Displacement measurement is a crucial application, with laser-based methods offering high precision and being well established in commercial settings. However, these methods often come with the drawbacks of significant size and exorbitant costs. We introduce a novel displacement measurement method that utilizes the missing-order Talbot effect. This approach circumvents the need to measure contrast in the Talbot diffraction field, opting instead to leverage the displacement within the missing-order Talbot diffraction pattern. Our method only requires parallel light, an amplitude grating, and a detector to achieve displacement measurement. The measurement dynamic range can be adjusted by altering the grating period and the wavelength of the incident light. Through careful simulation and experimental validation, our method exhibits a correlation coefficient R surpassing $0.999$ across a 30 mm dynamic range and achieves a precision superior to 3 $\mathsf{\mu }$m.

The precision of two-dimensional angular sensing is crucial for applications such as navigation, robotics, and optical alignment. Conventional methods often struggle to balance precision, dynamic range, and affordability. We introduce a novel method leveraging the Talbot effect, enhanced by 3D printing technology, to fabricate a grating prototype for high-precision angular measurements. The method detects amplitude grating displacement at the self-imaging position and employs a frequency filtering algorithm for improved accuracy. Rigorous validation through simulations and physical experiments demonstrates that our method achieves a detection resolution of 0.4 arcseconds and a dynamic range exceeding 1400 arcseconds. This research presents a cost-effective, high-precision angular detection solution with potential for broad application in precision mechanical assembly, optical alignment, and other relevant domains.