Equations for designing a dialyte with minimal TSA based on the caustic surfaces

A method to design a dialyte based on the caustic surfaces is presented, with correction at the third-order spherical aberration. We predefined, a priori, the value for the back focal length of the separated doublet for two different wavelengths to calculate their radii of curvature, which depend on all the parameters involved in the refraction process. Additionally, we approached the caustic surface by applying a Taylor series, which provided the analytical spherical aberration coefficient at the third order. Two alternatives to design achromatic separated doublets are provided. Finally, we extended the method to design achromatic cemented triplets.