Introduction

It has been stressed in previous chapters that perfection at more than one magnification is impossible. Even so, lenses are often used for a variety of object and image distances. Camera lenses as well as enlarger lenses need to form sharp images over a wide range of conjugates. Even high power microscope objectives, notoriously sensitive to variations in object distance, are occasionally pressed into use for three-dimensional imaging. How can we deal with this paradox?

The explanation is that images need not be perfect. All we need is images that are sharp enough to utilize fully the finite resolution of the recording medium. Photographic film is limited by the size of the grain; CCDs are limited by the finite gate size; the retina of the eye is limited by the size of the rods and cones; etc.

For an analysis of incompatible lens requirements it is convenient to describe a lens by one of its eikonal functions. This provides all the information needed to calculate its aberrations at any magnification. The calculations are straightforward, at least in principle: choose a set of rays originating in a specified object point, determine their continuation in the image space, and see where they intersect the image plane.

Unfortunately these calculations can hardly ever be carried out in closed form. The central problem is to calculate x′, y′, L′, and M′ when x, y, L, and M are specified.