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 timeinversion symmetry. In our experiments, we spatiotemporally resolve the motion of threedimensional chiral microstructured polymer metamaterials, with nanometer precision and under timeharmonic 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.
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Introduction
An object is geometrically chiral if it lacks centrosymmetry, mirror symmetries, and rotationreflection symmetries^{1,2,3}. Chirality, or handedness, has played an important role in physics, optics, chemistry, biology, and medicine for decades already. Examples are optical activity^{4,5,6,7} and circular dichroism^{8,9}, exclusively righthanded DNA doublehelix strands^{10}, exclusively lefthanded neutrinos^{11}, chiral topological fermions^{12}, stereochemistry^{13}, and the effect of drugs depending on the handedness of the underlying molecules^{14}.
Surprisingly, chirality did not play a major role in the context of mechanics for many years. This has started to change recently. For example, geometrical chirality is important in plant seedpod mechanics^{15}. Chirality has enabled topological effects at finite frequencies^{16,17}, and has opened degrees of freedom beyond Cauchy elasticity in the stationary regime^{18,19}. Handedness in shearing auxetics has created rigid and compliant structures^{20}. The reviews^{21,22,23,24,25} provide an overview on the status of the field of mechanical metamaterials in general. Chiral phonons in quasi twodimensional monolayers of noncentrosymmetric tungsten diselenide crystals have only recently been inferred from optical spectroscopy^{26}. However, one of the paradigms of chirality, namely “activity” has so far only been a theoretical possibility for generalized elastic continua^{27,28,29}.
In the presence of optical activity^{4,7} or acoustical activity^{27,28,29}, an incident transverse linear polarization of a propagating wave rotates: the eigenpolarizations of the chiral medium are not linear but rather circular, with a lifted degeneracy between left and righthanded circular modes. Therefore, a linear incident polarization must be decomposed into the two circularpolarization eigenmodes, which propagate with different phase velocities. The resulting phase difference accumulated during propagation leads to a continuous rotation of the linear polarization axis versus the propagation coordinate. Optical and acoustical activity therefore allow, for example, to convert an incident linear polarization into the orthogonal one. Both are reciprocal effects, hence distinct from the nonreciprocal Faraday effect, which requires broken timereversal symmetry, yet does not need chirality^{7}. Regarding mechanics, yet a different kind of mode conversion has recently been observed in twodimensional centrosymmetric metamaterials^{30}. There, the conversion between inplane longitudinal (compression) and inplane transverse (shear) modes did not require chirality.
In this Letter, we report the experimental observation of the phenomenon of acoustical activity in threedimensional (3D) chiral polymer metamaterials by direct spatiotemporal imaging at ultrasound frequencies, using crosscorrelations of optical microscopy images taken under synchronized stroboscopic illumination. The experimental results agree well with numerical phonon bandstructure calculations and simulations for the considered experimental geometry using finitesize samples.
Results and discussion
Beyond Cauchy elasticity
Let us start by recalling that effects of chirality are neglected in mechanics on the level of textbook linear Cauchy elasticity^{31}. Plainly speaking, Cauchy continuum mechanics can be seen as “point mechanics”^{32,33,34}. Therefore, Cauchy elasticity becomes exact if the wavelength of an elastic wave, λ, is much larger than the crystal lattice constant, a, and if the sample size, L, is much larger than a, too. Therefore, at least one of the two conditions, λ/a≫1 and L/a≫1, needs to be violated to observe acoustical activity. Violating both conditions simultaneously is expected to yield larger effects. This means that we need to investigate the phonon band structure in the middle between the Γpoint and the Brillouinzone edge (by symmetry the bands are degenerate at the Brillouinzone edge) for beams of finite and infinite lateral extent L_{x} = L_{y}, with the number of unit cells N_{x} = N_{y} = L_{x}/a = L_{y}/a in the two directions perpendicular to the wave propagation along z.
Bandstructure analysis
Figure 1a shows a snapshot of a calculated chiral phonon eigenmode (at wave number k_{z} = π/(4a), hence angular frequency ω_{1}≈2π × 107 kHz) propagating in an infinitely extended 3D chiral micropolar metamaterial with N_{x} = N_{y} = N_{z}→∞ in the linear elastic regime. For clarity, the displacements are largely exaggerated. The behavior is qualitatively similar for other parameters. Panel b exhibits a simplified representation and panel c the underlying chiral metamaterial crystal unit cell (also see ref. ^{19}), with geometrical parameters indicated.
