RESUMEN
Analytical wave tracing, including higher-order aberrations, is the state of the art, but at present it is only provided for single propagation or refraction. A complex optical system being a sequence of many refractive surfaces and gaps in between can only be treated by successive application of elementary wave-tracing steps, which is time-consuming for a large number of surfaces or even impossible if a system specification by surfaces is not available. Provided the ray transfer properties of a system are summarized as a nonlinear function f whose Jacobian is the ABCD matrix of the system, by multiple derivative we obtain wave-tracing equations for the wavefront's local derivatives of any desired order. The outgoing wavefront derivative of any order can be written as a sum of multinomials of derivatives of the incoming wavefront, weighted by system-dependent coefficients and by powers of the factor ß=(A-BE2)-1, where E2 is the second-order aberration of the incoming wavefront. Compared to stepwise wave tracing, this approach is extremely efficient when tracing many different wavefronts through one fixed optical system.
RESUMEN
From the literature the calculation of power and astigmatism of a local wavefront after refraction at a given surface is known from the vergence and Coddington equations. For higher-order aberrations (HOAs) equivalent analytical equations do not exist. Since HOAs play an increasingly important role in many fields of optics, e.g., ophthalmic optics, it is the purpose of this study to extend the "generalized Coddington equation" to the case of HOA (e.g., coma and spherical aberration). This is done by local power series expansions. In summary, with the results presented here, it is now possible to calculate analytically the local HOA of an outgoing wavefront directly from the aberrations of the incoming wavefront and the refractive surface.