Simulation of ray-traced fringes and Zernike-based reconstruction of elliptical optical surfaces

This study presents a comprehensive modeling and reconstruction framework for high-precision elliptical mirrors using simulated fringe patterns and Zernike polynomial fitting. A synthetic elliptical surface is generated, featuring a dome-shaped curvature with a radius of -1300 mm, major and minor axes of 229 and 124 mm, respectively, and sub-wavelength flatness and roughness characteristics. The surface is divided into 180 radial profiles using the non-interpolated profile rotation model (N-IPRM), which preserves native spatial sampling. To address discontinuities inherent in radial reconstruction, the interpolated profile rotation model (IPRM) applies shape-preserving interpolation to enable smooth surface recovery. Sag and slope equations derived from the interpolated profiles are used in a custom ray-tracing algorithm to simulate interference fringes at both visible (0.0006328 mm) and millimeter-wave (5.052296 mm) wavelengths. The algorithm incorporates planar, spherical, and oppositely curved spherical reference surfaces relative to the elliptical surface. The simulated fringe patterns undergo thinning and fringe order estimation, followed by surface reconstruction using a custom Zernike polynomial fitting algorithm implemented in MATLAB. Due to the challenges posed by incomplete circular fringes in Zernike-based reconstruction of elliptical surfaces, we propose a novel approach, to our knowledge: transforming the elliptical surface into an equivalent spherical surface, performing the reconstruction, and then converting it back to the original elliptical form. The reconstructed surface shows excellent agreement with the model, achieving an absolute error of 0.0161 mm and a relative error of 0.32% along the major axis at the longer wavelength. Symmetry deviations of approximately 0.5% further validate the reconstruction accuracy. This methodology provides a robust and versatile toolset for the simulation, analysis, and precise reconstruction of elliptical optical surfaces, with significant implications for optical fabrication and metrology.