{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T20:45:36Z","timestamp":1778705136618,"version":"3.51.4"},"reference-count":25,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,2,24]],"date-time":"2022-02-24T00:00:00Z","timestamp":1645660800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002261","name":"Russian Foundation for Basic Research","doi-asserted-by":"publisher","award":["20-32-90161"],"award-info":[{"award-number":["20-32-90161"]}],"id":[{"id":"10.13039\/501100002261","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["20-79-00264"],"award-info":[{"award-number":["20-79-00264"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>A simple method for reconstructing the spatial parameters of a laser beam, based on the transport-of-intensity equation, is presented. Registration of cross-section intensity distributions in several planes was carried out using a single CMOS camera. The processing of the experimental measurements with the help of specialized software helped to reconstruct all of the spatial parameters, namely, the radius and position of the waist, Rayleigh length, angular divergence, quality parameter M2 The method was compared with measurements made according to the international standard ISO 11146 and showed that the difference in the spatial parameters is 10% or less, which shows good agreement.<\/jats:p>","DOI":"10.3390\/s22051765","type":"journal-article","created":{"date-parts":[[2022,2,24]],"date-time":"2022-02-24T21:11:07Z","timestamp":1645737067000},"page":"1765","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Reconstructing the Spatial Parameters of a Laser Beam Using the Transport-of-Intensity Equation"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5074-0718","authenticated-orcid":false,"given":"Michael","family":"Kovalev","sequence":"first","affiliation":[{"name":"Laser and Optoelectronic Systems Department, Bauman Moscow State Technical University, 2nd Baumanskaya St. 5\/1, 105005 Moscow, Russia"},{"name":"Lebedev Physical Institute, Russian Academy of Sciences, Leninskiy Prospekt 53, 119991 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5874-2395","authenticated-orcid":false,"given":"Iliya","family":"Gritsenko","sequence":"additional","affiliation":[{"name":"Laser and Optoelectronic Systems Department, Bauman Moscow State Technical University, 2nd Baumanskaya St. 5\/1, 105005 Moscow, Russia"},{"name":"Lebedev Physical Institute, Russian Academy of Sciences, Leninskiy Prospekt 53, 119991 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8383-7396","authenticated-orcid":false,"given":"Nikita","family":"Stsepuro","sequence":"additional","affiliation":[{"name":"Laser and Optoelectronic Systems Department, Bauman Moscow State Technical University, 2nd Baumanskaya St. 5\/1, 105005 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0362-106X","authenticated-orcid":false,"given":"Pavel","family":"Nosov","sequence":"additional","affiliation":[{"name":"Laser and Optoelectronic Systems Department, Bauman Moscow State Technical University, 2nd Baumanskaya St. 5\/1, 105005 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6904-0911","authenticated-orcid":false,"given":"George","family":"Krasin","sequence":"additional","affiliation":[{"name":"Lebedev Physical Institute, Russian Academy of Sciences, Leninskiy Prospekt 53, 119991 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sergey","family":"Kudryashov","sequence":"additional","affiliation":[{"name":"Lebedev Physical Institute, Russian Academy of Sciences, Leninskiy Prospekt 53, 119991 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,24]]},"reference":[{"key":"ref_1","first-page":"7","article-title":"Restoration of the phase of the light field. Differential approach","volume":"3","author":"Abramochkin","year":"1987","journal-title":"Bull. Lebedev Phys. Inst."},{"key":"ref_2","first-page":"1319","article-title":"Phase problem, wavefront dislocations, and an equation for the intensity of a two-dimensional optical field","volume":"8","author":"Aksenov","year":"1995","journal-title":"Atmos. Ocean. Opt."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1070\/PU1988v031n04ABEH005755","article-title":"On the phase retrieval problem in optics","volume":"31","author":"Kuznetsova","year":"1988","journal-title":"Sov. Phys. Usp."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"614","DOI":"10.1070\/QEL17288","article-title":"Holographic wavefront sensors","volume":"50","author":"Venediktov","year":"2020","journal-title":"Quantum Electron."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Krasin, G., Kovalev, M., Stsepuro, N., Ruchka, P., and Odinokov, S. (2020). Lensless Scheme for Measuring Laser Aberrations Based on Computer-Generated Holograms. Sensors, 20.","DOI":"10.3390\/s20154310"},{"key":"ref_6","unstructured":"Volostnikov, V.G. (2014). Methods for the Analysis and Synthesis of Coherent Light Fields, Fizmatlit."