Wave-momentum shaping for moving objects in heterogeneous and dynamic media

The study addresses the challenge of non-contact manipulation of objects in complex, heterogeneous, and dynamically changing environments. Traditional optical and acoustic tweezers excel in controlled, low-reverberant, static media but their applicability is limited when disorder, multiple scattering, and time-varying conditions prevail, or when manipulation must occur at a distance. Prior work has showcased a variety of acoustic manipulation strategies—standing-wave trapping in 1D/2D/3D, acoustic vortices, and devices using lenses, metasurfaces, and holograms, as well as acoustofluidic platforms and microrobots—but these approaches typically require static, precisely controlled conditions and proximity to the target. The purpose of this study is to overcome these constraints by proposing and experimentally demonstrating wave-momentum shaping: an approach that uses only far-field, real-time measurements of the scattering matrix and a guide-star position signal to optimally transfer momentum (linear or angular) to an object as it moves. The importance lies in enabling robust manipulation—translation and rotation—of objects inside disordered and even dynamically changing scattering environments, without knowing the object's properties or modeling the interaction forces, potentially impacting biomedical applications, sensing, and manufacturing.

Extensive prior work has enabled contactless manipulation using standing acoustic fields that create potential wells at pressure nodes or antinodes, allowing collective or selective trapping and positioning in one, two, and three dimensions. Selectivity and functionality have been enhanced with acoustic vortices and with engineered elements such as lenses, metasurfaces, and holographic transducers. Parallel developments include on-chip acoustofluidic and acoustophoretic devices for lab-on-a-chip applications and wave-controlled microrobots for biomedical tasks. However, these methods generally presume controlled, static environments and often require proximity to the target, limiting their use in real-world, disordered or time-varying conditions. In photonics and adaptive optics, wavefront shaping methods have progressed to optimize focusing and compensate aberrations and multiple scattering using feedback or transmission/scattering matrix-based strategies. Generalized Wigner-Smith (GWS) operator techniques, derived from measured scattering matrices, have been shown to optimally focus waves in disorder and to maximize forces/torques on static objects, and even to cool ensembles of particles. The present work bridges these advances with dynamic manipulation: it adapts GWS-based optimality to moving targets, combining iterative guide-star feedback with real-time S-matrix updates to shape wave momentum in complex media.

Platform and setup: A macroscopic two-dimensional acoustic cavity is constructed by coupling a water-filled tank (100 × 100 × 3 cm) to a 2D air waveguide (width 104 cm, length 180 cm, height 8 cm) with anechoic terminations. At the operating frequency f = 1,590 Hz (λ ≈ 0.22 m), the waveguide supports N = 10 propagating modes on each side (2N = 20 total). Two linear arrays of 10 loudspeakers each are placed at both ends to excite controlled incident mode mixtures. Two columns of 10 microphones per side measure outgoing fields in the asymptotic regions. The movable target is a ping-pong ball (diameter 4 cm, mass 4.17 g) floating on the water surface; static disorder is implemented with plastic cylinders (2–4 cm diameter) protruding above the water surface. Surfaces are hydrophobically coated to avoid capillary sticking. A wide-angle camera (1,920 × 1,080 at 60 fps) provides the guide-star position of the object. For rotation experiments, three balls are glued in line and the center is pinned to a needle to allow rotation while preventing translation. For dynamic-media experiments, multiple additional balls with metallic nuts are loosely tethered and their positions randomized via a moving electromagnet. Scattering matrix acquisition: For each of the 2N orthogonal speaker excitations (one speaker at a time, 1,590 Hz), microphone data are processed to extract incident and outgoing modal amplitudes. Thus, the 2N × 2N scattering matrix S(t) is reconstructed in about 1.6 s total (≈80 ms per excitation). The raw S is regularized to be close to unitary by discarding a small antisymmetric part and rescaling subunitary eigenvalues while keeping phases. Wave-momentum shaping principle: The momentum transferred to the object along coordinate a (a ∈ {x, y} or a rotation angle θ) is linked to the expectation value of the GWS operator Q = −i S^{-1} (dS/da). For an input state that is an eigenvector of Q, the momentum transfer is proportional to the corresponding eigenvalue, enabling an optimal push along a. Discrete gradient estimation for translation: At time step m, three consecutive S-matrix measurements S_m, S_{m−1}, S_{m−2} are taken at nearby object positions (x_m, y_m), (x_{m−1}, y_{m−1}), (x_{m−2}, y_{m−2}), enabling a discrete approximation of ∂S/∂x and ∂S/∂y. The path’s intermediate checkpoints are arranged in a zigzag to form near-equilateral measurement triangles, minimizing gradient error. With ∂S/∂x and ∂S/∂y estimated, the translation GWS operators Q_x and Q_y are built and diagonalized to obtain eigenvectors and eigenvalues. Iterative motion algorithm (translation): 1) Initialize by sending three random wavefronts to move the ball slightly, measuring S at three nearby points and recording positions by camera. 2) Estimate ∂S/∂x and ∂S/∂y from the last three S matrices. 3) Construct and diagonalize Q_x and Q_y. 4) Form an input mode mixture as a superposition of the eigenvectors associated with the largest eigenvalues of Q_x and Q_y, weighted to push in the desired direction toward the next checkpoint. 5) Send the wavefront, allow the object to move and stabilize, measure S again, and iterate until the destination is reached. Rotation control: For angular manipulation, S is measured at successive angles with a fixed step (e.g., 20°). A backward three-point derivative approximates ∂S/∂θ, building Q_θ = −i S^{-1} (∂S/∂θ) S. The input state is chosen as the eigenvector of Q_θ with the desired sign of eigenvalue to realize anticlockwise or clockwise torque. Switching the sign of selected eigenvalues switches rotation direction. Dynamic medium: The same real-time S-based procedure is applied while additional scatterers move randomly under time-varying magnetic perturbations. The target object’s trajectory is controlled to follow a prescribed path despite environmental fluctuations. Field mapping: For insight, a perforated top plate allows scanning of acoustic pressure fields near the object with a robotic-arm-mounted microphone. Field maps for optimal eigenstates display pressure hotspots near the object aligned with the desired push, whereas eigenstates with small |eigenvalue| exhibit low pressure near the object, placing it in a quiet zone. Implementation details: Input eigenvectors are converted to speaker voltages using a calibrated coupling matrix, and the controller (Speedgoat + Matlab/Simulink) generates signals while acquiring microphone data for S estimation. The process assumes near-unitary S after regularization and relies on accurate, rapid camera-based position (or angle) measurements.

