Novel freeform optical design methods can be classified in two categories, relevant to the generation of a starting point or to the development of new optimization tools. In this paper, we design a freeform TMA optimized with different surface representations using either a differential ray-tracer as a new optimization tool, or commercial ray tracer (ANSYS-ZEMAX OpticStudio) for comparison. For differential ray tracing, we use FORMIDABLE (Freeform Optics Raytracer with Manufacturable Imaging Design cApaBiLitiEs): an optical design library with differential ray tracing and NURBS optimization capabilities under European Software Community License (ESCL). NURBS allow to represent a freeform surface without needing any prior knowledge on the surface, such as the polynomial degree in polynomial descriptions. However, commercial optical design software are designed to optimize polynomial surfaces, but are not well-suited to optimize NURBS surfaces. This requires to use a custom optical design library. To demonstrate the interest of using NURBS representation, freeform telescopes are independently optimized with NURBS using FORMIDABLE, or XY polynomials with OpticStudio over different iteration cycles. We then compare the resulting systems using their RMS Field Maps to assess the optimization quality of each surface representation. We also provide a full-system comparison, including mirror freeform departures. This study shows that NURBS can be a relevant alternative to XY polynomials for freeform optimization of reflective 3-mirror telescopes.
