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

Exact nonparaxial beams of the scalar Helmholtz equation

Gustavo Rodríguez-Morales and Sabino Chávez-Cerda
Opt. Lett. 29(5) 430-432 (2004)

Scalar field of nonparaxial Gaussian beams

Z. Ulanowski and I. K. Ludlow
Opt. Lett. 25(24) 1792-1794 (2000)

Geometrical representation of the fundamental mode of a Gaussian beam in oblate spheroidal coordinates

B. Tehan Landesman
J. Opt. Soc. Am. A 6(1) 5-17 (1989)

Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams

A. Wünsche
J. Opt. Soc. Am. A 9(5) 765-774 (1992)

Gaussian amplitude functions that are exact solutions to the scalar Helmholtz equation; Geometrical representation of the fundamental mode of a Gaussian beam in oblate spheroidal coordinates: comment

L. B. Felsen
J. Opt. Soc. Am. A 6(10) 1640-1641 (1989)