Encircled energy in systems with truncated-Gaussian apertures of different Fresnel numbers. I. Generalization of the Q2n function of E. Wolf and the Yn function of H. H. Hopkins.
In his works on diffraction [Proc. R. Soc. London Ser. A204, 533 (1951)], E. Wolf introduced the Q2n function, which enters his expressions for the encircled energy. This quantity specifies the fraction of the total energy within various rings in receiving planes parallel to the geometrical focal plane. In addition to the Q2n function, another special function, called the Yn function, was used in his formulation, which had been introduced by H. H. Hopkins [Proc. Phys. Soc. London Sect. B62, 22 (1949)]. The purpose of this study is to generalize both the Q2n and Yn functions for evaluating the encircled energy in systems of focused truncated Gaussian beams by apertures of different Fresnel numbers and different levels of beam truncation. The generalized Q and Y functions are functions of more than one variable and are applicable to all nonnegative integers m; they may therefore be called the Qm and the Ym functions. Computed results are shown graphically in the form of contour lines of the encircled energy. Part II of this study [J. Opt. Soc. Am. A24, 2033 (2007)] contains an analysis of maximizing beam energy concentration on a target.
Li
Yajun
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