{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,16]],"date-time":"2026-07-16T06:15:40Z","timestamp":1784182540746,"version":"3.55.0"},"reference-count":63,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,9,26]],"date-time":"2021-09-26T00:00:00Z","timestamp":1632614400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This article demonstrates the crucial importance of the symmetrization method for the formation of autofocusing beams. It is possible to impart autofocusing properties to rather arbitrary distributions, for example, truncated and inverted classical modes (such as Hermite\u2013Gaussian, Laguerre\u2013Gaussian, and Bessel modes) or shift the fundamental Gaussian beam by inserting mirror or circular symmetry. The most convenient for controlling autofocusing characteristics is the truncated sinus function with a power-law argument dependence. In this case, superlinear chirp beams (with power q &gt; 2) exhibit sudden and more abrupt autofocusing than sublinear chirp beams (with power 1 &lt; q &lt; 2). Comparison of the different beams\u2019 propagation is performed using fractional Fourier transform, which allows obtaining the field distribution in any paraxial region (both in the Fresnel and Fraunhofer diffraction regions). The obtained results expand the capabilities of structured beams in various applications in optics and photonics.<\/jats:p>","DOI":"10.3390\/sym13101794","type":"journal-article","created":{"date-parts":[[2021,9,27]],"date-time":"2021-09-27T23:08:31Z","timestamp":1632784111000},"page":"1794","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Mirror and Circular Symmetry of Autofocusing Beams"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6765-4373","authenticated-orcid":false,"given":"Svetlana N.","family":"Khonina","sequence":"first","affiliation":[{"name":"IPSI RAS\u2014Branch of the FSRC \u201cCrystallography and Photonics\u201d RAS, Molodogvardeyskaya 151, 443001 Samara, Russia"},{"name":"Department of Technical Cybernetics, Samara National Research University, Moskovskoye Shosse 34, 443086 Samara, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4045","DOI":"10.1364\/OL.35.004045","article-title":"Abruptly autofocusing waves","volume":"35","author":"Efremidis","year":"2010","journal-title":"Opt. 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