Figure 2 gives an overview on the theoretical expectations for the corresponding band structures ω_{n} = ω_{n}(k_{z}) = ω_{n}(−k_{z}), with the angular frequencies ω_{n} (n = 1,2,…) and the wave number k_{z}. All statements concerning the character of the bands made below are based on an investigation of the associated eigenmodes. Here, we assume Blochperiodic boundary conditions along the zdirection and open boundaries of the beams along the x and ydirection for finite N_{x} = N_{y} (left and center column in Fig. 2). For the limit N_{x} = N_{y}→∞ (right column in Fig. 2), we assume Blochperiodic boundary conditions along the x and ydirection as well. For a finite squareshaped crosssection N_{x} × N_{y}, four different bands emerge from the Brillouinzone center. All other bands (n ≥ 5) are of lesser importance here and are hence plotted in gray for clarity. The lowest two modes (n = 1,2) shown in red correspond to flexural waves with transverse circular polarization. Here, the center of mass of any given unit cell rotates clockwise or counterclockwise around the wave vector k, respectively, in circles around its rest position (cf. Fig. 1). In other words, these modes are chiral phonons (upper row of Fig. 2). As usual for thin plates and beams^{35}, the flexural modes start from the Γpoint with a parabolic dispersion. The behavior changes toward a linear dispersion if the crosssection of the beam is successively increased.
The absolute and the relative splitting between these two chiral phonon bands tends to zero near the Γ–point and, by symmetry, is also zero at the edge of the first Brillouin zone at k_{z} = π/a. Thus, maximum splitting occurs roughly around the middle of the onedimensional Brillouin zone, where k_{z} = π/(2a) or λ/a = 4 with k_{z} = 2π/λ. In analogy to optical activity, for a given frequency ω = ω_{1} = ω_{2}, the splitting in wave number, Δk_{z}, leads to a polarization rotation angle Δφ due to acoustical activity over propagation distance L_{z} given by Δφ = Δk_{z}L_{z}/2. The maximum splitting in Fig. 2 decreases with increasing beam crosssection N_{x} = N_{y} = L_{x}/a = L_{y}/a, but remains finite in the limit N_{x} = N_{y}→∞. This behavior is in line with our above intuitive reasoning starting from Cauchy elasticity as well as with our previous static experiments^{19}. In the remainder of this Letter, we will focus on these two chiral phonon bands, i.e., on n = 1,2.
Our analysis of phonons based on micropolar cubicsymmetry continua^{32} yields that these two transverse bands live in a subspace orthogonal to the other two bands (n = 3,4). This statement is true for wave vectors along the three cubic principal directions, e.g., for the zdirection. We, therefore, find several crossings of these bands (rather than avoided crossings). Without chirality, the character of the third (black) and fourth (blue) bands in Fig. 2 would be that of a pure twist and that of a pure compression wave, respectively. In the presence of chirality, their characters are mixed. This band mixing is the direct dynamic counterpart of our recent static experiments on metamaterials with cubic symmetry^{19}, in which an axial force imposed onto a beam led to a twist of the beam. In the limit N_{x} = N_{y} → ∞, the twist mode (black) is absent because it is disallowed for Blochperiodic boundary conditions along the x, y, and zdirections.
Connection to acoustical activity
The slopes of all four modes depend on the beam crosssection, hence on N_{x} = N_{y}. This behavior is the counterpart of the fact that the effective Young’s modulus in the stationary case of micropolar materials is no longer a constant “material parameter”, but rather depends strongly on the geometry, i.e., on the ratio N_{x} = N_{y} = L_{x}/a = L_{y}/a^{18}. If we switch off chirality (δ = 0 in Fig. 1a) as a control calculation and use otherwise identical geometrical and constituent material parameters (cf. Fig. 1), the behavior becomes ordinary again and the slopes no longer change very much (see lower row in Fig. 2).
