Landgrave and Berriel-Valdos presented axial and radial sampling expansions for three-dimensional light amplitude distribution around the Gaussian focal point. [J. Opt. Soc. Am. A 14, 2962 (1997)]. The expansions were obtained under the assumption that the pupil function was rotationally symmetric. We present a new derivation of the axial expansion that does not make use of arbitrary formal assumptions used by Landgrave and Berriel-Valdos and eliminates some faults of the derivation given by Arsenault and Boivin, who published this expansion in 1967 [J. Appl. Phys. 38, 3988 (1967)]. We also discuss generalizations of the axial expansion to the case of pupils that exhibit no symmetry with respect to the axis considered.
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