{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:33:16Z","timestamp":1760232796071,"version":"build-2065373602"},"reference-count":50,"publisher":"MDPI AG","issue":"23","license":[{"start":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T00:00:00Z","timestamp":1669248000000},"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":["51735002","XDA25020317"],"award-info":[{"award-number":["51735002","XDA25020317"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Strategic Priority Research Program of the Chinese Academy of Sciences","award":["51735002","XDA25020317"],"award-info":[{"award-number":["51735002","XDA25020317"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper presents a polarization grating based circular subaperture stitching interferometer. The system can be used for small F\/# concave surface tests with a large F\/# transmission sphere, where F\/# is the ratio of focal length to aperture. A polarization grating was employed to deflect the incident beam for subaperture scanning by its axial rotation instead of a multi-axis motion-control system. Compared with the traditional subaperture stitching interferometric system, the system proposed in this paper is smaller in size and reduces the measurement error introduced by mechanical adjustment. Using a virtual interferometer model and a virtual\u2013real combination algorithm to remove the retrace error, the full-aperture figure error can be directly obtained without the need for a complex stitching algorithm. The feasibility of the algorithm was verified, and the measurement error caused by the modeling error was analyzed by simulation. The capability of the polarization grating to scan subapertures was experimentally confirmed, and possible solutions to some engineering challenges were pointed out. The research in this paper has pioneering and guiding significance for the application of polarization grating in interferometry.<\/jats:p>","DOI":"10.3390\/s22239129","type":"journal-article","created":{"date-parts":[[2022,11,25]],"date-time":"2022-11-25T03:34:24Z","timestamp":1669347264000},"page":"9129","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Circular Subaperture Stitching Interferometry Based on Polarization Grating and Virtual\u2013Real Combination Interferometer"],"prefix":"10.3390","volume":"22","author":[{"given":"Yao","family":"Hu","sequence":"first","affiliation":[{"name":"Beijing Key Laboratory for Precision Optoelectronic Measurement Instrument and Technology, School of Optics and Photonics, Beijing Institute of Technology, Beijing 100081, China"}]},{"given":"Zhen","family":"Wang","sequence":"additional","affiliation":[{"name":"Beijing Key Laboratory for Precision Optoelectronic Measurement Instrument and Technology, School of Optics and Photonics, Beijing Institute of Technology, Beijing 100081, China"}]},{"given":"Qun","family":"Hao","sequence":"additional","affiliation":[{"name":"Beijing Key Laboratory for Precision Optoelectronic Measurement Instrument and Technology, School of Optics and Photonics, Beijing Institute of Technology, Beijing 100081, China"},{"name":"School of Opto-Electronic Engineering, Changchun University of Science and Technology, Changchun 130022, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"7254","DOI":"10.1364\/AO.428136","article-title":"Anamorphic Objective Design for Extreme Ultraviolet Lithography at the 5~1 Nm Technology Node","volume":"60","author":"Liu","year":"2021","journal-title":"Appl. Opt."},{"key":"ref_2","unstructured":"Ahn, J., and Kim, U. (2014). Optical Pickup Device and Optical Disk Drive Including the Same. (2014\/0010064A1), U.S. Patent."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1364\/OSAC.414835","article-title":"Design of a High Uniform Collimation Illumination System for near Field Measurement","volume":"4","author":"Yao","year":"2021","journal-title":"OSA Contin."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"8230","DOI":"10.1364\/AO.56.008230","article-title":"Optimized Design Method for the Inner Surface of a Conformal Dome Based on the Ray Tracing Approach","volume":"56","author":"Dang","year":"2017","journal-title":"Appl. Opt."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"103967","DOI":"10.1016\/j.infrared.2021.103967","article-title":"Conformal Dome Aberration Correction Based on the Integrated Design of Inner Surface and Fixed Corrector","volume":"120","author":"Liu","year":"2022","journal-title":"Infrared Phys. Technol."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Malacara, D. (2007). Optical Metrology of Diffuse Surfaces. Optical Shop Testing, Wiley-Interscience. [3rd ed.].","DOI":"10.1002\/9780470135976"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3242","DOI":"10.1364\/OE.17.003242","article-title":"Null Test of an Off-Axis Parabolic Mirror II Configuration with Planar Reference Wave and Spherical Return Surface","volume":"17","author":"Burke","year":"2009","journal-title":"Opt. Express"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Li, S., Zhang, J., Liu, W., Liang, H., Xie, Y., and Li, X. (2020). The Methods and Experiments of Shape Measurement for Off-Axis Conic Aspheric Surface. Materials, 13.","DOI":"10.3390\/ma13092101"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2278","DOI":"10.1364\/OL.456783","article-title":"Quasi-Absolute Interferometric Testing of Cylinders","volume":"47","author":"Chen","year":"2022","journal-title":"Opt. