A diffraction beam field expressed as the superposition of Gaussian beams

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.