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. 2021 May 20;12:2979. doi: 10.1038/s41467-021-23249-y

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© The Author(s) 2021

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Fig. 1 — a Schematic representation of the lens-axicon triplet. Gaussian-shaped laser line is transmitted to the optical unit from left to right. Lens-axicon doublet and a (second) lens can move translationally along the optical axis (red double-headed arrow). Distance between the doublet and the second lens, and that from the second lens are denoted as $z_{1}$ and $z_{2}$ , respectively. Phase shifts caused by transmission of the doublet and lens are indicated. b Illustration of the cylindrical coordinate system. z-Direction is set as the optical axis, and the xy plane is set transverse to the z-axis. c Transverse profiles through the optical unit ( $z_{1} = 125$ $mm$ and $z_{2} = 30$ mm). Parameters (thickness and radius) are measured to characterize the annulus pattern. Radius is defined as the distance between the origin and the maximum intensity point in the radial direction. Thickness is defined as the full width at the half maximum (FWHM) of the distribution in radial direction. d Lens and doublet position dependence of the annulus parameters, radius, and thickness estimated by the 2nd order multiple regression model.