Topologically Protected Twist Edge States for a Resonant Mechanical Laser-Beam Scanner

We design a one-dimensional chain of two different alternating three-dimensional elastic chiral unit cells. The chain’s topological band gap, a result of the alternation of unit cells combined with their chirality and an effective mirror symmetry, guarantees a protected edge state, corresponding to a localized twist mode with an eigenfrequency inside the one-dimensional band gap. A small axial modulation at the one end of the beam can excite this resonant twist mode at the other end of the beam, via evanescent modes in the gap. The topological robustness of the edge state allows us to add a micromirror to the other end of the beam, turning the arrangement into a resonant mechanical laser-beam scanner. Its scalable operation frequency makes it attractive for applications.

Figure 1

(a) A designed elastic beam can convert the weak time-harmonic axial excitation at one end of the beam into the excitation of an edge mode at the other end. The edge mode exhibits a large twist motion or rotation. A micromirror fixed to the other end of the beam can thus deflect a laser beam. (b) A magnified view onto the end with the mirror. The required 1D topological band gap is accomplished by an alternation of two different 3D chiral cells. The two red ellipses highlight the subtle structural difference between the two neighboring cube cells. One of the cells is depicted in (c), together with relevant geometrical parameters. In particular, the angle $\delta$ controls the level of chirality. The cell becomes achiral for $\delta =0$. Panel (c) has been adapted with permission from Ref. [31].

Figure 2

(a) The alternation of 3D cubic chiral cells as in Fig. 1. We map this complex microstructure onto two coupled Su-Schrieffer-Heeger (SSH) models for the displacement $u$ and the twist angle $\phi$ [Eqs. (1) and (2)]. (b) The chiral coupling is indicated by the gray dotted lines. The amplitudes of the coupling coefficients are illustrated qualitatively by the different Hooke’s springs. The grayish box shows the unit-cell convention used in this work.

Figure 3

(a) The band structure of the mass-spring model without chiral coupling ($D_1^{u\phi }=D_2^{u\phi }=0$) and (b) with chiral coupling ($D_1^{u\phi }\ne 0;D_2^{u\phi }\ne 0$). In (a), the red and blue lines correspond to the rotational and longitudinally polarized (pressure) bands, respectively. Each of the band pairs is separated by a band gap—which, however, is very small for the rotational bands and not visible within the line width. For the coupled case in (b), individual bands exhibit mixed rotational-longitudinal modes. The calculated associated Zak phases are indicated. The resulting band gap between the second and third band in (b) is topologically nontrivial. The normalized coupling parameters are as follows: ${\tilde{D}}_1^{uu}=D_1^{uu}{a}^2/m=1459\mathrm{N}\mathrm{m}/\mathrm{kg}$, ${\tilde{D}}_2^{uu}=0.354{\tilde{D}}_1^{uu}$, ${\tilde{D}}_1^{\phi \phi }=D_1^{\phi \phi }{a}^2/j=370\mathrm{N}\mathrm{m}/\mathrm{kg}$, ${\tilde{D}}_2^{\phi \phi }=1.081{\tilde{D}}_1^{\phi \phi }$, ${\tilde{D}}_1^{u\phi }=D_1^{u\phi }{a}^2/\sqrt{mj}=23.3\mathrm{N}\mathrm{m}/\mathrm{kg}$, and ${\tilde{D}}_2^{u\phi }=6{\tilde{D}}_1^{u\phi }$. (c) The band structure as obtained from finite-element calculations of the microstructure shown in Fig. 2. The lower red and blue bands are as in (b): their Zak phases agree with those in (b). The Zak phases have been obtained from the underlying modes’ parity (see Fig. 6). The bands shown in gray are shear modes (which do not couple to the colored bands), third or higher back-folded longitudinal or rotational bands, or originate from local resonances within the cells at frequencies above the colored bands. The structure parameters are as follows: $r_1=0.32l$, $r_2=0.4l$, $d=0.06l$, $b_1=0.098l$, $b_2=0.03l$, ${\delta }_1={7.2}^{\circ }$, and ${\delta }_2={19.6}^{\circ }$ [according to Fig. 1]. The frequency scale on the right-hand side corresponds to a cube-cell size of $l=a/2=150\mu \mathrm{m}$.

Figure 4

Topological “phase diagrams” of the mass-spring model illustrated in Fig. 2 for (a) the first and (b) the second longitudinal-torsional band. A trivial phase with a Zak phase of $\gamma =0$ is shown in white and a nontrivial phase with $\gamma =\pi$ is shown in red. On the horizontal axis, we vary the torsional spring-constant ratio ($D_2^{\phi \phi }/D_1^{\phi \phi }$), and on the vertical axis, we vary the longitudinal spring-constant ratio ($D_2^{uu}/D_1^{uu}$) for constant chiral coupling parameters (see Fig. 3). $D_1^{uu}$ and $D_1^{\phi \phi }$ are kept constant while $D_2^{uu}$ and $D_2^{\phi \phi }$ are varied. Due to the SSH mechanism combined with the anticrossing, either the first or the second band is nontrivial for most of the considered configurations. Thus, the band gap has a nontrivial topology over a wide parameter range. Only for very small ratios of the spring constants are both the first and the second band in the trivial phase. The parameter configuration that reproduces the microstructure’s band structure [compare Figs. 3 and 3] is highlighted by a black cross.

Figure 5

The eigenfrequencies (black dots) of a finite-length mass-spring chain (cf., Fig. 2) with $100$ lattice sites. The left boundary of the chain is fixed. The right boundary is open but approximates a fixed end by an additional mass $m_f=2m$ (moment of inertia $j_f=2j$) of the last element. The colored solid curves are identical to those in Fig. 3, with the same color coding. For the finite chain, the wave number of each eigenmode is obtained by a Fourier transformation of the spatial amplitude. Two edge modes with frequencies inside the band gap are found (green and yellow dots). The insets illustrate the corresponding twist amplitudes for the right end (left end) edge mode for the last (first) 15 lattice sites.

Figure 6

The eigenmodes of the first and second longitudinal-torsional bands at the center (wave number $k=0$) and the boundary ($k=\pm \pi /a$) of the first Brillouin zone. The red arrows show the displacement at the corners of the plates oriented along the $z$ axis. For the first band, the two plates in the unit cell are symmetrically displaced at both, $k=0$ and $k=\pm \pi /a$. This behavior indicates a trivial topology. For the second band, the plates are symmetrically displaced at $k=0$ but antisymmetrically displaced at $k=\pm \pi /a$. This difference in parity, also called band inversion, indicates a nontrivial topology of the band.

Figure 7

(a) The normalized squared twist amplitude (on a logarithmic scale) versus the excitation frequency for three different ratios of the imaginary to the real part of the constituent material’s Young’s modulus $E^{\mathrm{\prime }\mathrm{\prime }}/E^{\mathrm{\prime }}$: $2.5\times {10}^{-5}$ (blue), $2.5\times {10}^{-4}$ (red), and $2.5\times {10}^{-3}$ (yellow). For unity normalized twist amplitude, the azimuthal component of the displacement vector has the same length as the axial component of the displacement vector corresponding to the excitation on the left end. (b) Snapshots of the displacement fields for the four resonances, 1, 2, 3, and 4, indicated in panel (a). Resonances 2 and 3 correspond to edge modes. Mode 2 is the one of interest in this paper (see also Fig. 1). The displacement fields, which are computed within the linear elastic regime, have been exaggerated for clarity.