Spherical aberration and the information content of optical images

Abstract

The main purpose of the paper is to investigate the effect of spherical aberration on the information content of low-contrast photographic images with specified spread and noise characteristics. In section 1 an outline is given of the basic ideas and the relevant formalism. In section 2 the response functions of monochromats with spherical aberration and defocusing are considered, and their computed values in selected special cases are displayed in figures 3 to 5. A digression is made in section 3 in order to discuss, with the help of these values, a point of topical interest, namely, the variation of best focus with the line frequency of a sinusoidal test object. In section 4 the notion of the equivalent receiving surface of a low-contrast photographic process is introduced and two equivalent receiving surfaces are defined, with the help of some experimental results of Higgins & Jones (1952), which correspond to model photographic processes used with low-contrast objects. In section 5 the mean information density in the images of an aberration-free monochromat is calculated for two model emulsions, at selected noise levels, over a range of focal settings. In both models, the correctly focused images of a random object set are found to contain about one bit per Airy disk when the signal-to-noise ratio is 100 and the effect of defocusing are similar in the two cases. The effect of spherical aberration on information density at different focal settings is then examined in the second model. It appears that, for amounts of fourth-power aberration up to two fringes, the informationally best focus is approximately midway between paraxial and marginal foci, and that the acceptance of a 20% drop in information content corresponds to a focal tolerance of approximately $\pm {\textstyle\frac{1}{2}}$ fringe.