Coupled Soft-Clamped Polygon Resonators with Tunable Mechanical Coupling

Coupler geometry controls both the frequency splitting and the dissipation of coupled polygon resonators. We develop a continuum model for two identical square-shaped resonators coupled by a shared tether and compare the analytical predictions with finite element simulations. The model relates the coupler length and width to the frequency splitting and dissipation dilution of the two coupled modes. In the short-coupler limit, $k_cl_c\ll1$, the symmetric-mode coupler undergoes nearly uniform rotation, and its torsional strain energy vanishes asymptotically. The remaining loss contributions arise from polygon-side bending and supporting-tether torsion. The reduced supporting-tether contribution allows the symmetric-mode dissipation dilution to exceed that of an isolated polygon with the same side and supporting-tether dimensions. The antisymmetric mode retains torsional strain in the coupler and has an additional loss contribution. We also derive an expression for $Q\cdot f$ in the short-coupler and short-tether limits, where $Q$ is the mechanical quality factor and $f$ is the resonance frequency. This expression gives the dependence of $Q\cdot f$ on tensile stress, film thickness, and supporting-tether geometry. For a coupler length of $100~μ\mathrm{m}$ and width of $2~μ\mathrm{m}$, finite element calculations give a frequency splitting of $17.7~\mathrm{Hz}$ and a symmetric-mode dilution factor approximately $13\%$ above the analytical isolated-polygon value.