Encircled energy for systems with truncated-Gaussian apertures of different Fresnel numbers. II. Maximizing beam energy concentration on a target.

In Part I of this study [J. Opt. Soc. Am. A24, 2023 (2007)] the Q(2n) functions of E. Wolf and the Y(n) functions of H. H. Hopkins have been generalized for evaluating the fraction of the total energy in systems with focused truncated Gaussian beams by apertures of different Fresnel numbers and different ratios of aperture radius to beam radius. The generalized special functions provide a mathematical basis for a rigorous study of maximizing beam energy concentration on a target. This subject is addressed under two subtitles: (1) active focusing of a Gaussian beam onto a distant target and (2) optimizing photodetection in a focused field.