{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:01:15Z","timestamp":1760234475072,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,5,19]],"date-time":"2021-05-19T00:00:00Z","timestamp":1621382400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR-20-CE42-0010"],"award-info":[{"award-number":["ANR-20-CE42-0010"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In this paper, a method based on the inherent event-based sampling capability of laser optical feedback interferometry (OFI) is proposed to assess the optical feedback factor C when the laser operates in the moderate and strong feedback regimes. Most of the phase unwrapping open-loop OFI algorithms rely on the estimation of C to retrieve the displacement with nanometric precision. Here, the proposed method operates in open-loop configuration and relies only on OFI\u2019s fringe detection, thereby improving its robustness and ease of use. The proposed method is able to estimate C with a precision of &lt;5%. The obtained performances are compared to three different approaches previously published and the impacts of phase noise and sampling frequency are reported. We also show that this method can assess C on the fly even when C is varying due to speckle. To the best of the authors\u2019 knowledge, these are the first reported results of time-varying C estimation. In addition, through C estimation over time, it could pave the way not only to higher performance phase unwrapping algorithms but also to a better control of the optical feedback level via the use of an adaptive lens and thus to better displacement retrieval performances.<\/jats:p>","DOI":"10.3390\/s21103528","type":"journal-article","created":{"date-parts":[[2021,5,19]],"date-time":"2021-05-19T21:49:21Z","timestamp":1621460961000},"page":"3528","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Toward an Estimation of the Optical Feedback Factor C on the Fly for Displacement Sensing"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8889-0620","authenticated-orcid":false,"given":"Olivier D.","family":"Bernal","sequence":"first","affiliation":[{"name":"LAAS-CNRS, University of Toulouse, INP-ENSEEIHT, 31000 Toulouse, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6744-7371","authenticated-orcid":false,"given":"Usman","family":"Zabit","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, National University of Sciences and Technology, NUST, Islamabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1703-6029","authenticated-orcid":false,"given":"Francis","family":"Jayat","sequence":"additional","affiliation":[{"name":"LAAS-CNRS, University of Toulouse, INP-ENSEEIHT, 31000 Toulouse, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4016-0634","authenticated-orcid":false,"given":"Thierry","family":"Bosch","sequence":"additional","affiliation":[{"name":"LAAS-CNRS, University of Toulouse, INP-ENSEEIHT, 31000 Toulouse, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1002\/lpor.201100002","article-title":"Developing self-mixing interferometry for instrumentation and measurements","volume":"6","author":"Donati","year":"2012","journal-title":"Laser Photonics Rev."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"570","DOI":"10.1364\/AOP.7.000570","article-title":"Laser feedback interferometry: A tutorial on the self-mixing effect for coherent sensing","volume":"7","author":"Taimre","year":"2015","journal-title":"Adv. Opt. Photonics"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"822","DOI":"10.1364\/OL.36.000822","article-title":"High-resolution monitoring of the hole depth during ultrafast laser ablation drilling by diode laser self-mixing interferometry","volume":"36","author":"Mezzapesa","year":"2011","journal-title":"Opt. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1088\/0957-0233\/14\/1\/304","article-title":"Self-mixing laser diode vibrometer","volume":"14","author":"Giuliani","year":"2002","journal-title":"Meas. Sci. Technol."},{"key":"ref_5","first-page":"1","article-title":"Obtaining Scalable Fringe Precision in Self-Mixing Interference Using an Even-Power Fast Algorithm","volume":"9","author":"Wei","year":"2017","journal-title":"IEEE Photonics J."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/s11082-017-0951-5","article-title":"Multiple self-mixing interferometry algorithm based on phase modulation for vibration measurement","volume":"49","author":"Jiang","year":"2017","journal-title":"Opt. Quantum Electron."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1101","DOI":"10.1109\/TIM.2006.876544","article-title":"Displacement measurements using a self-mixing laser diode under moderate feedback","volume":"55","author":"Bes","year":"2006","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4962","DOI":"10.1109\/JSEN.2013.2276106","article-title":"Study of Laser Feedback Phase Under Self-Mixing Leading to Improved Phase Unwrapping for Vibration Sensing","volume":"13","author":"Bernal","year":"2013","journal-title":"IEEE Sens. J."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"5064","DOI":"10.1364\/AO.50.005064","article-title":"Improving the measurement performance for a self-mixing interferometry-based displacement sensing system","volume":"50","author":"Fan","year":"2011","journal-title":"Appl. Opt."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1109\/JSEN.2015.2478755","article-title":"Real-Time Algorithm for Versatile Displacement Sensors Based on Self-Mixing Interferometry","volume":"16","author":"Arriaga","year":"2016","journal-title":"IEEE Sens. J."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1109\/3.563379","article-title":"Reconstruction of displacement waveforms with a single-channel laser-diode feedback interferometer","volume":"33","author":"Merlo","year":"1997","journal-title":"IEEE J. Quantum Electron."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"7425","DOI":"10.1109\/JSEN.2017.2758440","article-title":"Analysis and Implementation of a Direct Phase Unwrapping Method for Displacement Measurement Using Self-Mixing Interferometry","volume":"17","author":"Ehtesham","year":"2017","journal-title":"IEEE Sens. J."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"11151","DOI":"10.1109\/JSEN.2019.2935087","article-title":"Spectral Processing of Self-Mixing Interferometric Signal Phase for Improved Vibration Sensing under Weak-and Moderate-Feedback Regime","volume":"19","author":"Zabit","year":"2019","journal-title":"IEEE Sens. J."