Optical systems can benefit strongly from freeform surfaces; however, the choice of the right surface representation is not trivial and many aspects must be considered. In this work, we discuss the general approach classical globally defined representations, as well as the basic mathematics and properties of the most commonly used descriptions and present a new description developed by us for describing freeform surfaces.

Optical systems can benefit strongly from freeform surfaces; however, the choice of the right representation is not trivial, and many aspects must be considered. Many possibilities to formulate the surface equations in detail are available, but the experience with these newer representations is rather limited. Therefore, in this work, the focus is to investigate the performance of several classical descriptions as well as one extended freeform surface description in their performance in concrete design optimization tasks. There are different influencing factors characterizing the surface representations, the basic shape, the boundary function, the symmetry, a projection factor, as well as the deformation term describing higher order contributions. We discuss some possibilities and the consequences of describing and using these options with success. These surface representations were chosen to evaluate their impact on all these aspects in the design process. As criteria to distinguish the various options, the convergence over the polynomial orders, as well as the quality of the final solutions, is considered. As a result, recommendations for the right choice of freeform surface representations for practical issues in the optimization of optical systems can be given under restrictions of the benchmark assumptions.

Freeform surfaces are becoming an increasingly exciting opportunity in optical design, in particular when correcting systems with off-axis geometries. Nevertheless, especially when coming to commercial use, the challenges for manufacturing are difficult to handle. The optical quality of a system is perturbed by typical deformations, such as localized figure errors and regular mid-spatial-frequency ripples, that come from the diamond-turning process. In this proposal, we investigated a workflow for analyzing the impact of real optical surfaces on the optical performance for even complex systems. Based on a simple and robust description, the surface is implemented back into the design. While the more localized deviations are analytically described by radial basis functions, the residual ripple structures are covered by a new approach based on the power spectral density. The reimport of optical surfaces back into the design software allows simple estimations for the requirements on manufacturing and the analysis of the realistic impact on system performance.

Three mirror anastigmats (TMA) are telescopic optical systems with only plane symmetry, that allow for good image quality without any central obscuration. The complexities of manufacturing and alignment can be reduced by fabricating the first mirror and the third mirror in one piece and defining a common axis of all the mirrors. It is attractive to use off-axis used aspheres and to come to an acceptable performance with the smallest number of freeform surfaces. In this paper, different types of freeform surfaces are considered to evaluate their potential. In the performed case study, the correction of spherical aberration and coma is best corrected in the pupil with the second mirror and to select the Zernike representation with remaining x-symmetry is one of the best ways to do this. The use of the Chebyshev polynomials also gives good results. Furthermore it is found, that the first mirror and the third mirror are quite beneficial to be modelled as off-axis aspheres of the Q-type. The result shows that a combination of two Q-aspheres with a Zernike surface at the second mirror is one of the most favorable combinations