- Successful far-field manipulation via wave-momentum shaping in a disordered medium: A floating sphere (radius 20 mm ≈ 0.1λ at 1,590 Hz, λ ≈ 0.22 m) was guided along a predefined S-shaped trajectory of total length ≈ 4 λ using only far-field S-matrix measurements and a guide-star camera. - Mode-resolved optimality: At multiple time steps, the net momentum expectation of the injected superposition (from Q eigenstates) closely matched the measured direction of the object’s velocity. Individual modes contributed constructively under the constraints imposed by the local speckle field, confirming the optimal collective action of the mode mixture. - Angular momentum control and reversal: A three-ball object pinned for rotation was driven anticlockwise by selecting eigenvectors of Q_θ with positive eigenvalues. Abruptly switching to eigenvectors with negative eigenvalues reversed the rotation direction, as confirmed by angle-versus-time measurements (20° steps between S measurements). - Robust control in dynamic disorder: With multiple additional moving scatterers (blue balls with metallic nuts randomly actuated magnetically), the target ball (orange) followed a prescribed sinusoidal path with tiny deviations, while the other scatterers exhibited large, unpredictable fluctuations. This demonstrates robust trajectory control under strongly time-varying scattering. - Field-level insight: Measured pressure maps for optimal eigenstates showed localized hotspots near the object aligned with desired push directions; eigenstates with minimal |eigenvalue| lacked such hotspots, effectively placing the object in a quiet zone. - Practical performance metrics: S-matrix acquisition took ≈1.6 s (2N = 20 orthogonal excitations at ≈80 ms each). The method required no knowledge of the target’s physical properties nor modeling of interaction forces, and remained effective despite absorption, which altered amplitudes more than phases without significantly changing predicted momentum directions.

The findings directly address the central question of whether objects can be moved and rotated in complex, disordered, and dynamically changing media using only far-field information. By establishing and exploiting the link Δp_a = ⟨in| −i S^{-1} (dS/da) |in⟩, the study shows that diagonalizing the GWS operator yields input states that optimally transfer linear or angular momentum to the object. Iteratively updating the S matrix and recomputing Q accommodates changes due to the object’s motion and to environmental dynamics, enabling continuous guidance along complex paths and controlled rotation. The experimental agreement between predicted momentum directions and measured velocities validates the theoretical framework and demonstrates robustness to absorption and disorder. The approach eliminates the need for potential traps or detailed force modeling, making it well suited to settings where the medium is heterogeneous or time-varying, and it is compatible with scaling to other wave types (e.g., ultrasound, optics) and sizes (micro- to macroscale).

In this Article, we report on the experimental control of an object's translation and rotation in a complex and dynamic scattering medium using wave-momentum shaping. An iterative manipulation protocol, based solely on knowledge about the far-field scattering matrix of the system and a position guide star, enables the optimal transfer of linear and angular momentum from an acoustic field to manipulate an object in both static and dynamic disordered media. The dynamically injected wavefronts generate the optimal field speckle near the object so that it moves, much like a hockey player guiding a puck, through successive momentum kicks. This method is free of potential traps, robust against disorder and is tolerant of changes in time to the surrounding medium throughout the manipulation. Remarkably, the method is rooted in momentum conservation and does not require any knowledge of the object being manipulated, but only a guide-star measurement of its position. In addition, it does not require any modelling of interaction forces, making the protocol very general and broadly applicable to many real-life scenarios (including different waves, scales, objects and so on). Future efforts will focus on developing methods for objects of various sizes, for example, by transposing the method to ultrasonic frequencies to manipulate smaller objects, as well as extensions to the microscopic scale and to other platforms such as light.

- Dependence on real-time S-matrix measurement: Requires rapid, repeated acquisition of the full scattering matrix with sufficient signal-to-noise and near-unitarity after regularization; performance is constrained by acquisition time (~1.6 s per update here) and hardware. - Guide-star requirement: Needs external measurement of position (or angle) to inform gradient estimation and path following; in this work a camera provided the guide star. - Local degrees of freedom limitations: At certain locations, available speckle degrees of freedom may not support pushing exactly in the desired direction; although the algorithm typically recovers at subsequent steps, instantaneous control can be limited. - Gradient approximation sensitivity: Accuracy of Q_x, Q_y (and Q_θ) depends on the geometry of the last three measurement points (best with near-equilateral triangles for translation and appropriately spaced angles for rotation). Poor geometry increases error. - Assumption of near-unitarity: The optimality relation is derived for unitary S; experimental absorption and noise are mitigated via regularization, but strong losses or non-reciprocity could degrade performance. - Environmental constraints of the demonstration: Experiments used an audible-frequency, macroscopic 2D waveguide and floating objects; translating to other scales and environments (e.g., microscopic, in vivo) requires suitable transduction and S-matrix access. - Throughput and speed: The closed-loop rate is limited by S-matrix measurement and object response times; very fast dynamics could outpace the update rate without further hardware acceleration or predictive control.