Analytical solutions for acoustic vortex beam radiation from planar and spherically focused circular pistons

Acoustic vortex beams are quasiplanar waves with helical wavefronts characterized by the orbital number ℓ. Although Gaussian amplitude distributions result in closed-form analytical solutions for the entire paraxial field [Gokani et al., J. Acoust. Soc. Am. 155, 2707–2723 (2024)], acoustic vortex beams are usually radiated by sources with a uniform circular amplitude distribution. In this talk, analytical solutions for the field radiated by unfocused and focused uniform circular vortex sources of radius a are derived. Evaluation of the Fresnel diffraction integral in the far field of an unfocused source and in the focal plane of a focused source leads to a solution in terms of an infinite series of Bessel functions for ℓ>−2. By calculating the first local maximum of this solution, it is found that the vortex ring radius is rℓ = ξ ℓ z/ ka in the far field of an unfocused source and rℓ = ξ ℓ d/ka in the focal plane of a focused source with focal distance d, where ξ ℓ = 1.23ℓ + 1.49 and k is the wavenumber. The solution given by the infinite series is reduced to closed forms for 0 ≤ ℓ ≤ 4, corresponding to orbital numbers commonly used in experiments. [CAG supported by the ARL:UT McKinney Fellowship in Acoustics.]