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"9637","DOI":"10.1364\/AO.54.009637","article-title":"Development of a design tool for closed-loop digital vibrometer","volume":"54","author":"Melchionni","year":"2015","journal-title":"Appl. Opt."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/j.optcom.2015.12.002","article-title":"Self-mixing digital closed-loop vibrometer for high accuracy vibration measurements","volume":"365","author":"Magnani","year":"2016","journal-title":"Opt. Commun."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"11433","DOI":"10.1364\/OE.419844","article-title":"Convolutional neural network for self-mixing interferometric displacement sensing","volume":"29","author":"Barland","year":"2021","journal-title":"Opt. Express"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1364\/AO.53.001001","article-title":"On the nature of Acket\u2019s characteristic parameter C in semiconductor lasers","volume":"53","author":"Taimre","year":"2014","journal-title":"Appl. Opt."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1109\/JSTQE.2014.2381494","article-title":"Robust Method of Stabilization of Optical Feedback Regime by Using Adaptive Optics for a Self-Mixing Micro-Interferometer Laser Displacement Sensor","volume":"21","author":"Bernal","year":"2015","journal-title":"IEEE J. Sel. Top. Quantum Electron."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"990","DOI":"10.1109\/LPT.2004.824631","article-title":"Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect","volume":"16","author":"Yanguang","year":"2004","journal-title":"IEEE Photonics Technol. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1109\/JQE.1980.1070479","article-title":"External optical feedback effects on semiconductor injection laser properties","volume":"16","author":"Lang","year":"1980","journal-title":"IEEE J. Quantum Electron."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"9582","DOI":"10.1364\/OE.19.009582","article-title":"Measuring the feedback parameter of a semiconductor laser with external optical feedback","volume":"19","author":"Yu","year":"2011","journal-title":"Opt. Express"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"5195","DOI":"10.1109\/JSEN.2020.2970599","article-title":"Sub-\u03bb\/2 Displacement Sensor with Nanometric Precision based on Optical Feedback Interferometry used as a Non-Uniform Event-Based Sampling System","volume":"20","author":"Bernal","year":"2020","journal-title":"IEEE Sens. J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"18423","DOI":"10.1364\/OE.26.018423","article-title":"Phase-resolved terahertz self-detection near-field microscopy","volume":"26","author":"Giordano","year":"2018","journal-title":"Opt. Express"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"7820","DOI":"10.1364\/AO.54.007820","article-title":"Sinusoidal phase-modulating self-mixing interferometer with nanometer resolution and improved measurement velocity range","volume":"54","author":"Xia","year":"2015","journal-title":"Appl. Opt."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"702","DOI":"10.1364\/AO.53.000702","article-title":"Classification of laser self-mixing interferometric signal under moderate feedback","volume":"53","author":"Bernal","year":"2014","journal-title":"Appl. Opt."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"5318","DOI":"10.1364\/AO.51.005318","article-title":"Self-mixing vibrometer with real-time digital signal elaboration","volume":"51","author":"Magnani","year":"2012","journal-title":"Appl. Opt."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1157","DOI":"10.1109\/JQE.2005.853364","article-title":"Behavioral model of a self-mixing laser diode sensor","volume":"41","author":"Plantier","year":"2005","journal-title":"IEEE J. Quantum Electron."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Marvasti, F. (2001). Nonuniform Sampling: Theory and Practice, Springer.","DOI":"10.1007\/978-1-4615-1229-5"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"798","DOI":"10.1109\/JQE.2013.2276894","article-title":"Speckle pattern errors in self-mixing interferometry","volume":"49","author":"Donati","year":"2013","journal-title":"IEEE J. Quantum Electron."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"5892","DOI":"10.1109\/JSEN.2017.2737472","article-title":"All analog processing of speckle affected self-mixing interferometric signals","volume":"17","author":"Siddiqui","year":"2017","journal-title":"IEEE Sens. J."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"798","DOI":"10.1109\/2944.892620","article-title":"Design of a phase-shifting optical feedback interferometer using an electrooptic modulator","volume":"6","author":"Servagent","year":"2000","journal-title":"IEEE J. Sel. Top. Quantum Electron."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Vezyrtzis, C., and Tsividis, Y. (2009, January 24\u201327). Processing of signals using level-crossing sampling. Proceedings of the 2009 IEEE International Symposium on Circuits and Systems, Taipei, Taiwan.","DOI":"10.1109\/ISCAS.2009.5118257"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2089","DOI":"10.1109\/TCSI.2008.2011586","article-title":"A Level-Crossing Analog-to-Digital Converter With Triangular Dither","volume":"56","author":"Wang","year":"2009","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"7903","DOI":"10.1109\/JSEN.2016.2599702","article-title":"Robust Detection of Non-Regular Interferometric Fringes From a Self-Mixing Displacement Sensor Using Bi-Wavelet Transform","volume":"16","author":"Bernal","year":"2016","journal-title":"IEEE Sens. J."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1028","DOI":"10.1109\/68.867997","article-title":"Laser diode linewidth measurement by means of self-mixing interferometry","volume":"12","author":"Giuliani","year":"2000","journal-title":"IEEE Photonics Technol. Lett."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/10\/3528\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:03:52Z","timestamp":1760162632000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/10\/3528"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,19]]},"references-count":35,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2021,5]]}},"alternative-id":["s21103528"],"URL":"https:\/\/doi.org\/10.3390\/s21103528","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2021,5,19]]}}}