On Perfect Optical Instruments

The angular characteristic function T is used to investigate optical systems of finite focal length in which one extended surface is imaged on another without aberration. The forms of T found earlier to correspond to perfect imagery are shown to be exhaustive. When perfect imagery is achieved the image surface is similar to the object surface, with the exception that it may be stretched uniformly by arbitrary amounts in each of three mutually perpendicular directions. The length of the optical path between conjugate points of these surfaces is proportional to the distance of either from a fixed plane. In general only one object surface can be imaged without aberration. In an exceptional group two pairs of conjugate surfaces - parts of centred second-order surfaces - can be perfectly imaged. The two surfaces of the same space have coincident principal axes, and in each of these directions the product of the two magnifications is equal to the square of the ratio of the refractive indices of the external media. In general the centre of the object is not perfectly imaged. The length of the path from any point of either object surface to its image is invariable. The only elementary example of this group is a single spherical refracting surface. Another exceptional case is when the conjugate surfaces are plane. It is shown that in addition to being free from all aberrations for a single magnification, a system such as a photographic lens can produce images free from curvature, astigmatism and distortion for all magnifications. The properties found for both these groups contradict some general conclusions reached by Clerk Maxwell. Though the existence of perfect instruments is not inconsistent with optical laws, material systems of reflecting or refracting surfaces, whatever their shapes, do not enable this ultimate standard to be fully realized. This shortcoming bears significantly on the methods used in designing optical instruments.