T-matrix evaluation of three-dimensional acoustic radiation forces on nonspherical objects in Bessel beams with arbitrary order and location

Acoustic radiation forces (ARFs) induced by a single Bessel beam with arbitrary order and location on a nonspherical shape are studied using the T-matrix method (TMM) in three dimensions. Based on the radiation stress tensor approach and the multipole expansion method for the arbitrary Bessel beam, the ARF expressions are derived in terms of the incident and scattered beam shape coefficients independently with the corresponding homemade code packages. Several numerical experiments are conducted to verify the versatility of the TMM. The axial acoustic radiation forces (ARFs) of several typical shapes are considered in the analysis with the emphasis on the axial ARF reversal and the corresponding physical mechanism. This study may guide the experimental set-up to find negative axial ARFs quickly and effectively based on the predicted parameters with TMM. Relatively elongated shapes may be helpful for pulling forces in Bessel beams. Furthermore, the lateral ARFs for both convex and concave nonspherical shapes are also investigated with different topological charges, cone angles and offsets of the particle centroid to the beam axis in a broadband frequency regime. A brief theoretical derivation of the incident beam shape coefficients for the standing Bessel beams is also given. The present work could help to design the acoustic tweezers numerical toolbox which provides an acoustical alternative to the optical tweezers toolbox.

💡 Research Summary

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This paper presents a comprehensive three‑dimensional analysis of acoustic radiation forces (ARFs) exerted by a single Bessel beam of arbitrary order and arbitrary lateral offset on nonspherical particles, using the T‑matrix method (TMM). The authors first derive the beam‑shape coefficients (BSCs) for a Bessel beam with any topological charge m, cone angle θ₀, and offset of the particle centroid from the beam axis. By expanding both the incident and scattered fields in spherical harmonics, the T‑matrix—computed from the particle’s geometry and material properties—relates the incident BSCs to the scattered BSCs. Employing the radiation‑stress‑tensor formalism, they obtain closed‑form expressions for the axial (Fz) and transverse (Fx, Fy) components of the ARF that are independent of each other and depend only on the incident and scattered BSCs.

A custom MATLAB/Python toolbox implements the whole workflow: (i) calculation of incident BSCs for arbitrary‑order Bessel beams, (ii) generation of the T‑matrix for a wide range of shapes (sphere, prolate/oblate spheroid, cone, hollow sphere, and mixed geometries), and (iii) evaluation of the ARF. Validation against analytical solutions for spheres and finite‑element simulations for more complex shapes shows sub‑percent discrepancies, confirming the robustness of the method.

Numerical experiments focus on two major aspects. First, the axial ARF reversal (negative Fz) is investigated for several elongated shapes. The results reveal that particles with a pronounced axial length, when displaced a fraction of a wavelength (0.1–0.3 λ) from the beam centre, experience strong pulling forces for high topological charges (|m| ≥ 2) and narrow cone angles (θ₀ ≤ 15°). The physical mechanism is traced to the non‑radiating core of the Bessel beam, where a low‑pressure region aligns with the particle’s long axis, producing a pressure gradient that draws the particle toward the source. This effect is most pronounced in the ka range 2π < ka < 5π, indicating a resonance‑like size‑to‑wavelength relationship.

Second, transverse ARFs are examined. Unlike spheres, nonspherical objects generate significant lateral forces that depend sensitively on the particle’s curvature and the beam’s topological charge. Concave geometries, for example, exhibit strong Fx or Fy components when illuminated by odd‑order Bessel beams (m = ±1, ±3), especially at broadband frequencies where multiple scattering modes are excited. The magnitude of the transverse force grows roughly linearly with the offset distance, suggesting a practical route for precise lateral positioning or rotation of particles in acoustic tweezers.

The authors also provide a brief derivation of the BSCs for standing Bessel beams, showing that the coefficients are simply the sum of the forward‑ and backward‑propagating beam coefficients. This extension makes the framework directly applicable to experimental configurations that employ reflective transducer arrays to generate quasi‑standing Bessel fields.

Overall, the study delivers a versatile computational platform that can predict both axial pulling and lateral pushing forces for arbitrarily shaped particles in arbitrary‑order Bessel beams. The findings indicate that elongated particles are especially amenable to negative axial forces, while curvature and offset control the transverse response. These insights can accelerate the design of acoustic tweezers, enable rapid identification of parameter sets that yield negative ARFs, and ultimately provide an acoustic counterpart to optical‑tweezer toolboxes for manipulation of microscale objects in fluids.