For example, on the basis of the data shown for N_{x} = N_{y} = 1 in the upper row of Fig. 2, we estimate from the wave number splitting at about k_{z} = π/(2a) an acoustical activity of 17° per unit cell. We have confirmed this estimate by independent numerical calculations for samples with a finite extent along the zdirection (see Supplementary Fig. 1). Here, we excite a linearly polarized transverse wave at the bottom of the metamaterial beam. N_{z} unit cells upwards, a plate with markers allows for monitoring the local displacement vectors versus time. Clearly, the open end of the sample leads to reflections of the elastic waves. However, on the way back, the polarization rotation is reversed due to timereversal symmetry (unlike for the Faraday effect, see section “Introduction”). Therefore, standing waves along the beam axis do not change the polarization rotation at the top of the beam. Standing waves may well influence the amplitude of the displacements at the top though. As the zaxis has fourfold rotational symmetry, the polarization rotation does not depend on the orientation of the incident linear polarization (see Supplementary Fig. 2).
Measurement setup
We have designed our experiments based on these calculations. We fabricate the polymer samples by 3D laser nanoprinting as described previously^{19}. The dedicated homebuilt measurement setup has not been described previously: timeharmonic transverse flexural waves are excited by a piezoelectric transducer (Physik Instrumente, PICMA Chip Actuator). The samples are directly glued onto a dedicated holder, which has been 3D printed onto the transducer (see electron micrograph in Fig. 3 and Supplementary Fig. 3). The samples are imaged from the top by a microscope objective lens (Carl Zeiss, LD Achroplan 20×, NA = 0.4). They are stroboscopically illuminated by two infrared (850 nm center wavelength) lightemitting diodes (Vishay, VSLY 3850), the current through which is pulsed with a constant duty cycle of 1.5% (e.g., corresponding to 125 ns pulse duration for a frequency of 120 kHz), and synchronized with respect to the sinusoidal piezoelectric excitation. In this fashion, we take snapshots of the displaced structure versus time. By varying the time delay between excitation and illumination, we acquire slowmotion movies of the markers on top of the plate or at the side of the bottom of the sample. The latter serve to control the incident wave polarization and behavior. Using image crosscorrelation analysis^{36}, we detect displacements that are much smaller than the wavelength of the light used for illumination and which are much smaller than one pixel of the complementary metal–oxide–semiconductor black/white camera (FLIR Systems BFLYPGE50S5MC) used to record the images.
Experimental data
Example measurements merged into an obliqueview scanning electron micrograph of a metamaterial structure are shown in Fig. 3. The red (blue) scale bar refers to the measured displacement vectors in the middle of the top plate (at the side of the bottom of the sample). These data are shown for five consecutive oscillation periods, demonstrating the reproducibility of the measurements. In this particular example, we derive a polarization rotation of 44° (also see Supplementary Fig. 4). These and further results are summarized in Fig. 4 (full dots) and compared with results from the bandstructure calculations (dashed curves) and results from frequencydomain finitesample calculations (solid curves) already discussed above. The overall agreement between experiment and theory is very good. All qualitative trends are reproduced (as can be seen in Fig. 4a–c), the rotation angle generally increases with increasing frequency. This increasing deviation from Cauchy elasticity is expected based on our reasoning that the fixed unit cell size increases with respect to the wavelength with increasing frequency, moving away from the limit of “point mechanics”. This behavior must not be misinterpreted in terms of a resonance. It also occurs in generalized continuum theories^{27,28,29}, which do not account for resonances. Figure 4a additionally demonstrates that the rotation angle is maximum for slender beams composed of only a singleunit cell in the crosssection (N_{x} = N_{y} = 1). It decreases toward a certain finite level versus increasing beam crosssection toward a 3D bulk metamaterial (N_{x} = N_{y} = L/a ≫ 1). The rotary power stays finite in the bulk (see dashed black curve) because the unit cell size a is finite compared to the wavelength λ, again deviating from the limit of “point mechanics”. Figure 4b, c shows that the rotation angle increases versus beam length or height N_{z}, regardless of the beam crosssection (which is different in b and c). This behavior is analogous to the rotation angle in optical activity increasing proportionally with the propagation coordinate.
