Optimal Multi-Objective Wave-Momentum Shaping in Scattering Media

TL;DR

The work develops a comprehensive, unified framework for optimizing radiation forces and torques in complex, inhomogeneous media by linking parametric medium changes to scattering-field variations through the Generalized Wigner-Smith matrix. Central to the method is an immittance-based scattering formalism that yields variational identities and a Hermitian generator $\boldsymbol{Q}_{\Delta \boldsymbol{r}}$ whose Rayleigh quotient prescribes force and torque optimizations. For a single object, extremal GWS eigenstates maximize projections of force or torque; for multiple objects, the framework treats multi-objective optimization via Pareto fronts and exposes fundamental uncertainty-like trade-offs bounding simultaneous objective improvements. The paper also develops constrained optimization strategies for selective manipulation and demonstrates, with simulations and supplemental material, that GWS-based Pareto-optimal strategies outperform naive random sampling. Together, these insights provide a principled route to coordinated wavefront shaping in strongly scattering media with potential applications in precision therapies and targeted manipulation.

Abstract

Radiation forces and torques are key to manipulating objects with acoustic or electromagnetic waves. An important concept in this context is the Generalized Wigner-Smith (GWS) matrix, which has previously been primarily studied for optimizing radiation forces and torques on single objects embedded inside complex scattering environments. Here, we develop a unified scattering framework that rigorously establishes this connection for arbitrary inhomogeneous, lossless electromagnetic and acoustic media, as well as for controlling multiple objects individually. Variational identities relate parametric changes of the medium to perturbations of the scattering fields, from which the GWS matrix emerges as a natural generator of radiation forces and torques. For a single object, its extremal eigenstates yield maximal force or torque along a chosen direction; for multiple objects, the same framework defines Pareto-optimal compromises among competing objectives and reveals uncertainty relations for their simultaneous optimization. This establishes a comprehensive foundation towards collective and selective manipulation of objects in complex media.

Figures (4)

• Figure 1: Two examples of 2D inhomogeneous media described by the scattering matrix formalism, together with two optimization goals. The volume $V$ is represented in gray, comprising both the background and the inclusions, and is delimited by the dashed boundary $\bold{A}$ oriented away from the scattering region. a) Multimode disordered waveguide in which two objects must be displaced. b) Multiport open disordered cavity with monomode waveguides (here plane waves as in acoustics) in which one object must be displaced in the $y$-direction only, and one object must not be displaced at all.
• Figure 2: Collective and selective manipulation of objects in scattering media. a) Bi-objective optimization: probability distribution of the force projections on two objects for $10^7$ random input states. The marginal distributions form B-splines. The Pareto front (red solid line) is compared to a heuristic weighted sum of eigenstates (green dashed line). b) Full wave simulation of the squared acoustic pressure field corresponding to the Pareto-optimal solution for which the force projections are equal, i.e., $\braket* [1] {\bold{Q}_{\Delta \bold{r}_1}}=\braket* [1] {\bold{Q}_{\Delta \bold{r}_2}}$. The blue arrows are the objectives and the black arrows are the net forces exerted on the objects. c) Constrained optimization: the object is displaced along the vertical direction only (net force in black perfectly overlapping with the objective in blue), while the object experiences local forces (small red arrows with their width and length logarithmically scaling with the local force magnitude) which globally compensate to obtain zero net force. The wavelength $\text{}$ is shown on the right.
• Figure 3: Probability density function of expectation values of Hermitian matrices of size $N$. Comparison between $10^5$ random input states (in grey) and the theoretical M-spline (black curves).
• Figure 4: Further examples of bi-objective optimization for which $\braket* [1] {\bold{Q}_{\Delta \bold{r}_1}}=\braket* [1] {\bold{Q}_{\Delta \bold{r}_2}}$. As in \ref{['fig:subfigs']}b, blue arrows represent the objectives and black arrows represent the realized Pareto-optimal net equal-projection forces on the scatterers.