For optimization and for image analysis of optical systems it is necessary to calculate derivatives of aberrations. The main task is to calculate a derivative matrix. This has to be done repeatedly during optimization. The conventional finite difference method suffers form two major drawbacks, i.e. inefficiency and inaccuracy. Since some years a new elegant differential technique for analytical calculation of the derivative matrices was developed here first by the senior author and later competed by co-workers. We represent the central parts of the differential technique in this paper, and the concrete application possibilities will not be given in details. A comparison of differential derivative and the conventional finite different method is made at last. The analytical formulas have been derived for the following types of variables: surface radius, axial distance, refractive indices and aspherical coefficients. With the help of an optimally organized program structure we realized an efficient tool for optimization and system analysis.
"Differential ray tracing for optical design", Proc. SPIE 3737, Design and Engineering of Optical Systems II, (27 August 1999); doi: 10.1117/12.360003; https://doi.org/10.1117/12.360003