Fractal-like Mechanical Resonators with a Soft-Clamped Fundamental Mode

Self-similar structures occur naturally and have been employed to engineer exotic physical properties. We show that acoustic modes of a fractal-like system of tensioned strings can display increased mechanical quality factors due to the enhancement of dissipation dilution. We describe a realistic resonator design in which the quality factor of the fundamental mode is enhanced by as much as 2 orders of magnitude compared to a simple string with the same size and tension. Our findings can open new avenues in force sensing, cavity quantum optomechanics, and experiments with suspended test masses.

Figure 1

(a) Binary tree with six branching levels. Local $x$ coordinates are shown for the first three levels. One tree defines half a resonator, and the complete structure is formed by adding its mirror reflection in the $yz$ plane. Hatched rectangles indicate clamping points. (b) Two branching points of a binary-tree resonator with the definitions of the segment widths, $w$, and lengths, $l$. The dashed blue contour is used to derive the transformation of the mode derivative.

Figure 2

(a) FEM simulation of the fundamental mode of a stress-preserving binary-tree resonator. (Inset) Schematic display of a cut view of one segment (marked with orange) and illustration of the torsion created by the previous segment (marked with green). (b) The displacement of the mode shown in (a) plotted over the local $x$ coordinates. (c) Quality factors and frequencies of out-of-plane modes of the resonator shown in (a). Blue diamonds correspond to the theory presented in this Letter, red dots to the result of FEM simulation. Filled red circles, symmetrically branched modes; empty circles, other modes. Green squares show out-of-plane modes of a doubly clamped beam resonator with the same $l_{\mathrm{tot}}$.

Figure 3

Loss coefficients for stress-preserving binary-tree resonators with $r_l=r_{l,\mathrm{crit}}(\theta )$ and different numbers of branchings $N$. (a) Boundary loss coefficient. (b) Distributed bending loss coefficient. (c) Distributed torsional loss coefficient. The region where ${\beta }_{\mathrm{tors}}$ converges with increasing $N$ is shaded gray. The direction of increasing $N$ is also indicated by arrows in the plots.