{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T00:48:16Z","timestamp":1771462096531,"version":"3.50.1"},"reference-count":41,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,1,17]],"date-time":"2023-01-17T00:00:00Z","timestamp":1673913600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62005252"],"award-info":[{"award-number":["62005252"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51875532"],"award-info":[{"award-number":["51875532"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Here, we report an ultracompact angular displacement sensor based on the Talbot effect of optical microgratings. Periodic Talbot interference patterns were obtained behind an upper optical grating. By putting another grating within the Talbot region, the total transmission of the two-grating structure was found to be approximatively in a linear relationship with the relative pitch angle between the two gratings, which was explained by a transversal shift of the Talbot interference patterns. The influence of the grating parameters (e.g., the grating period, the number of grating lines and the gap between the two gratings) was also studied in both a simulation and an experiment, showing a tunable sensitivity and range by simply changing the grating parameters. A sensitivity of 0.19 mV\/arcsec was experimentally obtained, leading to a relative sensitivity of 0.27%\/arcsec within a linear range of \u00b1396 arcsec with the 2 \u03bcm-period optical gratings. Benefitting from tunable properties and an ultracompact structure, we believe that the proposed sensor shows great potential in applications such as aviation, navigation, robotics and manufacturing engineering.<\/jats:p>","DOI":"10.3390\/s23031091","type":"journal-article","created":{"date-parts":[[2023,1,18]],"date-time":"2023-01-18T01:33:26Z","timestamp":1674005606000},"page":"1091","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["An Ultracompact Angular Displacement Sensor Based on the Talbot Effect of Optical Microgratings"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2320-8088","authenticated-orcid":false,"given":"Zhiyong","family":"Yang","sequence":"first","affiliation":[{"name":"School of Instrument and Electronics, North University of China, Taiyuan 030051, China"},{"name":"School of Mechanical Engineering, North University of China, Taiyuan 030051, China"},{"name":"School of Instrument and Intelligence, North University of China, Taiyuan 030051, China"}]},{"given":"Xiaochen","family":"Ma","sequence":"additional","affiliation":[{"name":"School of Instrument and Electronics, North University of China, Taiyuan 030051, China"}]},{"given":"Daguo","family":"Yu","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, North University of China, Taiyuan 030051, China"}]},{"given":"Bin","family":"Cao","sequence":"additional","affiliation":[{"name":"School of Instrument and Electronics, North University of China, Taiyuan 030051, China"}]},{"given":"Qianqi","family":"Niu","sequence":"additional","affiliation":[{"name":"School of Instrument and Electronics, North University of China, Taiyuan 030051, China"}]},{"given":"Mengwei","family":"Li","sequence":"additional","affiliation":[{"name":"School of Instrument and Electronics, North University of China, Taiyuan 030051, China"},{"name":"School of Instrument and Intelligence, North University of China, Taiyuan 030051, China"}]},{"given":"Chenguang","family":"Xin","sequence":"additional","affiliation":[{"name":"School of Instrument and Electronics, North University of China, Taiyuan 030051, China"},{"name":"School of Instrument and Intelligence, North University of China, Taiyuan 030051, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Zhou, S., Le, V., Mi, Q., and Wu, G. (2020). Grating-Corner-Cube-Based Roll Angle Sensor. Sensors, 20.","DOI":"10.3390\/s20195524"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Yuan, J., Dai, P., Liang, D., Zhou, S., Xiao, S., and Liang, X. (2019). Grid Deformation Real-Time Measurement System of Ion Thruster Based on Videometrics. Appl. Sci., 9.","DOI":"10.3390\/app9091759"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"045011","DOI":"10.1088\/1361-6501\/ab41ae","article-title":"Visual measurement method for large-space dynamic angles","volume":"31","author":"Wang","year":"2020","journal-title":"Meas. Sci. Technol."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Chang, D., Xing, X., Hu, P., Wang, J., and Tan, J. (2019). Double-Diffracted Spatially Separated Heterodyne Grating Interferometer and Analysis on its Alignment Tolerance. Appl. Sci., 9.","DOI":"10.3390\/app9020263"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"065116","DOI":"10.1063\/1.4811526","article-title":"Beam splitting target reflector based compensation for angular drift of laser beam in laser autocollimation of measuring small angle deviations","volume":"84","author":"Zhu","year":"2013","journal-title":"Rev. Sci. Instrum."