A diffraction beam field expressed as the superposition of Gaussian beams

TLDR

The diffraction field of a Gaussian planar velocity distribution behaves as a Gaussian beam when (ka)²≫1, making Gaussian functions attractive as a base function set. The study proposes a method to represent any axisymmetric beam field as a superposition of Gaussian beams with varying waists and positions along the same axis. The method uses computer optimization to determine coefficients, waists, and positions, and is tested on a piston radiator. The Gaussian beam superposition accurately reproduces the beam field across far, transition, and near regions, with discrepancies only in the extreme nearfield; smoother velocity distributions require fewer terms, and a Gaussian radiator needs only one term, simplifying analysis of any axisymmetric beam field.

Abstract

The diffraction field of a Gaussian planar velocity distribution is a Gaussian beam function under the condition (ka)2≫1. This property makes a series of Gaussian functions attractive as a possible base function set. The new approach presented enables one to express any axisymmetric beam field in a simple analytical form—the superposition of Gaussian beams about the same axis but with beam waists of different sizes located at different positions along the axis. A computer optimization is used to evaluate the coefficients, as well as the beam waists and their positions. The extreme case of a piston radiator is used to test the approach. Good agreement between a ten-term Gaussian beam solution and the results of numerical integration (or analytical solution on axis) is obtained throughout the beam field: in the farfield, the transition region, and the nearfield. Discrepancies exist only in the extreme nearfield (<0.1 times the Fresnel distance). For surface velocity distributions that are less discontinuous (smoother), the number of terms in the Gaussian beam solution is reduced. In the extreme case of a Gaussian radiator, only one term is needed. The approach, then, reduces the study of any axisymmetric beam field to the study of the much simpler Gaussian beam.