Objective Pinhole point-diffraction interferometry (PDI) is a core technique for nanometre-scale displacement measurement. This system utilizes pinhole diffraction to generate a high-precision spherical reference wave, which propagates coaxially with the light beam reflected from the test plane. The interference of these waves at the detector produces stable concentric fringes, establishing a highly reliable and self-referenced metrological benchmark. This principle allows the PDI system to facilitate accurate, controllable, and reproducible performance evaluations of nanometre-positioning stages and precision motion mechanisms, with applications in ultra-precision machining, chip manufacturing, and metrological traceability. Nevertheless, the ongoing trend toward higher integration and miniaturization in precision equipment has exacerbated challenges like temperature drift and environmental vibration. To eliminate the dependency on piezoelectric ceramic (PZT) phase shifters—a source of temperature drift and timing synchronization problems in multi-frame techniques—this study introduces a two-frame image-processing method that requires no phase-shifting, based on the analysis of pinhole spherical-wave interference fringes, to achieve high-precision nanometre displacement measurement.

Methods This study develops a displacement measurement system utilizing pinhole point-diffraction interferometry (PDI). The system employs a polarization-maintaining fiber-collimated laser, which is focused by a microscope objective onto a diffraction mask to produce a high-quality spherical reference wave. This wave interferes with the light reflected from the test plane, forming stable concentric fringes on the image sensor (Fig.3). The subsequent processing pipeline (Fig.5) performs automatic multi-ring detection in the spatial domain, precise center localization, and radial profiling. The procedure then conducts outlier rejection under a normality assumption and employs statistical fusion across the rings to estimate the normal displacement. This phase-shift-free methodology improves both measurement efficiency and system robustness.

Results and Discussions The pinhole point diffraction interferometry displacement measurement system established in this paper employs two frames of phase-shift-free spatial domain multi-ring fringe processing chains for calibration and testing. Shapiro–Wilk normality tests are performed on each set of experimental data, yielding W statistics greater than 0.85 and p-values greater than 0.05. Therefore, subsequent analyses were based on the assumption of normal distribution. Outliers were automatically removed using a criterion, and the remaining valid displacement results were averaged using a mean fusion method to obtain the final displacement results. As shown in Fig.11, Fig.11(a) displays the displacement measurement curves from three independent experiments, with each set of data points showing high consistency with the theoretical displacement values. To further evaluate the system's linear response, linear fitting is performed on the experimental data and theoretical values for the three groups, as shown in Fig.11(b), (c), and (d).The coefficients of determination (R²) from the linear regression analysis were 0.9995, 0.9993, and 0.9981, respectively, all significantly greater than 0.99, indicating that the experimental method exhibits excellent linearity and reliability. The displacement error of this system mainly stems from the measurement error of the radius of curvature (R), the resolution of the nano-positioning stage, and the hysteresis effect of the piezoelectric material. Through the uncertainty analysis of the three types of errors, it is obtained that the final combined uncertainty is 2.94 nm. The displacement error of this system mainly stems from the measurement error of the radius of curvature (R), the resolution of the nano-positioning stage, and the hysteresis effect of the piezoelectric material. Table1 presents the average displacement measurements and corresponding absolute errors at each step point for the three experimental groups. The results show that within the 400 nm displacement range, the maximum absolute error at each step point is 5.8 nm, further validating the high measurement accuracy and reliability of this experimental method.

Conclusions This study introduces a phase-shift-free, two-frame method for nanometre-scale displacement measurement based on pinhole spherical-wave interference fringes. In this approach, a spherical reference wave generated by pinhole diffraction interferes with the light reflected from the test plane, producing concentric interference fringes for out-of-plane sensing. The computational pipeline involves: 1) Automatic detection of multiple fringe rings and sub-pixel center localization in the image domain; 2) Extraction of each ring's radius via local extremum detection; 3) Outlier removal guided by the Shapiro–Wilk normality test; and 4) Statistical fusion of the remaining ring-specific measurements to compute the final displacement. The system's displacement error originates mainly from three factors: the measurement error of the radius of curvature (R), the resolution limit of the nano-positioning stage, and the hysteresis effect of the piezoelectric material. An uncertainty analysis integrating these three error types results in a combined uncertainty of 2.94 nm. By eliminating the need for phase-shifting, this method enhances measurement efficiency and system robustness, providing a high-precision solution for small-range displacement measurement in applications like ultra-precision machining and chip manufacturing.