Spatiotemporally decoupled iterative algorithm using multi-order Taylor expansion for random phase-shifting interferometry with finite bandwidth illumination

Phase-shifting interferometry with finite bandwidth light sources is a key technique for high-precision measurement of ultra-smooth optical components. However, influenced by vibrations, moving devices, and spectral bandwidth, the interference fringes exhibit not only random phase-shifting errors but also intra- and inter-frame nonlinear and non-uniform variations in modulation amplitude, which degrades the phase retrieval accuracy of existing random phase-shifting algorithms. To solve this problem, we propose a spatiotemporally decoupled iterative algorithm (SDIA). This method decouples the spatiotemporally varying modulation amplitude via multi-order Taylor expansion, stepwise separating the time-dependent and space-dependent variables in two distinct iterative steps to construct linear equations without variable coupling, thereby achieving high-precision phase retrieval via the least-squares alternating iteration. Simulations and experiments validated the feasibility and high accuracy of the proposed algorithm. The experimental results demonstrate that, for randomly phase-shifted interferograms under 40 nm bandwidth illumination, the retrieval residual errors of a planar smooth mirror are below 1 nm (PV) and 0.1 nm (RMS), respectively. To the best of our knowledge, this is the first time that such a spatiotemporal decoupling iterative algorithm using multi-order Taylor expansion has been proposed, which can address both random phase-shifting errors and spatiotemporally coupled modulation variations under finite bandwidth illumination conditions.