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4699","DOI":"10.1364\/AO.455315","article-title":"Solving Optimal Carrier Frequencies of a CGH Null Compensator through a Double-Constrained Searching Method Based on Iterative Ray-Tracings","volume":"61","author":"Liang","year":"2022","journal-title":"Appl. Opt."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"31172","DOI":"10.1364\/OE.26.031172","article-title":"Near-Null Interferometry Using an Aspheric Null Lens Generating a Broad Range of Variable Spherical Aberration for Flexible Test of Aspheres","volume":"26","author":"Xue","year":"2018","journal-title":"Opt. Express"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"4040","DOI":"10.1364\/AO.452951","article-title":"Mapping Distortion Correction in Off-Axis Aspheric Mirror Testing with a Null Compensator","volume":"61","author":"Hao","year":"2022","journal-title":"Appl. Opt."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Xie, F., Hao, Q., Zhu, Q., Tang, L., and Hu, Y. (2010, January 18\u201320). Compensation Ability Evaluation of Part-Compensation Lens Based on Slope Asphericity. Proceedings of the Optical Design and Testing IV, Beijing, China.","DOI":"10.1117\/12.869062"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"18107","DOI":"10.1364\/OE.25.018107","article-title":"Vertex Radius of Curvature Error Measurement of Aspheric Surface Based on Slope Asphericity in Partial Compensation Interferometry","volume":"25","author":"Hao","year":"2017","journal-title":"Opt. Express"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"10111","DOI":"10.1364\/OE.19.010111","article-title":"High-Speed, Sub-Nyquist Interferometry","volume":"19","author":"Wu","year":"2011","journal-title":"Opt. Express"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"4521","DOI":"10.1364\/AO.21.004521","article-title":"Polynomial Fit of Interferograms","volume":"21","author":"Kim","year":"1982","journal-title":"Appl. Opt."},{"key":"ref_17","unstructured":"Thunen, J.G., and Kwon, O.Y. (1982, January 24\u201327). Full Aperture Testing with Subaperture Test Optics. Proceedings of the 26th Annual Technical Symposium, San Diego, CA, USA."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1921","DOI":"10.1364\/AO.23.001921","article-title":"Subaperature Optical System Testing","volume":"23","author":"Negro","year":"1984","journal-title":"Appl. Opt."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"740","DOI":"10.1364\/AO.23.000740","article-title":"Subaperture Testing Approaches: A Comparison","volume":"23","author":"Jensen","year":"1984","journal-title":"Appl. Opt."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1364\/AO.25.000503","article-title":"Numerical Processing of Dynamic Subaperture Testing Measurements","volume":"25","author":"Erteza","year":"1986","journal-title":"Appl. Opt."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1117\/12.152248","article-title":"Measurement of Large Plane Surface Shapes by Connecting Small-Aperture Interferograms","volume":"33","author":"Okada","year":"1994","journal-title":"Opt. Eng"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"4504","DOI":"10.1364\/AO.27.004504","article-title":"Subaperture Testing of Aspheres with Annular Zones","volume":"27","author":"Liu","year":"1988","journal-title":"Appl. Opt."},{"key":"ref_23","first-page":"251","article-title":"Iterative Stitching Algorithm for Annular Subaperture Testing and Its Experimental Verification","volume":"17","author":"Dai","year":"2009","journal-title":"Opt. Precis. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"7073","DOI":"10.1364\/AO.58.007073","article-title":"Measurement of Rotationally Symmetric Aspherical Surfaces with the Annular Subaperture Stitching Method","volume":"58","author":"Yan","year":"2019","journal-title":"Appl. Opt."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"29047","DOI":"10.1364\/OE.23.029047","article-title":"Subaperture Stitching Test of Large Steep Convex Spheres","volume":"23","author":"Chen","year":"2015","journal-title":"Opt. Express"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"126162","DOI":"10.1016\/j.optcom.2020.126162","article-title":"Research on Compensating Stitching Testing for Convex Aspherical Surface","volume":"474","author":"Yan","year":"2020","journal-title":"Opt. Commun."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"468","DOI":"10.1364\/OL.8.000468","article-title":"Method for Subaperture Testing Interferogram Reduction","volume":"8","author":"Chow","year":"1983","journal-title":"Opt. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Stuhlinger, T.W. (1986, January 15\u201318). Subaperture Optical Testing: Experimental Verification. Proceedings of the 1986 International Symposium\/Innsbruck, Innsbruck, Austria.","DOI":"10.1117\/12.938444"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Chen, M., Cheng, W., and Wang, C.-W.W. (1991, January 21). Multiaperture Overlap-Scanning Technique for Large-Aperture Test. Proceedings of the Laser Interferometry IV: Computer-Aided Interferometry, San Diego, CA, USA.","DOI":"10.1117\/12.135344"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1219","DOI":"10.1364\/JOSAA.23.001219","article-title":"Iterative Algorithm for Subaperture Stitching Test with Spherical Interferometers","volume":"23","author":"Chen","year":"2006","journal-title":"J. Opt. Soc. Am. A"},{"key":"ref_31","unstructured":"Geary, J.M., Yoo, K.M., and Si, G. (1992, January 22). Retrace Error: A Case Study. Proceedings of the Interferometry: Surface Characterization and Testing, San Diego, CA, USA."