Aperture and far-field distributions expressed by the Debye integral representation of focused fields

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Abstract
We study the anomalous asymptotic behavior of the Debye integral far from focus that occurs in the vicinities of the axis of the focusing system and the boundary of the geometrical-optics shadow. The first terms in the asymptotic power series of the far field valid on the axis, on the shadow boundary, in the shadow, and in the geometrical illuminated region off axis are obtained to show how they change discontinuously as the field point passes from one region to another. We obtain the second-order term in the asymptotic power series valid in the last-named region to show how it grows without limit as the field point approaches the axis or the shadow boundary. We then derive an approximation valid far from focus that remains continuous as the field point approaches the axis and the shadow boundary. This approximation agrees with the asymptotic power-series results where they are valid. The continuous approximation is applied to determine the sizes of the regions where the field does not approximate the geometrical-optics field.