedback with PMUT1 motion (left column) and PMUT2 motion (right column). Equally shown is the onset of oscillations as the gain is increased (bottom row). (b) 2D amplitude plots showing the onset of oscillations in the pitch-gain parameter space which is easily recognizable by the sudden increase in amplitude, the border is highlighted with a white dashed line for convenience, shown for the 100 $\mu$m devices (top) and the 80$\mu$m devices (bottom). For the used loop gain range, the feedback of PMUT1 motion shows constant (absent) oscillations for the (80) 100$\mu$m devices, therefore most of the coupling information is obtained from PMUT2 feedback data. The unequal gain step is visible in the plots.
• Figure 4: Coupling extracted from closed-loop measurements. (a) Magnitude of coupling for the 100$\mu$m (right) and the 80$\mu$m (left) devices shown as discrete points are overlaid with values extracted from the open-loop measurements (continuous lines) For the 100 $\mu$m devices the Hopf bifurcation and the EP measurements show excellent agreement, the 80 $\mu$m devices have only Hopf-type data. (b) Extracted coupling phase showing a good agreement between Hopf and EP data, as well as the open-loop data.