From Spherical Harmonics to Gaussian Beampatterns

The use of multipoles, otherwise called spherical wavefunctions, has been explored for acoustic fields that can be omnidirectional, for example, in scattering theory. Less developed is the use of spherical harmonic multipoles for the construction of directed beams, such as the Gaussian unfocused beampattern, which is an important reference beam in many practical applications. We develop the straightforward construction of a Gaussian unfocused beam using the special properties of the sum of spherical harmonics; these include the use of an imaginary offset in directing the forward propagation to the desired beampattern. Examples are given for narrowband and broadband pulse propagation in the ultrasound MHz range, with comparisons against a classical acoustics formulation of the Gaussian beam. The use of spherical harmonics forms an alternative framework for devising beampatterns, with apodization and concentration issues of the beam linked to an array of a limited number of discrete multipoles at the source.