High-order pseudo-Gaussian scalar acoustical beams

Exact solutions of the scalar Helmholtz equation describing tightly spherically-focused beams are introduced without any approximations using the complex source point method in spherical coordinates. The generalized solutions, valid for any integer degree n and order m, describe high-order pseudo-Gaussian vortex, intermediate (vortex), hollow (nonvortex), and trigonometric (non-vortex) beams having an arbitrary beam waist w0. A very useful property of these beams is the efficient and fast computational modeling of tightly focused or quasi-collimated wave-fronts depending on the dimensionless waist parameter kw0, where k is the wave number of the acoustical radiation. Examples that illustrate hollow vortex and non-vortex beams are provided, and numerical simulations for the magnitude, isosurface, and phase plots of the pressure wave field of higher-order quasi-Gaussian beams are evaluated with particular emphasis on kw0 for strongly (kw0 = 3) to weakly focused (i.e., quasi-collimated) beams (kw0 = 7). Potential applications are in beam-forming design, imaging, particle sizing and manipulation in acoustical tweezers, and phenomena related to scattering, radiation force, and torque.