Ultrasound experiments on acoustical activity in chiral mechanical metamaterials

Abstract

Optical activity requires chirality and is a paradigm for chirality. Here, we present experiments on its mechanical counterpart, acoustical activity. The notion "activity" refers the rotation of the linear polarization axis of a transversely polarized (optical or mechanical) wave. The rotation angle is proportional to the propagation distance and does not depend on the orientation of the incident linear polarization. This kind of reciprocal polarization rotation is distinct from nonreciprocal Faraday rotation, which requires broken time-inversion symmetry. In our experiments, we spatiotemporally resolve the motion of three-dimensional chiral microstructured polymer metamaterials, with nanometer precision and under time-harmonic excitation at ultrasound frequencies in the range from 20 to 180 kHz. We demonstrate polarization rotations as large as 22° per unit cell. These experiments pave the road for molding the polarization and direction of elastic waves in three dimensions by micropolar mechanical metamaterials.

Fig. 1

Chiral phonon eigenmode and unit cell. a Snapshot of a calculated chiral phonon eigenmode with eigenfrequency ω₁ propagating along the z-direction in an infinitely extended 3D chiral micropolar metamaterial crystal. For clarity, all unit cells except for one column along the propagation axis (z-axis) are shown semitransparent and the displacements are largely exaggerated. b Simplified representation only showing the behavior of the unit cells’ centers of mass, forming a helix that is moving along the z-axis versus time. An animation of a and b is shown in Supplementary Video 1. c Single-unit cell (cf. ref. ). Geometrical parameters are cubic lattice constant a = 250 μm, d/a = 0.06, r₁/a = 0.32, r₂/a = 0.4, and δ = 34.8°. For the bulk constituent polymer material we have chosen the Young’s modulus E = 4.18 GPa, mass density ρ = 1.15 g cm⁻³, and Poisson’s ratio ν = 0.4. The wave number is k_z = π/(4a) and the angular frequency is ω₁ = 2π × 107 kHz, equivalent to a/λ₀ = 0.0095, with the wavelength λ₀ of pressure waves in the bulk constituent polymer material (cf. Fig. 2)

Fig. 2

Calculated phonon band structures for chiral metamaterial beams. The left vertical scales are in absolute units, the right scales, a/λ₀, are normalized and scalable to other parameters (cf. Fig. 1). We consider wave propagation with wave number k_z in a beam which is infinitely extended along the z-direction and which contains N_x × N_y unit cells (with N_x = N_y) in the two orthogonal directions. The three panel columns correspond to N_x = N_y = 1, 3, and ∞, respectively. The upper row is for chiral (δ = 34.8°) metamaterials beams, the lower row for achiral (δ = 0) ones. Transverse or shear-like bands are shown in red (a lifting of degeneracy reflects acoustical activity), pressure-like modes in blue, and twist-like modes in black. The latter are absent for N_x = N_y = ∞. For clarity, higher bands not relevant here are plotted in gray. The insets illustrate the corresponding geometries. The spatial mode depicted in Fig. 1 is highlighted by a circle. Parameters are as in Fig. 1

Fig. 3

Oblique-view electron micrograph with overlaid measurement. The sample contains N_x = N_y = 3 and N_z = 12 unit cells (scale bar: 400 μm). A piezoelectric transducer excites the sample bottom. Top-view optical micrographs are taken under delayed stroboscopic illumination versus time delay. Displacement vectors are extracted by using image cross-correlation analysis. Results for the bottom side of the sample (blue, multiplied by a factor of 5 × 10³) and for the middle of the top of the sample (red, multiplied by a factor of 10⁴) are blended into the electron micrograph. For both cases, five oscillation periods are depicted to emphasize the reproducibility of the experiments (also see Supplementary Fig. 4). From these data, we derive a rotation of linear polarization due to acoustical activity of 44°. Here, the excitation frequency is 160 kHz, all other parameters are as in Figs. 1 and 2. Results from many different similar experiments on different samples are summarized in Fig. 4

Fig. 4

Measurements and calculations of acoustical activity. The rotation angle of an incident linear polarization upon propagation of a transverse elastic (flexural) wave is obtained from measurements (full dots) analogous to the example shown in Fig. 3 and plotted versus excitation frequency f. Results from band-structure calculations (cf. Fig. 2) are additionally shown as dashed curves, results from frequency-domain finite-sample calculations as solid curves (here E = E^' + E^'' = 4.18 GPa + i0.20 GPa). a Rotation angle versus excitation frequency f for different beam cross-sections N_x = N_y and fixed beam height N_z = 12. b Same as a, but for fixed beam cross-section N_x = N_y = 3 and different heights N_z. c As b, but N_x = N_y = 1. The miniatures on the right-hand side, which are colored according to the experimental data points, illustrate the parameter variations. The largest measured rotation is 22° per unit cell for N_x = N_y = 1 and f = ω/2π = 180 kHz. Achiral control samples show zero polarization rotation within the experimental error (see open blue dots in (b))