Comparison with theory
The calculations for finite metamaterial samples shown by the solid curves in Fig. 4 exhibit weak wiggles superimposed on a monotonous behavior. We have traced their origin back to massspring wobbling resonances, with the mass given mainly by the top plate. The wobbling eigenfrequency decreases with increasing total weight of the top plate, hence it decreases with increasing beam crosssection. Due to the sample chirality, the wobbling motion induces an additional polarization rotation, which is an artifact in the sense that it does not occur in the bandstructure calculations (which refer to N_{z}→∞). The behavior near the wobbling resonances depends on whether damping is accounted for. In the calculations shown in Fig. 4, we have added a finite imaginary part to the polymer’s Young’s modulus according to E = E^{'} + iE^{''} = 4.18 GPa + i0.20 GPa. This complex value has been obtained from independent experiments on tuning forks made by the same 3D laser nanoprinting process under comparable conditions and measured at comparable excitation frequencies (see Supplementary Fig. 5). For comparison, Supplementary Fig. 6 shows computations for E^{''} = 0. Apart from the vicinity of the artifact resonances, the observed polarization rotation does not depend on E''. Intuitively, damping reduces the amplitudes of all polarization components alike; therefore, it changes neither the polarization state nor its evolution, making acoustical activity a robust phenomenon.
In conclusion, we have experimentally demonstrated the phenomenon of acoustical activity, the mechanical counterpart of optical activity. Based on generalizations of Cauchy elasticity, acoustical activity in chiral continua has previously been predicted theoretically. Our ultrasound experiments on 3D chiral mechanical metamaterials reveal linear polarization rotations as large as 22° per metamaterial unit cell. These chiral degrees of freedom provide an unprecedented control of the polarization of elastic waves in three dimensions. By scaling the metamaterial lattice constant a, a large range of operation frequencies is accessible.
Methods
Sample fabrication
We fabricate the polymer samples by 3D laser nanoprinting (Nanoscribe GmbH, Photonics Professional GT) using a commercially available photoresist (IPS, Nanoscribe GmbH). During the printing process a 25× objective lens (numerical aperture NA = 0.8, Carl Zeiss) was dipped directly into the photoresist. Underlying 3D models were created in STLfile format using COMSOL Multiphysics (COMSOL Inc.) and further processed using Describe (Nanoscribe GmbH). Therein, the 3D models were split into lines with a horizontal distance between adjacent lines of 0.3 μm and a vertical distance of 0.7 μm between adjacent planes. The scanning of the galvanometric mirrors led to a laser focus speed of 0.1 ms^{−1}, and the laser power was set to 45% of the maximum output power. To remove the unexposed photoresist, the samples were first placed in a bath of mrDev 600 (20 min), followed by a bath of acetone (2 min) and finally placed in a bath of isopropanol (20 min). Afterwards, the sample was ultraviolet cured for 20 min.
Finiteelement method (FEM) bandstructure and eigenmode calculations
For the finiteelement bandstructure calculations ω(k_{z}), we solved the eigenvalue problem using COMSOL multiphysics. We used a threedimensional model typically meshed into a few million tetrahedra and applied secondorder ansatz functions (MUMPS solver). In the case of the infinite crystal (N_{x} = N_{y} = N_{z} = ∞), the cubic unit cell as shown in Fig. 1c was implemented with Blochperiodicity on all sides of the cube. In the cases of finite footprints (N_{x} = N_{y} = 1,2,3,5 and N_{z} = ∞) N_{x} × N_{y} × 1 unit cells were implemented. Bloch boundary conditions were only applied along the zdirection, whereas all other boundaries were left open (i.e., traction free). A linear elastic material was assumed with Young’s modulus E = 4.18 GPa, a Poisson’s ratio ν = 0.4, and a mass density ρ = 1.15 g cm^{−3}. In order to isolate the structural properties of the unit cell from the ones of the constituent material (Young’s modulus, Poisson’s ratio, and mass density), the left vertical frequency scale f was converted to a/λ_{0} (right vertical scale). Here, λ_{0} is the wavelength of the constituent’s longitudinal mode, given by c/f, with c being the phase velocity of the longitudinal mode \(c = \sqrt {\frac{{E(1  \nu )}}{{\rho (1  2v)(1 + \nu )}}}\).