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"11391","DOI":"10.1364\/OE.21.011391","article-title":"Common-path design criteria for laser datum based measurement of small angle deviations and laser autocollimation method in compliance with the criteria with high accuracy and stability","volume":"21","author":"Zhu","year":"2013","journal-title":"Opt. Express"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Saito, Y., Gao, W., and Kiyono, S. A micro-angle sensor based on laser autocollimation, In Proceedings of the International Symposium on Optomechatronic Technologies, Sapporo, Japan, 5\u20137 December 2005.","DOI":"10.1117\/12.647981"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"084011","DOI":"10.1088\/0957-0233\/27\/8\/084011","article-title":"Simultaneous multiple degrees of freedom (DoF) measurement system","volume":"27","author":"Molnar","year":"2016","journal-title":"Meas. Sci. Technol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"043604","DOI":"10.1117\/1.2188945","article-title":"Two-dimensional small-rotation-angle measurement using an imaging method","volume":"45","author":"Suzuki","year":"2006","journal-title":"Opt. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1016\/j.optlastec.2018.12.029","article-title":"Dual-axis optoelectronic level based on laser auto-collimation and liquid surface reflection","volume":"113","author":"Zhang","year":"2019","journal-title":"Opt. Laser Technol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"6389","DOI":"10.1364\/OE.27.006389","article-title":"Cube-corner autocollimator with expanded measurement range","volume":"27","author":"Li","year":"2019","journal-title":"Opt. Express"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"015010","DOI":"10.1088\/1361-6501\/abaf6f","article-title":"Improved performance of a refurbished photoelectric autocollimator","volume":"32","author":"Astrua","year":"2021","journal-title":"Meas. Sci. Technol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"10114","DOI":"10.1364\/AO.438457","article-title":"New type of autocollimator based on normal tracing method and Risley prisms","volume":"60","author":"Peng","year":"2021","journal-title":"Appl. Opt."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"120929","DOI":"10.1109\/ACCESS.2019.2937919","article-title":"A Novel Enhanced Roll-Angle Measurement System Based on a Transmission Grating Autocollimator","volume":"7","author":"Ren","year":"2019","journal-title":"IEEE Access"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"055102","DOI":"10.1063\/5.0089964","article-title":"A three-dimensional small angle measurement system based on autocollimation method","volume":"93","author":"Ren","year":"2022","journal-title":"Rev. Sci. Instrum."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"9986","DOI":"10.1364\/AO.55.009986","article-title":"Design, fabrication, and verification of a three-dimensional autocollimator","volume":"55","author":"Yin","year":"2016","journal-title":"Appl. Opt."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2262","DOI":"10.1364\/AO.384124","article-title":"Three-degree-of-freedom autocollimator based on a combined target reflector","volume":"59","author":"Guo","year":"2020","journal-title":"Appl. Opt."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"17185","DOI":"10.1364\/OE.26.017185","article-title":"Low cost, compact 4-DOF measurement system with active compensation of beam angular drift error","volume":"26","author":"Huang","year":"2018","journal-title":"Opt. Express"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"011203","DOI":"10.3788\/COL202220.011203","article-title":"Analysis and suppression of thermal effect of an ultra-stable laser interferometer for space-based gravitational waves detection","volume":"20","author":"Wang","year":"2022","journal-title":"Chin. Opt. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"25587","DOI":"10.1364\/OE.22.025587","article-title":"Laser heterodyne interferometer for simultaneous measuring displacement and angle based on the Faraday effect","volume":"22","author":"Zhang","year":"2014","journal-title":"Opt. Express"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1400108","DOI":"10.1109\/JQE.2015.2497237","article-title":"Single Channel Self-Mixing Interferometer Measures Simultaneously Displacement and Tilt and Yaw Angles of a Reflective Target","volume":"51","author":"Donati","year":"2015","journal-title":"IEEE J. Quantum Electron."},{"key":"ref_22","first-page":"20","article-title":"The small angle generator based on a laser angle interferometer","volume":"8","author":"Eom","year":"2007","journal-title":"Int. J. Precis. Eng. Manuf."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/s11018-016-0930-4","article-title":"A Study of an Angle Examiner Based on the Fizeau Interferometer with Expanded Measurement Range","volume":"59","author":"Chekirda","year":"2016","journal-title":"Meas. Tech."