Abstract
Analyses of beam propagation have generally proceeded either by applying paraxial approximations to rigorous wave formulations or by ray tracing. Relationships between the two approaches have heretofore been established in an almost empirical manner—for example, the correspondence between the caustics of the rays and the variation of the spot size of the beam distribution. A rigorous analytical connection between the ray and wave approaches has not been fully developed. In this paper we describe how a uniform asymptotic description of a scalar two-dimensional beam may be obtained. This description is valid at any distance from the beam axis and for any degree of beam divergence. When the divergence is small and the paraxial region is considered, the description reduces to the usual Gauss–Hermite beam function; furthermore, it is applicable to the deep shadow where the latter is inappropriate. For highly divergent beams, the geometrical-optics rays and field amplitudes are obtained. The constructs of geometrical optics and the beam functions are thereby rigorously related. In addition, the asymptotic description applies to intermediate geometries for which neither of the usual methods is entirely adequate.
© 1971 Optical Society of America
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