Calculation of the Aberrations and Image Assessment for a General Optical System

Formulae are developed which apply equally to any image-forming system: these may include, for example, crossed cylindrical lenses, eccentric portions of non-spherical surfaces, refracting prisms, gradient-index lenses, holographic images and diffraction gratings. Given, in any particular case, the necessary ray-tracing formulae for the optical elements, the methods and formulae described are always of good accuracy and never involve indeterminacies. Such indeterminacies usually arise when the object or entrance pupil is at infinity, or when the image or exit pupil is at infinity. This is achieved by the use of auxiliary variables, akin to paraxial variables, which are introduced for the object and image spaces. Formulae for the calculation of the exit pupil coordinates, the wavefront aberration, the chromatic aberration, and for the components of the reduced transverse aberration are given, which never need to be modified to meet any special case. In many systems, for example crossed cylinders with an aperture stop between them, there is no unique position for either the entrance or the exit pupil. All such cases are, nevertheless, included in the same formulae. Opening formulae for tracing rays through such systems are given, and these are again always of good accuracy and never need modifications to meet special cases. These formulae are useful both for exploring the domain of the effective pupil, and for all subsequent ray-tracing. In conclusion, a summary of image assessment techniques is given, together with an account of how the point spread function, the line spread function, and the optical transfer function, may be calculated for a general optical system. The methods and formulae described are well suited to form modules of a computer program of very wide applicability.