Universal approach for local higher-order wavefront tracing equations for complex optical systems

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 β=(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.