Results and Discussions Ten repeated experiments are carried out to validate the subnano-scale measurement method. The piezoelectric ceramics nanopositioner is used for unidirectional scanning at an interval of 50 nm for 10 scans. The mean of the displacement measurements for the 10 sets of data is 50.0254 nm, with a standard deviation of 0.114 nm, which demonstrates that the proposed method can achieve subnano-scale precision displacement measurement. Moreover, 40 validation experiments are performed for 10 μm and 30 μm displacements separately to validate the method over a large measurement range. For the 10 μm displacement, the results display a mean of 10.0149 μm, with an average relative error of 1.577% and a standard deviation of 0.2040 μm. In the case of the 30 μm displacement, the mean is 30.0057 μm, with an average relative error of 0.368% and a standard deviation of 0.1453 μm. It is proved that the proposed method can achieve precision displacement measurement over a large range. Simulations are carried out to validate the aberration correction scheme based on the differential evolution algorithm. For a standard displacement of 30 nm, the displacement measurements before and after correction are 32.033 nm and 30.332 nm, with relative errors of 6.78% and 1.11%, respectively. The results demonstrate the effectiveness of the correction method.

Accurate precision displacement measurement systems are of great importance to the revolution in human scientific research and the iterative upgrade of industrial manufacturing. The research on vortex beams is developing rapidly, with promising applications. The vortex beam has a spiral phase, and each photon of the beam carries orbital angular momentum. With the continuous improvement of the production and detection technology for vortex beams, the research on their applications in precision displacement measurement has been on the increase. In this study, to address the contradictory problems of a large range and high accuracy in precision displacement measurement, a precision displacement measurement system with interference of conjugated vortex beams is designed and built. It is expected that this solution can provide new research ideas and technical ways for high-accuracy and large-range displacement measurement, which is of positive significance to the development of contemporary science, technology, and industry.

A precision displacement measurement method based on the interference of conjugated vortex beams is proposed in this study, with the interference pattern of conjugated vortex beams as the source of displacement data. By establishing the mathematical relationship between rotational angle radian of the pattern and displacement and designing the experimental data processing algorithm for subnano-scale displacement and large-measurement-range displacement, accurate precision displacement measurement results can be obtained by accurate extraction of the rotational angle radians. Then, an optical system is designed and simulated according to the modified Mach-Zehnder structure for the measurement scheme (Fig. 5). An experimental system is developed and experimentally tested to verify the effectiveness of the proposed method (Fig. 6). In addition, aberrations in the optical system can be corrected by differential evolution algorithms to improve the accuracy of the displacement measurement system.

In this study, a new precision displacement measurement method is established on the basis of interference of conjugated vortex beams. The interference pattern of conjugated vortex beams is used as the source of displacement data, and a mathematical relationship between rotational angle radian of the pattern and displacement is established. An optical design is carried out, an experimental system is set up, and the proposed method is validated. The results of the 10 sets of experimental measurements demonstrate a mean of 50.0254 nm, with a standard deviation of 0.114 nm at a displacement of 50 nm, which indicates the validity of the proposed method. Meanwhile, the proposed experimental system can also perform precision displacement measurements over a large measurement range, and the experimental results show that the proposed method can be used for a measurement range of at least 30 μm. In addition, the system aberrations in the vortex beam interference process are fitted, and the measurement accuracy of the proposed method can be improved by reasonable compensation of the optical system. The proposed system renders a new research idea and technical approach for displacement measurement with high accuracy and a large range, and provides the basic theoretical and technical support for precision displacement measurement in fields such as lathe processing, the semiconductor industry, and aerospace.