Reconstruction of freeform optical surfaces based on radial Zernike expansion and ray-traced fringes

Abstract

Freeform optical surfaces are essential for advanced optical systems due to their design flexibility, but accurate reconstruction is challenging because of geometric complexity and inflection regions. This work numerically models a circular freeform surface with a 111 mm diameter, a peak-to-valley (PV) sag of 3.5 mm, an RMS flatness error of 76 nm, and an RMS roughness of 0.5 nm. The surface is represented using XY polynomial functions, with sag data extracted from 180 diametrical profiles via a Profile Rotation Model (PRM) and fitted with eighth-order polynomials. A ray-tracing algorithm simulates one-dimensional fringe intensity distributions at a terahertz wavelength of 0.284 mm. Reconstruction employs a custom Zernike-based fitting routine with 48 terms, enhanced by fringe-thinning, and a radial Zernike strategy that mitigates ambiguities by independently reconstructing azimuthally replicated radial profiles. Accuracy is assessed using shading error analysis and the area between reconstructed and simulated profiles. Results from eight representative profiles show excellent agreement, with an average relative error of 2.29% of the maximum surface height. For larger surfaces, increased sag and slope reduce ray-detection sensitivity, limiting effective measurement to ~174 mm for geometry.