Phase Distribution near Focus in an Aberration-free Diffraction Image

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A satisfactory picture of the three-dimensional intensity distribution near focus in the aberration-free diffraction image of a monochromatic point object was first given by Zernike and Nijboer in 1949. No corresponding general picture of the phase relations between different parts of the image seems to have been worked out, although Gouy's discovery in 1890 of the so-called phase anomaly at focus aroused an interest in this topic which, as a succession of publications shows, has not decreased in the intervening 66 years. In the present paper a sharpened version of Lommel's classical analysis is applied to obtain a general picture of the phase distribution near focus and to examine in detail, in the special case of an F/3.5 pencil, its peculiarities near the geometrical focal point, the Airy dark rings and the axial nodes (points of zero intensity) of the diffraction image. The `phase anomaly' near focus and the singular behaviour of the phase along the axis of the pencil become more readily intelligible when they are considered against the background of this general picture.