Single-beam heterodyne three-degree-of-freedom interferometric measurement method based on sparse sampling using a fiber array

To address the demand for synchronous axial displacement and orientation change measurements in precision manufacturing and high-precision motion stages, this paper proposes a single-beam heterodyne three-degree-of-freedom (3-DOF) interferometric measurement method based on sparse sampling with a fiber array. Here, a four-channel fiber array is used to sparsely sample the single-beam interference spot. Then, axial displacement, pitch angle, and yaw angle are jointly retrieved from multiple heterodyne beat signals. To clarify the signal formation and measurement mechanism, a dual-frequency Gaussian beam interference model and a four-circular-aperture integrated reception model are established. The effects of beam waist radius, fiber-core spacing, and fiber-core diameter on the angular measurement range, interference contrast, and angular noise are analyzed. The beam waist radius mainly affects the effective angular measurement range, the geometric baseline mainly determines the angular noise and resolution, and the fiber-core diameter affects the received signal and phase extraction through the finite-aperture integration effect. Accordingly, an integrated prototype is developed for experimental validation. The angular measurement range reaches 1.55 mrad, the angular resolution is better than 1 μrad, and the displacement resolution reaches 3 nm. The displacement measurement agrees well with that of a commercial interferometer over a travel range of 30 μm. During a long-term vacuum stability test lasting approximately 80 h, both the displacement and angular outputs remained stable. A static noise comparison under different array parameter combinations further verifies the influence of the geometric baseline on angular noise. The proposed method provides a theoretical basis and experimental support for parameter design and integrated implementation of compact single-beam heterodyne 3-DOF interferometric systems.