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1630","DOI":"10.1364\/AO.9.001630","article-title":"Interferometric measurement of angles","volume":"9","author":"Malacara","year":"1970","journal-title":"Appl. Opt."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"045103","DOI":"10.1063\/1.3374550","article-title":"A heterodyne interferometer for angle metrology","volume":"81","author":"Hahn","year":"2010","journal-title":"Rev. Sci. Instrum."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2451","DOI":"10.1364\/OE.23.002451","article-title":"Development of a grating-based interferometer for six-degree-of-freedom displacement and angle measurements","volume":"23","author":"Hsieh","year":"2015","journal-title":"Opt. Express"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/j.optlastec.2018.03.005","article-title":"High displacement resolution encoder by using triple grating combination interferometer","volume":"105","author":"Hsu","year":"2018","journal-title":"Opt. Laser Technol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4111","DOI":"10.1364\/AO.38.004111","article-title":"Effects of unparallel grating planes in Talbot interferometry","volume":"38","author":"Liu","year":"1999","journal-title":"Appl. Opt."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"115159","DOI":"10.1109\/ACCESS.2020.3004365","article-title":"Design of a Roll Angle Measuring Sensor","volume":"8","author":"Yang","year":"2020","journal-title":"IEEE Access"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"5897","DOI":"10.1364\/AO.48.005897","article-title":"Light field image sensors based on the Talbot effect","volume":"48","author":"Wang","year":"2009","journal-title":"Appl. Opt."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1873","DOI":"10.1109\/JLT.2019.2894365","article-title":"Discrete Self-Imaging in Small-Core Optical Fiber Interferometers","volume":"37","author":"Lian","year":"2019","journal-title":"J. Lightwave Technol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1677","DOI":"10.1631\/FITEE.2000432","article-title":"Fused-like angles: Replacement for roll-pitch-yaw angles for a six-degree-of-freedom grating interferometer","volume":"22","author":"Chang","year":"2021","journal-title":"Front. Inf. Technol. Electron. Eng."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.measurement.2014.08.018","article-title":"Sensitivity of transverse shift inside a double-grating Talbot interferometer","volume":"58","author":"Deachapunya","year":"2014","journal-title":"Measurement"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1253","DOI":"10.1364\/JOSAA.25.001253","article-title":"Diffraction analysis of blazed transmission gratings with a modified extended scalar theory","volume":"25","author":"Wang","year":"2008","journal-title":"J. Opt. Soc. Am. A."},{"key":"ref_35","unstructured":"Goodman, J.W. (2007). Introduction to Fourier Optics, W.H. Freeman & Company. [4th ed.]."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1346","DOI":"10.1364\/OL.23.001346","article-title":"Apparent superresolution in near-field optical imaging of periodic gratings","volume":"23","author":"Smolyaninov","year":"1998","journal-title":"Opt. Lett."},{"key":"ref_37","unstructured":"Stark, H. (2012). Applications of Optical Fourier Transforms, Academic Press."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/j.optlastec.2016.10.007","article-title":"Simulation and experimental studies of a double-fiber angular displacement sensor","volume":"89","author":"Zhu","year":"2017","journal-title":"Opt. Laser. Technol."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1642","DOI":"10.1002\/mop.32814","article-title":"Theoretical and experimental study on intensity modulation differential optical fiber angular displacement sensor","volume":"63","author":"Zha","year":"2021","journal-title":"Microw. Opt. Technol. Lett."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"7007509","DOI":"10.1109\/TIM.2022.3213005","article-title":"A Reflective-Type Heterodyne Grating Interferometer for Three-Degree-of-Freedom Subnanometer Measurement","volume":"71","author":"Zhu","year":"2022","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1016\/j.precisioneng.2019.06.012","article-title":"Real-time displacement calculation and offline geometric calibration of the grating interferometer system for ultra-precision wafer stage measurement","volume":"60","author":"Ye","year":"2019","journal-title":"Precis. Eng."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/3\/1091\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:08:39Z","timestamp":1760119719000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/3\/1091"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,17]]},"references-count":41,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["s23031091"],"URL":"https:\/\/doi.org\/10.3390\/s23031091","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,17]]}}}