## 1 Introduction  
Research on high-order transverse modes of laser cavities was mainly conducted by early pioneers after the invention of the laser in 1960 [1, 2, 3, 4, 5, 6] and was very quickly abandoned. It should be noted, however, that a renewed interest in the study of high-order transverse modes of lasers which can be found, without being exhaustive, in references [7, 8, 9, 10, 11]. The high-order transverse modes involve Hermite (Laguerre) polynomials for rectangular (cylindrical) coordinates. In the following text, we will emphasise radial Laguerre-Gauss \(LG_{p0}\) modes. The mathematical description of the high-order transverse mode is available in the standard textbooks in photonics [12, 13, 14]. The electric field associated with a \(LG_{p0}\) beam is given by  
\[ \begin{array}{ll}E_{p}(\rho,z)&=E_{0}\frac{W_{0}}{W(z)}L_{p}\left(\frac{2p^{ 2}}{W^{2}(z)}\right)\times\exp\left(-\frac{\rho^{2}}{W^{2}(z)}\right)\\ &\times\exp\left(i(2p+1)\Phi(z)-\frac{i\mu p}{2R_{c}(z)}\right)\end{array}  \tag{1}\]  
where \(\rho\) and \(z\) are the radial and longitudinal coordinates, respectively, and \(L_{p}\) is the Laguerre polynomial of order \(p\)[15]. The quantities \(W\) and \(R_{c}\) are the width of the Gaussian mode (\(p=0\)), and the wavefront radius of curvature, respectively, are \(z\)-dependent as well as the Gouy phase shift \(\Phi\):  
\[W^{2}(z)=W_{0}^{2}\left[1+\left(\frac{z}{z_{R}}\right)^{2}\right] \tag{2}\]  
\[R_{c}(z)=z\left[1+\left(\frac{z_{R}}{z}\right)^{2}\right] \tag{3}\]  
\[\Phi(z)=Arctg\left(\frac{z}{z_{R}}\right) \tag{4}\]  
The \(LG_{p0}\) beams are made of a central peak surrounded by \(p\) rings of light as shown in Figure 1.  
The Gaussian \(LG_{00}\) beam is characterised by its beam-waist radius \(W_{0}\) and a Rayleigh distance of \(z_{R}=\pi W_{0}^{2}/\lambda\). The beam waist plane defines the origin \(z=0\) of the longitudinal coordinate. The lateral spreading of the \(LG_{p0}\) beams are described by the width \(W_{p}\), based on the second moment radius, and the far-field angular divergence \(\theta_{p}\), given by [16]:  
\[W_{p}(z)=W(z)\sqrt{2p+1} \tag{5}\]  
\[\theta_{p}=\theta_{0}\sqrt{2p+1} \tag{6}\]  
where \(\theta_{0}=\lambda/(\pi W_{0})\) is the angular divergence of the Gaussian beam (\(p=0\)). Another important property of \(LG_{p0}\) beams is that they have the same on-axis (\(\rho=0\)) intensity whatever the mode order \(p\). This contrasts with the usual scale law stating that beam spreading results in a decrease in the on-axis intensity as the beam propagates [17]. Another quantity of importance that characterises the beam quality, called the beam propagation factor, was popularised by [PERSON] [18,19] and is equal to \(M^{2}=(2p+1)\) for an \(LG_{p0}\) beam. This enables us to introduce a fundamental quantity known as brightness, \(B\)[16]  
\[B=\frac{\pi^{2}P}{\lambda^{2}\left(M^{2}\right)^{2}} \tag{7}\]  
The brightness \(B\) describes the potential of a laser beam carrying a power \(P\) and with a beam propagation factor \(M^{2}\) in order to realise high intensities in combination with a large Rayleigh range. Note that a beam with a beam propagation factor of equal to one is Gaussian but the reverse is not true [19,20]. Equation (7) allows us to understand why the studies of higher-order transverse modes slowed after the invention of the laser, on the  
Figure 1: Transverse intensity distribution of the 4-first radial Laguerre–Gauss \(LG_{p0}\) beams. pretext that only the Gaussian \(LG_{00}\) mode is useful because of its higher brightness over the higher-order \(LG_{p0}\) beam. The consequence of this rejection is that most of commercial lasers still deliver a Gaussian beam (GB). However, recent laser studies on higher-order transverse modes have demonstrated numerous advantages in several laser applications. Recently, a review about the advantages and disadvantages of using structured \(LG_{p0}\) beams for certain applications (3D microfabrication, optical tweezers, spherical aberration mitigation) was published [21].