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"7025","DOI":"10.1364\/OE.17.007025","article-title":"Practical Methods for Retrace Error Correction in Nonnull Aspheric Testing","volume":"17","author":"Liu","year":"2009","journal-title":"Opt. Express"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"5152","DOI":"10.1364\/AO.43.005152","article-title":"Iterative Reverse Optimization Procedure for Calibration of Aspheric Wave-Front Measurements on a Nonnull Interferometer","volume":"43","author":"Gappinger","year":"2004","journal-title":"Appl. Opt."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"5755","DOI":"10.1364\/AO.53.005755","article-title":"Non-Null Annular Subaperture Stitching Interferometry for Steep Aspheric Measurement","volume":"53","author":"Zhang","year":"2014","journal-title":"Appl. Opt."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"19176","DOI":"10.1364\/OE.23.019176","article-title":"Aspheric Subaperture Stitching Based on System Modeling","volume":"23","author":"Zhang","year":"2015","journal-title":"Opt. Express"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/j.optcom.2017.07.004","article-title":"Validation of Simultaneous Reverse Optimization Reconstruction Algorithm in a Practical Circular Subaperture Stitching Interferometer","volume":"403","author":"Zhang","year":"2017","journal-title":"Opt. Commun."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"9992","DOI":"10.1364\/AO.55.009992","article-title":"Virtual Interferometer Calibration Method of a Non-Null Interferometer for Freeform Surface Measurements","volume":"55","author":"Hao","year":"2016","journal-title":"Appl. Opt."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"27014","DOI":"10.1364\/OE.435252","article-title":"Interferometric Measurement of High-Order Aspheric Surface Parameter Errors Based on a Virtual-Real Combination Iterative Algorithm","volume":"29","author":"Hao","year":"2021","journal-title":"Opt. Express"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1016\/j.matlet.2012.05.010","article-title":"Polarization-Dependent Laser-Light Structured Directionality with Polymer Composite Materials","volume":"81","author":"Shalit","year":"2012","journal-title":"Mater. Lett."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"031211","DOI":"10.1117\/1.OE.56.3.031211","article-title":"Large-Aperture, Wide-Angle Nonmechanical Beam Steering Using Polarization Gratings","volume":"56","author":"Serati","year":"2016","journal-title":"Opt. Eng."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"100636","DOI":"10.1016\/j.apmt.2020.100636","article-title":"Near-Frequency Photons Y-Splitter","volume":"19","author":"Castagna","year":"2020","journal-title":"Appl. Mater. Today"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/j.optcom.2014.11.075","article-title":"Non-Null Testing for Standard Quadric Surfaces with Subaperture Stitching Technique","volume":"340","author":"Yan","year":"2015","journal-title":"Opt. Commun."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2280","DOI":"10.1364\/AO.45.002280","article-title":"Lattice Design for Subaperture Stitching Test of a Concave Paraboloid Surface","volume":"45","author":"Chen","year":"2006","journal-title":"Appl. Opt."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"584","DOI":"10.1364\/OL.24.000584","article-title":"Measuring Stokes Parameters by Means of a Polarization Grating","volume":"24","author":"Gori","year":"1999","journal-title":"Opt. Lett."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Rutkowska, K.A., and Kozanecka-Szmigiel, A. (2020). Design of Tunable Holographic Liquid Crystalline Diffraction Gratings. Sensors, 20.","DOI":"10.3390\/s20236789"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Nikolova, L., and Ramanujam, P.S. (2009). Polarization Holography, Cambridge University Press.","DOI":"10.1017\/CBO9780511581489"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"2206","DOI":"10.3390\/s130202206","article-title":"Frequency Noise Properties of Lasers for Interferometry in Nanometrology","volume":"13","author":"Hrabina","year":"2013","journal-title":"Sensors"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Yang, H., Yin, Z., Yang, R., Hu, P., Li, J., and Tan, J. (2020). Design for a Highly Stable Laser Source Based on the Error Model of High-Speed High-Resolution Heterodyne Interferometers. Sensors, 20.","DOI":"10.3390\/s20041083"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"14084","DOI":"10.3390\/s121014084","article-title":"Refractive Index Compensation in Over-Determined Interferometric Systems","volume":"12","author":"Lazar","year":"2012","journal-title":"Sensors"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"(2021). Optical Phase-Shifting Methods Based on Low Coherence Laser for Large Aperture Fizeau Interferometer. Opt. Lasers Eng., 140, 106513.","DOI":"10.1016\/j.optlaseng.2020.106513"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/23\/9129\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:26:09Z","timestamp":1760145969000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/23\/9129"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,24]]},"references-count":50,"journal-issue":{"issue":"23","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["s22239129"],"URL":"https:\/\/doi.org\/10.3390\/s22239129","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2022,11,24]]}}}