Shaping contactless radiation forces through anomalous acoustic scattering

Waves impart momentum and exert force on obstacles in their path. The transfer of wave momentum is a fundamental mechanism for contactless manipulation, yet the rules of conventional scattering intrinsically limit the radiation force based on the shape and the size of the manipulated object. Here, we show that this intrinsic limit can be broken for acoustic waves with subwavelength-structured surfaces (metasurfaces), where the force becomes controllable by the arrangement of surface features, independent of the object’s overall shape and size. Harnessing such anomalous metasurface scattering, we demonstrate complex actuation phenomena: self-guidance, where a metasurface object is autonomously guided by an acoustic wave, and tractor beaming, where a metasurface object is pulled by the wave. Our results show that bringing the metasurface physics of acoustic waves, and its full arsenal of tools, to the domain of mechanical manipulation opens new frontiers in contactless actuation and enables diverse actuation mechanisms that are beyond the limits of traditional wave-matter interactions.


Experimental setup and the acoustic force balance
Experimental arrangement and the relative orientation of the metasurface, the acoustic source, and the pendulum ensure the deflection measurement is a result of the metasurface force (as shown in Fig. S1a,d, and the figures in the main text): • Z-axis is the pendulum rotation axis. Only a z-torque can lead to a deflection signal.
• Only the forces in the horizontal X-Y plane contribute to the z-torque. Forces along the zaxis are therefore not relevant. • Metasurface unit cell profile varies in the X-direction, leading to a horizontal force metasurface which contributes to the z-torque. This force is the focus of this work.
• Forces in the horizontal plane (but perpendicular to the metasurface variation along the Xdirection), i.e., Y-forces, are not expected (theoretically, in simulation, or experiment), either from anomalous or conventional scattering. Even if such forces were present, the corresponding torque arm-length would be significantly shorter than for metasurface (i.e., a fraction of the metasurface size ≪ the length of the pendulum rod), therefore minimizing their contribution to the z-torque. Figure S1. Experimental setup. a Photo of the experimental setup and the definition of the axes in the system (pink). The axis of rotation is the z-axis (gravity is also along z). Metasurface (white rectangular slab) and its counterweight (green block) are on the opposite ends of the pendulum rod (limited visibility from this angle). b, c Raw (b) and processed (c) camera image of the screen for real-time deflection tracking. The image is acquired by a horizontal-viewing camera on the opposite side of the screen (panel a, top right). Laser beam path is in the X-Y plane (horizontal plane). d X-Z plane schematic. The metasurface is facing down (-z), while the acoustic source faces up (+z).

Metasurface designs synthesized and fabricated in this work
*All lengths are in mm Metasurface 1 (Figure 2

Alternative metasurface topologies for shaping acoustic radiation forces
The acousto-mechanical metasurface physics that we report is general and not unique to a particular unit cell design. As a demonstration, in addition to the unit cell profile depicted in the main text and Fig. S4, we investigate a metasurface composed of a coiled unit cell topology. Figure  S5a shows an example implementation where a unit cell consists of five horizontal arms (three on the left side and two on the right side). For operation at the target frequency of 20 kHz, we chose the following fixed parameters = 0.25 mm, = 2.14 mm, = 2.14 mm, and the wall thickness is the variable parameter. Figure S5b shows the relative phase variation of the reflected wave as is varied.
Following the same design process outlined in the Methods section, we arrive at the analytical and the refined/optimized metasurface 1 with the space-coiled unit cell profile. Because the unit cell width is smaller than before, we increased the number of unit cells to =36 to maintain the same overall size of the metasurface. As in the case of the groove-like metasurface of the main text, the optimization step for the space-coiled design resulted in a substantial lateral acoustic force enhancement (1.39x increase). The analytical designs of the two unit cell topologies-groove and space-coiled-otherwise show comparable performance (i.e., the optimized coiled metasurface has a 1.24x stronger lateral force relative to the optimized grooved metasurface). The coiled design has a smaller minimal feature size as well as a region of sensitive phase vs wall thickness relationship (steep slope around ~0.7 mm in Fig. S5b) which could pose potential fabrication challenges. The schematics of coiled metasurfaces and the tabulated values for the unit cell wall thicknesses (t) are presented below.   Figure S6. Relationship between the beam diameter and the self-guiding force for the metasurface 2 of the main text ( Figure 3). For each beam diameter (indicated by the full-width half-maximum value in mm), the force is numerically evaluated for a range of source positions (relative displacements between the metasurface center and the beam axis -see schematic in Figure 3b of main text). Forces are normalized to the peak force ( * ) of the narrowest/first analyzed beam width. Initially, increasing the beam diameter helps enhance the peak restoring acoustic radiation force. However, as the beam diameter becomes broader, the countering contribution from the opposing metasurface symmetric side becomes stronger, and the overall self-guiding force is weakened. This implies that the relative source displacement for which the force is the strongest (for a given diameter) would increase as the beam diameter increases, and this trend is indeed observed (i.e., the vertical dashed lines shift from left to right).