IN Chapter V we studied the effects of aberrations on the basis of geometrical optics. In that treatment the image was identified with the blurred figure formed by the points of intersection of the geometrical rays with the image plane. Since geometrical optics gives an approximate model valid in the limit of very short wavelengths, it is to be expected that the geometrical theory gradually loses its validity as the aberrations become small. For example, in the limiting case of a perfectly spherical convergent wave issuing from a circular aperture, geometrical optics predicts for the focal plane an infinite intensity at the focus and zero intensity elsewhere, whereas, as has been shown in §8.5.2, the real image consists of a bright central area surrounded by dark and bright rings (the Airy pattern). In the neighbourhood of the focal plane the light distribution has also been seen to be of a much more complicated nature (see Fig. 8.41) than geometrical optics suggests. We are thus led to the study of the effects of aberrations on the basis of diffraction theory.

The first investigations in this field are due to Rayleigh. His main contribution was the formulation of a criterion (discussed in §9.3) which, in an extended form, has come to be widely used for determining the maximum amounts of aberrations that may be tolerated in optical instruments. The subject was carried further by the researches of many writers who investigated the effects of various aberrations, and we may mention, in particular, the more extensive treatments by Steward, Picht, and Born.