Modeling and compensation of periodic nonlinear error in micro-displacement measurement using a reflective-sphere-based low-finesse Fabry-Pérot cavity.

Abstract

Low-finesse Fabry-Pérot (F-P) cavities are attractive for high-precision micro-displacement measurement (MDM), but residual multiple reflections introduce periodic nonlinear error during quadrature demodulation. In this work, a reflective-sphere-based low-finesse F-P cavity and a model-based periodic nonlinear error compensation method are proposed. The reflective sphere is designed to provide a low effective return reflectivity and an approximately cosine-shaped interference response. Spectral measurements at working distances of 4-12 mm yield an average finesse of 0.492, close to the theoretical value of 0.5, and the measured free spectral range (FSR) values agree well with theoretical predictions. The compensation model parameters are identified from the sector-averaged radial distribution of the measured Lissajous trajectories. Under the parameter-identification condition of an initial cavity length of 2.0 cm, a nominal 4 µm peak-to-peak displacement, and a motion frequency of 3 Hz, the absolute displacement error is reduced from 150 to 30 nm after compensation. The same identified parameter set is directly applied without refitting to independent measurements and remains effective over the experimentally tested displacement-amplitude range of 2-6 µm and motion-frequency range of 3-9 Hz when the optical configuration is unchanged. A deliberate change in the initial cavity length requires parameter reidentification.