Radial Zernike-based reconstruction of freeform surfaces from ray-traced fringe simulations

Freeform surfaces are increasingly used in high-performance optical systems, but accurate reconstruction is challenging due to complex geometries and inflection points. Here, a circular freeform surface (111 mm diameter, 3.5 mm PV difference, 76 nm RMS flatness, 0.5 nm RMS roughness) is modeled using XY-type polynomials. Sag data from 180 profiles are extracted via a Profile Rotation Model and fitted with eighth-degree polynomials. A custom ray-tracing algorithm generates one-dimensional intensity maps at a 0.284 mm wavelength, and surface reconstruction is performed using a 48-term Zernike polynomial fitting algorithm enhanced by fringe thinning. To resolve ambiguities from diametrical profiles, a radial Zernike profile approach extracts radial intensity profiles, azimuthally replicates each profile, and reconstructs each independently. Reconstruction of eight radial profiles yields an average relative profile error of 2.29%, while a 25.4 mm experimental partial freeform surface is reconstructed with 1.9% relative error. For large surfaces with steep slopes, the ray-tracing algorithm's sensitivity limits the effective measurable diameter to approximately 174 mm.