Soft-Clamped Perimeter Modes of Polygon Resonators

TL;DR

This paper develops an analytical framework that extends the Timoshenko-Gere theory to polygon perimeter modes by including tensile stress and tether torsion, identifying bending of polygon sides and torsional deformation of tethers as the dominant dissipation channels. It derives closed-form expressions for resonance frequencies and dissipation-dilution factors, revealing a fundamental 1/\lambda^2 scaling and demonstrating that tether torsion can be tuned to optimize Q by adjusting the tether length to achieve optimal coupling (e.g., at kl_t = \pi/2). The model is validated against finite-element simulations, showing accurate predictions across most modes and guiding design tradeoffs between side width, tether length, and polygon geometry. The results offer a practical blueprint for engineering high-Q polygon resonators for cavity optomechanics and precision sensing, with implications for multimode and topological phononics.

Abstract

Polygon resonators are promising candidates for nanomechanical applications due to their compact architecture and high force sensitivity. Here, we develop an analytical framework to predict the resonance frequencies and dissipation dilution factors $D_Q$ of polygon perimeter modes by extending the Timoshenko-Gere equation to incorporate the tensile stress. The model identifies two dominant dissipation mechanisms: distributed bending in the polygon sides and torsional deformation in the supporting tethers. We reveal that dissipation dilution in these resonators scales as $1/λ^2$, distinct from the conventional $1/λ$ dependence associated with boundary bending loss. Furthermore, we demonstrate that the torsional loss can be suppressed by tailoring the torsion angle of the supporting tethers. The analytical predictions are validated by finite element simulations, providing a predictive framework for designing high-$Q$ polygon resonators for cavity optomechanics and precision sensing.

Figures (4)

• Figure 1: Schematic representation of the polygon resonator (light blue) with index number $N = 4$ and clamps (light blue). The substrates are represented in black. $l_{s}$, $l_{t}$, $w_{s}$, $w_{t}$ represent polygon side length, tether length, polygon side width, and tether width, respectively.
• Figure 2: FEM simulation of the fundamental mode of a stress preserving polygon resonator. The inset illustrates the torsion deformation caused by the out-of-plane displacement of two neighboring polygon sides. $\tau$ is the torsion angle of the supporting tethers.
• Figure 3: Dissipation dilution factors and frequencies of the perimeter modes of the resonator with dimensions shown in the simulation part. Blue filled circles and red open circles correspond to FEM results and analytical results, respectively. Green squares correspond to out-of-plane modes of normal strings with the same dimensions and mechanical properties as the polygon sides. The four insets show the fundamental perimeter mode (a), and higher order perimeter modes where bending-torsion coupling exists (b, c, and d).
• Figure 4: Dissipation dilution factors of fundamental perimeter modes with $N = 4$. (a) $D_Q$ dependence on the tether length, with $l_s = 700~\upmu \mathrm{m}$, $w_s = 300~\mathrm{nm}$, and $w_t = 400~\mathrm{nm}$. The orange triangles and light blue circles represent the FEM simulation results and the analytical predictions, respectively. (b) Torsion angle profiles for the tether at lengths of 3920, 7890, and $11810~\upmu \mathrm{m}$. The results from the FEM simulation are shown in orange and are compared against sinusoidal references (light blue). (c) $D_Q$ dependence on the polygon width, with $l_s = 700~\upmu \mathrm{m}$, $l_t = 700~\upmu \mathrm{m}$, and $w_t = 400~\mathrm{nm}$. (d) Distribution of $D_Q$ for $l_s = 700~\upmu \mathrm{m}$ and $w_s = 300~\mathrm{nm}$. (e) Distribution of $D_Q$ for $l_t = 700~\upmu \mathrm{m}$ and $w_t = 400~\mathrm{nm}$.