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3649","DOI":"10.1364\/OL.391823","article-title":"On a universal solution to the transport-of-intensity equation","volume":"45","author":"Zhang","year":"2020","journal-title":"Opt. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Martinez-Carranza, J., Falaggis, K., Kozacki, T., and Malgorzata, K. (2013, January 13). Effect of Imposed Boundary Conditions on the Accuracy of Transport of Intensity Equation based Solvers. Proceedings of the SPIE-The International Society for Optical Engineering, San Diego, CA, USA.","DOI":"10.1117\/12.2020662"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1364\/OPTICA.2.000411","article-title":"Holographic Shack\u2013Hartmann wavefront sensor based on the correlation peak displacement detection method for wavefront sensing with large dynamic range","volume":"2","author":"Saita","year":"2015","journal-title":"Optica"},{"key":"ref_10","unstructured":"(2005). ISO 11146\u20131\/2\/3."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Ross, T.S. (2013). Laser Beam Quality Metrics, SPIE Press.","DOI":"10.1117\/3.1000595"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"053103","DOI":"10.1063\/1.2198795","article-title":"Hartmann-Shack wave front measurements for real time determination of laser beam propagation parameters","volume":"77","author":"Mann","year":"2006","journal-title":"Rev. Sci. Instrum."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"10180","DOI":"10.1364\/AO.55.010180","article-title":"Complex amplitude reconstruction for dynamic beam quality M2 factor measurement with self-referencing interferometer wavefront sensor","volume":"55","author":"Du","year":"2016","journal-title":"Appl. Opt."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2765","DOI":"10.1364\/AO.58.002765","article-title":"Fast measurement of the laser beam quality factor based on phase retrieval with a liquid lens","volume":"58","author":"Ji","year":"2019","journal-title":"Appl. Opt."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"9347","DOI":"10.1364\/OE.17.009347","article-title":"Complete modal decomposition for optical fibers using CGH-based correlation filters","volume":"17","author":"Kaiser","year":"2009","journal-title":"Opt. Express"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"C146","DOI":"10.1364\/JOSAA.398712","article-title":"Modal analysis of structured light with spatial light modulators: A practical tutorial","volume":"37","author":"Pinnell","year":"2020","journal-title":"J. Opt. Soc. Am. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"20142","DOI":"10.1364\/OE.25.020142","article-title":"Real-time complex amplitude reconstruction method for beam quality M2 factor measurement","volume":"25","author":"Pan","year":"2017","journal-title":"Opt. Express"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"20898","DOI":"10.1364\/OE.394633","article-title":"Beam characterization by phase retrieval solving the transport-of-intensity-equation","volume":"28","author":"Merx","year":"2020","journal-title":"Opt. Express"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Gritsenko, I., Kovalev, M., Krasin, G., Konoplyov, M., and Stsepuro, N. (2021). Computational Method for Wavefront Sensing Based on Transport-of-Intensity Equation. Photonics, 8.","DOI":"10.3390\/photonics8060177"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"106187","DOI":"10.1016\/j.optlaseng.2020.106187","article-title":"Transport of intensity equation: A tutorial","volume":"135","author":"Zuo","year":"2020","journal-title":"Opt. Lasers Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"9546","DOI":"10.1364\/AO.55.009546","article-title":"Recursive method for phase retrieval using transport of intensity and its applications","volume":"55","author":"Basunia","year":"2022","journal-title":"Appl. Opt."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"5105","DOI":"10.1364\/OE.382550","article-title":"Combined laser variosystems paraxial design for longitudinal movement of a Gaussian beam waist","volume":"28","author":"Nosov","year":"2020","journal-title":"Opt. Express"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"012006","DOI":"10.1088\/1742-6596\/584\/1\/012006","article-title":"Investigation of the spatial structure of a high-power fiber laser beam","volume":"584","author":"Nosov","year":"2015","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1434","DOI":"10.1364\/JOSA.73.001434","article-title":"Deterministic phase retrieval: A Green\u2019s function solution","volume":"73","author":"Teague","year":"1983","journal-title":"J. Opt. Soc. Am."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/S0030-4018(01)01556-5","article-title":"Phase retrieval from series of images obtained by defocus variation","volume":"199","author":"Allen","year":"2001","journal-title":"Opt. 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