Analytical solution for a focused vortex beam radiated by a Gaussian source

Abstract

Current interest in focused vortex beams is motivated by the ability to trap particles axially and laterally using the resulting radiation force. A simple closed-form solution is obtained in the Fresnel approximation for a sound beam radiated by a Gaussian source with time dependence e−iωt , focal length d, amplitude distribution exp(− r 2/ a 2), and azimuthal phase dependence einθ , where θ is the angle in the plane perpendicular to the beam axis, and the integer n is the topological charge, referred to here as the vorticity. The solution is in good agreement with the pressure field predicted in the paraxial region by numerical evaluation of the Rayleigh integral. Of interest in optics as well as acoustics is the distance from the minimum along the beam axis to the first local maximum, referred to as the vortex ring. The present solution yields rn = ηn d/ ka for the vortex ring radius in the focal plane, where k is the wavenumber, and ηn = 1.69 + 0.965( n−1) for vorticities in the range 1 ≤ n < O(20). Within this range the radius rn thus increases linearly with the vorticity. Results are also presented for dependence of the focusing gain on the vorticity. [CAG was supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]