We present a new formalism, based on Richards–Wolf theory, to rigorously model nonparaxial focusing of radially polarized electromagnetic beams with freeform wavefront. The beams can be expressed in terms of Zernike polynomials. Our approach is validated by comparing known results obtained by Richards–Wolf theory. Our integral representation is compliant with diffraction theory, is thoroughly discussed and solved for various freeform wavefront that, so far, have not been treated analytically. The extension of the method to other polarization states is straightforward.

The exact partial differential equation to design aplanatic freeform-mirror-based optical systems is presented. The partial differential equation is not limited by the number of freeform surfaces or their orientations. The solutions of this partial differential equation can be useful as initial setups that can be optimized to meet higher criteria. One of these solutions is tested as an example of the initial setup, and the results are as expected by the theory.

A method to design catadioptric systems from scratch based on optimizing an element of the power set of stigmatic catadioptric systems is presented. This set contains all possible stigmatic catadioptric systems. The deduction of the set is also presented in this paper, and its derivation is totally analytical. Additionally, an illustrative example of optimization of an element of the mentioned set is presented. The results are as expected.

The set of all possible stigmatic systems made by mirrors is presented. The derivation of the set is analytical, and it is based on the Fermat principle. The properties of the set are properties that all possible stigmatic systems made by mirrors share. The set is tested here with a practical example of optical design, and the results are as expected by theory. This example works with a large field of view rather than a single field, and the volume of the example is several times less than similar systems reported in the literature.

This paper presents a novel method, to the best of our knowledge, to design three-freeform-mirror systems from scratch. The technique consists of getting an initial setup, before optimization, which is directly obtained from the set of all possible stigmatic three-freeform-mirror systems. Then, deformation coefficients are added to each surface and optimized to reduce aberration produced by additional fields. The method has been tested and the results are as expected.