Modeling and decoupling phase-shift-induced wavefront distortion in high NA spherical Fizeau interferometry via aberration-related iterative algorithm

In high numerical-aperture (NA) spherical Fizeau interferometry, axial mechanical phase shifting leads to a spatially nonuniform phase-shift distribution due to variations in the incidence angle across the pupil. This nonuniformity violates the uniform phase-shift assumption of conventional phase-shifting interferometry (PSI) and introduces systematic wavefront distortion, predominantly in the form of defocus and induced spherical aberrations. Furthermore, multiple reflections within the Fizeau cavity introduce high-order harmonics, which couple with the phase-shift errors and further degrade measurement accuracy. To address these issues, we propose the aberration-related iterative algorithm (ARIA). A model of the phase-shift field is established based on the geometrical optics of the interferometer cavity and parameterized using a low-dimensional set of Zernike polynomials. This physical constraint is integrated into a two-level iterative framework, where the surface figure and the second-harmonic components are rigorously decoupled at the pixel layer using an extended linear least-squares approach. Simultaneously, the spatially varying phase-shift parameters are updated at the frame layer, ensuring numerical stability and fast convergence. Experimentally, a reciprocal phase-shifting scheme is introduced to construct a reference with substantially reduced errors and to validate the algorithm using both forward and reverse phase-shifting datasets. Compared with conventional PSI, ARIA reduces the reconstructed RMS from 17.93 nm to 4.20 nm for the forward dataset and to 3.98 nm for the reverse dataset, with residual RMS values of about 1.00 nm and 1.25 nm, respectively, relative to the reciprocal-average reference. These results demonstrate that ARIA effectively suppresses both phase-shift-induced wavefront distortion and harmonic-induced artifacts, enabling high-precision measurements without hardware modification.