Light distribution near focus in an error-free diffraction image

Expressions are derived for the fraction of the total illumination present within certain regions in receiving planes near focus of spherical waves issuing from a circular aperture. The derivation is based on Lommel's treatment of Fresnel diffraction, and the solution takes the form of rapidly convergent series involving Bessel functions. Numerical results are illustrated by contour lines. The distribution of the illumination in a number of selected planes near focus is studied in greater detail. A comparison with the predictions of geometrical optics is also made. As a special case the fraction of the total illumination present in the geometrical shadow is discussed. The results provide a mathematical basis for a discussion of the imaging properties in optical system where, as for example in a well-corrected refracting telescope objective, the chromatic variation of focus is the only appreciable aberration.