Weighted order doubling in the computation of chromatic aberration coefficients

Abstract

The essentially Hamiltonian process of order doubling is incorporated into a Lagrangian method for the determination of the chromatic aberration coefficients of symmetric systems composed of homogeneous lenses and mirrors. In dealing with chromatic aberration coefficients, the prescription for weighted truncation of power series has been found to be of great value. Accordingly, in this paper we are initially concerned with the general principles of order doubling when weighted truncation is employed. These considerations give rise to the idea of weighted order doubling, which follows simply from a condition corresponding to Fermat’s principle. In the context of aberration theory, it is shown that, by using only the knowledge of a general ray path that includes terms up to order m in a chromatic coordinate, say ω, the configuration of this general ray at the initial or final base surface can be determined (by means of a characteristic function) including terms of order 2m + 1 in ω. A process is devised that incorporates these ideas into an efficient scheme for the determination of the Taylor coefficients of characteristic functions. As expected, the explicit inclusion of the chromatic dependence gives methods that are far superior to the corresponding monochromatic procedures.