Multi-Objective Tweezers in Scattering Media

1. Problem Statement

Radiation forces and torques generated by acoustic and electromagnetic waves enable the contactless manipulation of microscopic objects (e.g., cells, microrobots). While "wavefront shaping" has successfully demonstrated focusing and trapping in complex, scattering media (such as biological tissue), significant limitations remain:

• Single-Object Focus: Existing methods primarily optimize for a single object or a single focal point.
• Competing Objectives: Controlling multiple objects simultaneously is challenging because the objectives often conflict (e.g., maximizing force on Object A while minimizing it on Object B, or moving Object A in a specific direction while keeping Object B stationary).
• Lack of Theoretical Bounds: There is no general framework to quantify the fundamental trade-offs (incompatibility) between manipulating multiple objects in a disordered medium, nor a method to find the optimal compromise.
• Complex Media: High-fidelity manipulation in strongly scattering, inhomogeneous media (like living tissue) remains elusive, with current in vivo applications relying on simple focal targeting rather than programmable momentum transfer.

2. Methodology

The authors develop a unified theoretical framework based on variational relations linking parametric changes in the scattering medium to modifications in far-field scattering amplitudes.

• Generalized Wigner-Smith (GWS) Matrix:
  The core tool is an extension of the GWS matrix, defined as $Q_\alpha = jS^\dagger \partial_\alpha S$, where $S$ is the scattering matrix and $\alpha$ is a parameter (e.g., position or orientation of an object).

  • The authors derive a variational relation showing that the expectation value of the GWS matrix for a specific input state $s_+$ is directly proportional to the radiation force (or torque) projected along the direction of the parameter change:
    $$\frac{1}{\omega} \langle Q_{\delta r} \rangle_{s_+} = \mathbf{F} \cdot \hat{e}_r$$
  • This establishes that extremal eigenstates of the GWS matrix correspond to input wavefronts that maximize or minimize radiation forces.

• Multi-Objective Formulation:

  • For $N_d$ objects, the total quantity is a weighted sum of individual GWS matrices: $Q_{\delta r_{tot}} = \sum |\delta r_i| Q_{\delta r_i} / |\delta r_{tot}|$.
  • Since GWS matrices for different objects generally do not commute ($[Q_{\alpha}, Q_{\beta}] \neq 0$), they cannot be simultaneously diagonalized. This implies that no single input state can maximize all forces simultaneously.
  • The problem is framed as a Multi-Objective Optimization problem. The authors utilize the concept of Pareto optimality, where a solution is optimal if no objective can be improved without degrading another.

• Uncertainty Relation:
  By applying the Schrödinger-Robertson uncertainty relation and the Bhatia-Davis inequality to the GWS matrices, the authors derive a rigorous bound on the trade-offs between incompatible objectives:
  $$\epsilon_\alpha \epsilon_\beta \geq \frac{1}{4} \frac{|\langle [Q_\alpha, Q_\beta] \rangle|^2 + \text{Cov}(Q_\alpha, Q_\beta)^2}{(\langle Q_\alpha \rangle - \lambda_1)(\langle Q_\beta \rangle - \lambda_1)}$$
  This quantifies the fundamental "uncertainty" in simultaneously optimizing forces on multiple objects.

• Constrained Optimization:
  The framework is extended to handle hard constraints (e.g., "move Object 1 up, but keep Object 2 still"). This is formulated as a complex-valued quadratically constrained quadratic program (QCQP), solved numerically (e.g., using Gurobi) to find input wavefronts that satisfy specific force constraints while maximizing others.

3. Key Contributions

1. Unified Theory: A general theory applicable to both acoustics and electromagnetism that links far-field wavefront shaping to local radiation forces/torques on arbitrary objects in scattering media.
2. Pareto-Optimal Control: Demonstration that the eigenstates of the multi-objective GWS matrix provide Pareto-optimal solutions, enabling the coordinated manipulation of multiple objects where naive random search fails.
3. Fundamental Bounds: Derivation of an uncertainty relation that rigorously defines the limits of simultaneous control, proving that trade-offs are inherent to the physics of scattering.
4. Constrained Manipulation: A method to enforce specific constraints (e.g., zero net force on an object while maximizing force on another), allowing for selective and precise control.

4. Results

• Simulation of Bi-Objective Optimization: In a 20-port disordered 2D acoustic cavity, the authors simulated the simultaneous translation of two objects.
  • Pareto Front: The GWS-derived solutions formed a convex Pareto front that significantly outperformed $10^7$ random input states.
  • Comparison: A heuristic method (linear combination of single-object eigenstates) was consistently outperformed by the full multi-objective GWS optimization.
• Constrained Manipulation: The authors successfully demonstrated a scenario where Object 1 was moved vertically while Object 2 was held completely stationary.
  • The optimized wavefield generated local forces that canceled out globally on Object 2, while creating a net vertical force on Object 1.
  • This was validated using the Brillouin stress tensor, confirming that the far-field shaping translated to the desired local mechanical action.
• Tri-Objective Optimization: The framework was extended to three objects of arbitrary shapes (square, triangle, star) in a turbid medium, successfully imposing independent force directions on all three simultaneously.
• Symmetry Analysis: The paper analyzes cases where objectives commute (e.g., in mirror-symmetric systems), showing that the Pareto front collapses to a line segment, whereas in generic asymmetric systems, it fills a 2D region.

5. Significance and Applications

• Biomedical Applications: This work paves the way for in vivo precision therapies, such as targeted drug delivery, sorting of cells/organoids, and the navigation of microrobots through deep, scattering biological tissues (e.g., the brain).
• Beyond Focusing: It moves the field beyond simple "focusing" or "trapping" to programmable momentum transfer, allowing for complex assembly and manipulation tasks in opaque environments.
• Fundamental Physics: The discovery of an uncertainty relation for classical wave momentum shaping draws a profound parallel between quantum mechanics (incompatible observables) and classical wave control in complex media.
• Scalability: The method is platform-agnostic, applicable to acoustic, electromagnetic, and potentially other wave systems, offering a path toward collective and selective manipulation in lossy, strongly scattering media.

In summary, the paper establishes a rigorous mathematical and practical framework for "multi-objective tweezers," transforming the challenge of controlling multiple objects in chaos into a solvable optimization problem with defined physical limits.