Soft-Clamped Perimeter Modes of Polygon Resonators

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 $1/\lambda^2$, distinct from the conventional $1/\lambda$ 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.