Order doubling in the computation of aberration coefficients

In the context of ray tracing, the extremal nature of optical path lengths has largely been ignored. However, by using this property is it possible to derive the following: If a general ray, specified by its initial position and direction (which are considered as variables), is traced through a system of homogeneous media so that the coordinates of the intercepts with the refracting surfaces are known as power series accurate to a certain order in the initial variables, then the characteristic function of the system (and hence all the geometrical optical information) may be directly determined accurately to twice this order. Subsequently, a routine is devised that allows the power series of the characteristic function of a specified optical system (composed of homogeneous lenses and mirrors) to be computed to arbitrary orders. This routine requires significantly less computing time than the existing programs for analytic (as opposed to numerical) analysis of optical systems.