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Gaussian amplitude functions that are exact solutions to the scalar Helmholtz equation

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Abstract

A new family of exact solutions of 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 and can easily by transformed into 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 an amplitude consisting of a finite number of higher-order terms with a zero-order term that is Gaussian.

© 1988 Optical Society of America

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