Image formation by a general optical system 2: Computing methods

Angle and height variables, akin to the paraxial variables for the marginal and pupil (principal) rays in an axially symmetric system, are introduced. These give simple forms of opening and closing formulas for ray tracing, which are always determinate and of good accuracy. Preliminary ray tracing establishes the pupil periphery and base ray for any given object point, and thence the principal azimuths around the base ray are determined. For cases where the object and/or image surfaces are curved, formulas are given for the necessary coordinate transformations from the base systems used for ray tracing to the local and canonical coordinate systems used for aberration and image assessment calculations. Generalized numerical apertures are used for the two principal azimuths for any object image point, formulas for which, and also for the two corresponding principal magnifications, are given.