Generalized propagation of light through optical systems I Mathematical basics

The propagation of an input field through tilted and curved surfaces is presented and applied for a field tracing routine. This routine employs a ray tracing method for optimal coordinate customization, which is done due to the strong linear phase terms that arise because of tilted coordinates on one hand and the possibility of an application of fast numerical routines on the other. Several methods are proposed on how to optimize the field sampling during the propagation process, as for the astigmatic phase front problem and the curvature of the surface itself. Two solution methods are given to propagate from the curved surface through the homogeneous space to another two-dimensional submanifold. The approach is fully vectorial, Maxwell exact excluding only surface curvature terms for the Fresnel coefficients. A simple numerical propagation example is given.