The Spherical Harmonic Family of Beampatterns

Abstract

The free space solution to the wave equation in spherical coordinates is well known as a separable product of functions. Re-examination of these functions, particularly the sums of spherical Bessel and harmonic functions, reveals behaviors which can produce a range of useful beampatterns from radially symmetric sources. These functions can be modified by several key parameters which can be adjusted to produce a wide-ranging family of beampatterns, from the axicon Bessel beam to a variety of unique axial and lateral forms. We demonstrate that several special properties of the simple sum over integer orders of spherical Bessel functions, and then the sum of their product with spherical harmonic functions specifying the free space solution, lead to a family of useful beampatterns and a unique framework for designing them. Examples from a simulation of a pure tone 5 MHz ultrasound configuration demonstrate strong central axis concentration, and the ability to modulate or localize the axial intensity with simple adjustment of the integer orders and other key parameters related to the weights and arguments of the spherical Bessel functions.