Partial wave series expansions in spherical coordinates for the acoustic field of vortex beams generated from a finite circular aperture

TL;DR

This paper derives exact partial-wave series expansions for acoustic vortex beams from finite circular sources, enabling precise analysis of their scattering and mechanical effects in various applications.

Contribution

It introduces a novel mathematical framework for representing finite vortex beams using partial-wave series expansions in spherical coordinates, based on fundamental integral theorems.

Findings

Exact solutions for various vortex beam types

Enhanced accuracy in acoustic scattering calculations

Potential for optimized acoustic device design

Abstract

Stemming from the Rayleigh-Sommerfeld surface integral, the addition theorems for the spherical wave and Legendre functions, and a weighing function describing the behavior of the radial component of the normal velocity at the surface of a finite circular radiating source, partial-wave series expansions are derived for the incident field of acoustic spiraling (vortex) beams in a spherical coordinate system centered on the axis of wave propagation. Examples for vortex beams, comprising \rho-vortex, zeroth-order and higher-order Bessel-Gauss and Bessel, truncated Neumann-Gauss and Hankel-Gauss, Laguerre-Gauss, and other Gaussian-type vortex beams are considered. The mathematical expressions are exact solutions of the Helmholtz equation. The results presented here are particularly useful to accurately evaluate analytically and compute numerically the acoustic scattering and other mechanical effects of finite vortex beams, such as the axial and 3D acoustic radiation force and torque components on a sphere of any (isotropic, anisotropic etc.) material (fluid, elastic, viscoelastic etc.). Numerical predictions allow optimal design of parameters in applications including but not limited to acoustical tweezers, acousto-fluidics, beam-forming design and imaging to name a few.