Performance comparison of polynomial representations for optimizing optical freeform systems

Optical systems can benefit strongly from freeform surfaces, however the choice of the right representation isn` t an easy one. Classical representations like X-Y-polynomials, as well as Zernike-polynomials are often used for such systems, but should have some disadvantage regarding their orthogonality, resulting in worse convergence and reduced quality in final results compared to newer representations like the Q-polynomials by Forbes. Additionally the supported aperture is a circle, which can be a huge drawback in case of optical systems with rectangular aperture. In this case other representations like Chebyshev-or Legendre-polynomials come into focus. There are a larger number of possibilities; however the experience with these newer representations is rather limited. Therefore in this work the focus is on investigating the performance of four widely used representations in optimizing two ambitious systems with very different properties: Three-Mirror-Anastigmat and an anamorphic System. The chosen surface descriptions offer support for circular or rectangular aperture, as well as different grades of departure from rotational symmetry. The basic shapes are for example a conic or best-fit-sphere and the polynomial set is non-, spatial or slope-orthogonal. These surface representations were chosen to evaluate the impact of these aspects on the performance optimization of the two example systems. Freeform descriptions investigated here were XY-polynomials, Zernike in Fringe representation, Q-polynomials by Forbes, as well as 2-dimensional Chebyshev-polynomials. As a result recommendations for the right choice of freeform surface representations for practical issues in the optimization of optical systems can be given.