Abstract

A new family of exact closed-form solutions to the scalar Helmholtz equation is presented. The 0,0 order of this family represents a new mathematical model for the fundamental mode of a propagating Gaussian beam. The family consists of nonseparable functions in the oblate spheroidal coordinate system that are separable in a complex coordinate system and can easily be transformed to a different set of solutions in the prolate spheroidal coordinate system where the 0,0 order is a spherical wave. This transformation consists of two substitutions in the coordinate system parameters and represents a more general method of obtaining a Gaussian beam from a spherical wave than assuming a complex point source on-axis. Finally, each higher-order member of the family of solutions possesses a Gaussian amplitude along with a finite number of higher-order terms, all of which vary on propagation.

© 1988 Optical Society of America