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

Analytical solutions for acoustic vortex beams radiated by sources with uniform circular amplitude distributions are derived in the paraxial approximation. 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 solutions in terms of an infinite series of Bessel functions for orbital numbers ℓ>-2. These solutions are reduced to closed forms for 0≤ℓ≤4, which correspond to orbital numbers commonly used in experiments. A scaling law for the vortex ring radius is derived, and its relevance is characterized using ray theory.