Finitestructure FEM calculations
We deduced the polarization rotation angle of the finite structures containing N_{x} × N_{y} × N_{z} unit cells by solving the eigenvalue problem for a given frequency ω using COMSOL multiphysics. As in the experiment, a plate with the same footprint N_{x}a × N_{y}a as the structure and with a thickness of 10 μm was attached to the top. We prescribed a displacement at the bottom of the metamaterial structure. All other boundaries were left open (i.e., traction free). The polarization of the eigenmode was calculated by averaging the displacement field components u_{x} and u_{y} over a square with side length \(\frac{3}{4} \times a\) at the center of the top plate. This averaging size is comparable to the distance between the positions of the markers tracked in the experiment. A linear elastic material was assumed with a complexvalued Young’s modulus E = 4.18 GPa + i0.20 GPa, a Poisson’s ratio ν = 0.4, and a mass density ρ = 1.15 g cm^{−3}. The only exception are the dots in Fig. S6, where we have used E = 4.18 GPa (with zero imaginary part) for comparison.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
Kelvin, T. W. The second Robert Boyle lecture. J. Oxf. Univ. Jr. Sci. Club 18, 25 (1894).
Whyte, L. L. Chirality. Nature 180, 513 (1957).
Whyte, L. L. Chirality. Nature 182, 198 (1958).
Lindman, K. F. Über eine durch ein isotropes System von spiralförmigen Resonatoren erzeugte Rotationspolarisation der elektromagnetischen Wellen. Ann. Phys. 63, 621–644 (1920).
Lindell, I. V., Sihvola, A. H., Tretyakov, S. A. & Viitanen, A. J. Electromagnetic Waves in Chiral and BiIsotropic. (Media. Artech House Inc., Norwood, 1994).
Serdyukov, A., Semchenko, I., Tertyakov, S. & Sihvola, A. Electromagnetics of BiAnisotropic Materials—theory and Application. Vol. 11 (Gordon and Breach Science Publisher, Amsterdam, 2001).
Hecht, E. Optics. 4th ed. (Pearson Education Limited, Harlow, 2013).
Sharma, V., Crne, M., Park, J. O. & Srinivasarao, M. Structural origin of circularly polarized iridescence in jeweled beetles. Science 325, 449–451 (2009).
Gansel, J. K. et al. Gold helix photonic metamaterial as broadband circular polarizer. Science 325, 1513–1515 (2009).
Watson, J. D. & Crick, F. H. C. Molecular structure of nucleic acids. Nature 171, 737–738 (1953).
Griffiths, D. Introduction to Elementary Particles. (WileyVCH Verlag GmbH & Co. KGaA, Weinheim, 2004).
Takane, D. et al. Observation of chiral fermions with a large topological charge and associated fermiarc surface states in CoSi. Phys. Rev. Lett. 122, 076402 (2019).
Gal, J. Pasteur and the art of chirality. Nat. Chem. 9, 604–605 (2017).
Nguyen, L. A., He, H. & PhamHuy, C. Chiral drugs: an overview. Int. J. Biomed. Sci. 2, 85–100 (2006).
Armon, S., Efrati, E., Kupferman, R. & Sharon, E. Geometry and mechanics in the opening of chiral seed pods. Science 333, 1726–1730 (2011).
Süsstrunk, R. & Huber, S. D. Observation of phononic helical edge states in a mechanical topological insulator. Science 349, 47–50 (2015).
Ni, X., Weiner, M., Alù, A. & Khanikaev, A. B. Observation of higherorder topological acoustic states protected by generalized chiral symmetry. Nat. Mater. 18, 113–120 (2019).
Lakes, R. S. & Benedict, R. L. Noncentrosymmetry in micropolar elasticity. Int. J. Eng. Sci. 20, 1161–1167 (1982).
Frenzel, T., Kadic, M. & Wegener, M. Threedimensional mechanical metamaterials with a twist. Science 358, 1072–1074 (2017).
Lipton, J. I. et al. Handedness in shearing auxetics creates rigid and compliant structures. Science 360, 632–635 (2018).
Bertoldi, K., Vitelli, V., Christensen, J. & Van Hecke, M. Flexible mechanical metamaterials. Nat. Rev. Mater. 2, 17066 (2017).
