Among a host of familiar laser beams, most popular and well-understood are Laguerre-Gaussian (LG) beams (Kogelnik and Li in Appl. Opt. 5:1550–1567, 1966; Zauderer in J. Opt. Soc. Am. A 3:465–469, 1986; Wünsche in J. Opt. Soc. Am. A 6:1320–1329, 1989). Initially, these beams were looked upon as intra-cavity modes, but later on out-of-resonator LG beams were generated from Hermite-Gaussian (HG) beams using an astigmatic converter (Abramochkin and Volostnikov in Opt. Commun. 83:123–135, 1991). Particular interest in the LG beams was provoked by Allen’s et al. paper (Allen et al. in Phys. Rev. A 45:8185–8189, 1992) in which the LG beams were found to carry the angular orbital momentum (OAM). Extensively studied beams include generalized LG beams in the form of Hermite-Laguerre-Gaussian beams (Abramochkin and Volostnikov in J. Opt. A Pure Appl. Opt. 6:S157–S161, 2004; Abramochkin et al. in J. Opt. Soc. Am. A 27:2506–2513, 2010), elegant (Zhou and Ru in Prog. Electromagnet. Res. 141:751–768, 2013) and elliptic (Kotlyar et al. in J. Opt. Soc. Am. A 23:43–56, 2006) LG beams. These days have seen no signs of waning interest in studying the LG beams thanks to their wide use in telecommunications, micromanipulation, probing atmospheric turbulence, quantum information, and atom cooling. By way of illustration, a comparative analysis of LG beams and Bessel-Gaussian (BG) beams has been conducted (Mendoza-Hernández et al. in Opt. Lett. 40:3739–3742, 2015). Various approaches to generating LG modes discussed in Refs. (Wang et al. in Opt. Express 29:27783–27790, 2021; Rafayelyan and Brasselet in Opt. Lett. 42:1966–1969, 2017; Mao et al. in Photon. Res. 9:1689–1698, 2021) included the use of a specialized laser utilizing intra-cavity spherical aberration (Wang et al. in Opt. Express 29:27783–27790, 2021), q-plates (Rafayelyan and Brasselet in Opt. Lett. 42:1966–1969, 2017), and a special metasurface (Mao et al. in Photon. Res. 9:1689–1698, 2021). Reciprocal HG-to-LG and LG-to-HG mode conversion was studied in Ref. (Liang and Wang in Opt. Express 27:10684–10691, 2019). Of essential significance is the study of elegant LG beams that have shown outstanding characteristics for many application areas such as optical communications and optical trapping (Longman and Fedosejevs in J. Opt. Soc. Am. A 37:841–848, 2020). A method for measuring a topological charge of a partially coherent elegant LG beam has been proposed (Dong et al. in Opt. Express 26:33035–33043, 2018). The LG beams have formed a basis for developing new types of optical beams with a variety of promising properties. A family of asymmetric LG laser beams has been discussed (Kovalev et al. in Phys. Rev. A 93, 2016) and a technique for generating high-power asymmetric LG beams has been proposed (Hsieh et al. in Opt. Express 26:31738–31749, 2018). Using the LG beams, a vector beam with space dependent transverse polarization has been generated by a method of nonlinear magnetooptic rotation (Ghaderi Goran Abad and Mahmoudi in Sci Rep 11:5972, 2021). A new class of composite vortex beams generated by coaxially superimposing LG beams with identical waist location and parameters has been proposed (Huang et al. in Opt. Lasers Eng. 78:132–139, 2016). A new type of a partially coherent beam with a peculiar correlation function, which has been given the name an elliptic correlated Laguerre–Gauss-Shell model, has been theoretically and experimentally studied (Chen et al. in Opt. Express 22:13975–13987, 2014). In the far field, the intensity pattern of such beams is in the form of an ellipse-shaped ring. The LG beams are of great practical value for optical communication (Pang et al. in Opt. Lett. 43:5639–5642, 2018; Doster and Watnik in Appl. Opt. 55:10239–10246, 2016; Zeng et al. in Opt. Express 27:25342–25356, 2019; Cox et al. in J. Lightwave Technol. 37:3911–3917, 2019;), micromanipulation (Otsu et al. in Sci. Rep. 4:4579, 2014), and photo-induced atom excitation (Peshkov et al. in Phys. Rev. A 96, 2017).