Inverse problems for reflector systems

A comprehensive solution to the three-dimensional inverse problem of transforming wavefronts of arbitrary configuration after two reflections within the geometrical optics (GO) approximation is presented. We establish necessary and sufficient conditions that determine whether a given wavefront mapping can be realized using two mirrors, either prior to or without explicitly computing the mirror surfaces. The cases of spherical and plane wavefronts are analyzed separately, where we demonstrate that the governing equations simplify significantly, reducing to a single equation. Additionally, two-dimensional variants of the problem are considered, and it is shown that, in these cases, the equations for determining the mirror profiles can be solved in quadrature.