Bauer, J. et al. Nanolattices: an emerging class of mechanical metamaterials. Adv. Mater. 29, 1701850 (2017).
Kadic, M., Milton, G. W., Van Hecke, M. & Wegener, M. 3D metamaterials. Nat. Rev. Phys. 1, 198–210 (2019).
Nieves, M. J., Carta, G., Jones, I. S., Movchan, A. B. & Movchan, N. V. Vibrations and elastic waves in chiral multistructures. J. Mech. Phys. Solids 121, 387–408 (2018).
FernandezCorbaton, I. et al. New twists of 3D chiral metamaterials. Adv. Mater. 31, 1807742 (2019).
Zhu, H. et al. Observation of chiral phonons. Science 359, 579–582 (2018).
Portigal, D. L. & Burstein, E. Acoustical activity and other firstorder spatial dispersion effects in crystals. Phys. Rev. 170, 673–678 (1968).
Bhagwat, K. V. & Subramanian, R. Acoustical activity of crystals: a comparative study of three descriptions. Acta Cryst. A 44, 551–554 (1988).
Lakhtakia, A., Varadan, V. V. & Varadan, V. K. Elastic wave propagation in noncentrosymmetric, isotropic media: dispersion and field equations. J. Appl. Phys. 63, 5246–5250 (1988).
Kweun, J. M., Lee, H. J., Oh, J. H., Seung, H. M. & Kim, Y. Y. Transmodal FabryPérot resonance: theory and realization with elastic metamaterials. Phys. Rev. Lett. 118, 205901 (2017).
Sommerfeld, A. Mechanics of Deformable Bodies: Lectures on Theoretical Physics 2 (Academic Press, New York, 1950).
Eringen, A. C. Microcontinuum Field Theories: I. Foundations and Solids. (Springer Science & Business Media, New York, 1999).
Kadic, M., Frenzel, T. & Wegener, M. Mechanical metamaterials: when scale matters. Nat. Phys. 14, 8–9 (2018).
Coulais, C., Kettenis, C. & Van Hecke, M. A characteristic length scale causes anomalous size effects and boundary programmability in mechanical metamaterials. Nat. Phys. 14, 40–44 (2018).
Miklowitz, J. The Theory of Elastic Waves and Waveguides. 1st ed. (North Holland, Amsterdam, 2015).
Bückmann, T. et al. Tailored 3D mechanical metamaterials made by dipin directlaserwriting optical lithography. Adv. Mater. 24, 2710–2714 (2012).
Acknowledgements
We thank Andreas Naber and Alexander Münchinger (both KIT) for discussions. We acknowledge support by the Excellence Cluster “3D Matter Made to Order” (3DMM2O) funded through the German Excellence Strategy via Deutsche Forschungsgemeinschaft (DFG) and by the Carl Zeiss Foundation. We acknowledge additional support by the Helmholtz program “Science and Technology of Nanosystems” (STN) and through the associated “Virtual Materials Design” (VIRTMAT) project as well as by the Karlsruhe Nanostructure Service Laboratory (NSL). T.F. has been supported by the Karlsruhe School of Optics & Photonics (KSOP). T.F. and J. K. have been supported by the Heidelberg Karlsruhe Strategic Partnership (HEiKA) graduate school Functional Materials. J. K. acknowledges support by the Hector Fellow Academy. M. K. acknowledges support by the EIPHI Graduate School (contract “ANR17EURE0002”) and by the French Investissements d’Avenir program, project ISITEBFC (contract “ANR15IDEX03”). We acknowledge support by the KITPublication Fund of the Karlsruhe Institute of Technology.
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T.F. and M.W. conceived the experiment. M.W. supervised the work. T.F. conducted the main experiments; J.K. and E.J. performed the tuning fork experiments. T.F. performed numerical simulations with the help of M.K. T.F. analyzed all the experimental and numerical data. T.F. took the scanning electron micrographs. All authors discussed the results. M.W. wrote a first draft of the paper. All authors contributed to the final version of the paper.
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Frenzel, T., Köpfler, J., Jung, E. et al. Ultrasound experiments on acoustical activity in chiral mechanical metamaterials. Nat Commun 10, 3384 (2019). https://doi.org/10.1038/s41467019113668
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DOI: https://doi.org/10.1038/s41467019113668
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