Robust Fano resonance in a topological mechanical beam

The advances in topological condensed matter physics enable the manipulation of classic waves in different ways, such as unidirectional propagation featuring the suppression of backscattering and the robustness against impurities and disorder, making it possible to endow classical phenomena with topological properties. Fano resonance, a widely spread and basic kind of resonance, features an asymmetric line shape with an ultrahigh quality factor $Q$ that usually requires delicate designs and precise fabrication. In this work, we achieve a robust Fano mechanical resonance with topological protection by engineering band inversion of two different vibrating symmetries of a pillared beam that gives rise to dark and bright edge modes. The Fano resonance results from the constructive and destructive interferences between topological dark and bright modes. It is further demonstrated that the Fano asymmetric shape of the transmission peak and its frequency are robust against random perturbations in the pillars’ position as long as the symmetry is conserved. If random perturbations break the symmetry and only band inversion is involved, the asymmetric line shape of the Fano resonance weakens until disappearing before the closure of the bulk band gap, since the excitation will couple all fundamental modes of the beam. The analysis of the robustness of Fano resonance originating from band inversion and symmetry protection reveals the nature of topological protection which can be applied to design topological high-$Q$ resonance in sensing application.

Figure 1

(a) Unit cell of a phononic beam with the pillar spacing as $d=100\mu \mathrm{m}$. Periodic boundary conditions along the $x$ axis are applied to the two beams’ boundaries. (b) Dispersion curves of the pillared phononic beam where each color refers to a particular symmetry of the corresponding branch: band 1 (AS in blue), band 2 (SA in green), band 3 (AA in red), and band 4 (SS in black). Accidental double Dirac cones of symmetric and antisymmetric shear modes appear at 1.66 MHz for $k_x=\pi /a$. (c) Real part of the displacement field $u_y$ for the four eigenstates at the double Dirac points as indicated by the red and green arrows.

Figure 2

Dispersion curves for the crystal with short interval $d=60\mu \mathrm{m}$ (left-hand panel) between the pillars in the unit cell and large interval $d=140\mu \mathrm{m}$ (right-hand panel). They show exactly the same band structure. The eigenstates of the symmetric and antisymmetric shear modes above and below the gap (shown as the green and red arrows) are presented beside the dispersion. From $d=60\mu \mathrm{m}$ to $d=140\mu \mathrm{m}$, the eigenstates shown in red and blue dotted boxes flip as band inversion.

Figure 3

A finite-length stripe consists of topological $\left(10a\right)$ and trivial $10a$ beams with an interface in the middle (red dotted line). The total length of the stripe is $20a=8000\mu \mathrm{m}$, which is sufficiently large to reveal the interface modes when periodic boundary conditions are applied to the two beam's edges along the $x$ axis. (a) The dispersion curves show bright and dark shear edge modes (in green and red) in the topological band gap, respectively. (b) The eigenstates of the two edge modes (absolute displacement $u_y$). (c) Normalized displacements in pillars along the beam for bright (left) and dark (right) edge modes.

Figure 4

Left: Dark and bright transmission spectra with single symmetric and antisymmetric shear vibration excitation, respectively. Right: When both shear vibrations are excited, the dark and bright modes will couple and interfere to generate the high-$Q$ Fano asymmetric shape transmission.

Figure 5

Robustness of topological (a), (d) dark mode, (b), (e) bright mode, and (c), (f) Fano resonance against perturbation in the pillars’ position with random degree δ along the $x$ (a)–(c) and $y$ (d)–(f) axes. The position profiles of ${\delta }_x=20\mu \mathrm{m}$ and ${\delta }_y=20\mu \mathrm{m}$ are shown at the top and bottom, respectively.