Fractional-Fourier-transform-based Michelson interferometric measurement

In this paper, we propose an automated, high-precision analytical method based on the fractional Fourier transform (FRFT) for Michelson interferometric fringes. Unlike traditional techniques that depend on manual fringe counting or fringe-spacing analysis, the proposed approach overcomes the limitations of human enumeration and distance measurement, enabling direct retrieval of key physical parameters—such as the refractive index and thickness of the test glass—from the interference pattern. Starting from the imaging principle of Michelson interferometry, we develop a mathematical model of the fringe phase and reveal that it corresponds to a class of linear-chirp, non-stationary signals whose instantaneous frequency (chirp rate) is directly determined by the glass’s refractive index and thickness. By analyzing the chirp characteristics, we derive an explicit analytical relationship between the chirp rate and the FRFT rotation angle α. When α matches the signal’s chirp rate, energy is focused in the fractional domain, producing a sharp, easily identifiable peak in the amplitude spectrum. Exploiting this focusing effect via a search over α allows precise estimation of the chirp rate—and thus accurate determination of refractive index or thickness. Simulation and experimental results validate the feasibility and accuracy of the proposed method.