Highly-symmetric silicon dioxide shallow shell resonators with angstrom-level roughness

A highly-symmetric silicon dioxide shallow shell resonator with angstrom-level roughness (Ra=4.3Å) is presented. Curvature discontinuities in shallow shells, at a controlled distance from the base anchor of the shell, provide acoustic decoupling and significantly reduce the anchor contribution to the frequency split in degenerate wineglass modes as well as their damping. The Finite Element Method (FEM) simulations, performed in COMSOL 4.4, show increasing strain energy density at the vicinity of curvature discontinuities, confirming acoustic decoupling. The fabricated resonant shells are symmetric structures with largely asymmetric supports to force the anchoring stem to be the main contributor to the frequency split. Conformal Imaging Vibrometer (CIV) measurement characterization indicate that the fabricated shallow shell resonators have as-born sub-Hz frequency splits (Δf/f~60ppm), without any electrical frequency tuning, for the lowest elliptical mode shapes, while having asymmetric anchoring stems.