## 1 Introduction
In addition to this, there are numerous studies on the applications of \(LG_{p0}\) beams in the gravitational wave detectors [22; 23; 24; 25; 26]. First of all, let us clarify the concept of fundamental mode of a laser cavity since it is often confused in the literature with the Gaussian mode. The fundamental mode of a laser cavity is the mode with the lowest losses and is thus susceptible to reaching the laser oscillation first. The objective of this paper is to present a model (see Appendix A) of a laser cavity, including a phase or amplitude mask aiming to force the fundamental mode of the laser cavity to become a pure \(LG_{p0}\) beam. \"Pure\" means that the mode is a single mode, i.e., not a mix of several transverse modes. The amplitude and phase masks impose the position of zeros of the intensity of the desired \(LG_{p0}\) mode. The amplitude mask takes the form of thin annular absorbing rings with a radius which follows closely the location of the Laguerre polynomial zeros given in Table 1 for the three first high-order \(LG_{p0}\) modes. For convenience and clarity, we limited ourselves to \(p=3\), but it may be perfectly feasible to achieve a laser oscillation on higher-order transverse modes when forced by a binary amplitude or phase mask. The use of absorbing rings for forcing a \(LG_{p0}\) mode has been already attempted [27]. For instance, Hermite-Gauss modes have been obtained by inserting straight wires inside a laser cavity aligned along the nodes of the desired mode [28; 29]. In Section 2, we will consider the selection of a transverse \(LG_{p0}\) mode based on the insertion of a binary amplitude mask inside the cavity, which induced high losses in all modes except the specific desired mode. The second way that masks are able to force laser oscillation in \(LG_{p0}\) mode is through a phase mask, and more precisely, a binary phase mask, which is made up of a transparent material etched on annular zones and introduces a phase shift equal to 0 or \(\pi\) and consequently a transmittance equal to +1 or \(-1\). The radii of the phase discontinuities are (0 to \(\pi\)), and (\(\pi\) to 0) corresponds to the node radii of the \(LG_{p0}\) mode given in Table 1. In the following text, the phase mask will be known as a binary annular phase plate (BAPP), and it will be shown in Section 3 that, for the amplitude mask, the mechanism of transverse mode selection differs from that mentioned above. It is important to note that, for several decades, the selection of the fundamental mode of a laser cavity consisted of suppressing all high-order transverse modes except the Gaussian \(LG_{00}\) mode by inserting a circular aperture into the resonator [1; 2; 4; 30; 31; 32; 33; 34]. In this case, the selected \(LG_{00}\) mode is distorted by the diffraction and is no longer Gaussian in shape, as shown theoretically and experimentally [35]. Note that the selection of the fundamental \(LG_{00}\) mode by the circular internal \begin{table}
\begin{tabular}{c c c c} \hline \hline \(p\) & & & \\ \hline
1 & 0.707106 & & \\
2 & 0.541195 & 1.306562 & \\
3 & 0.455946 & 1.071046 & 1.773407 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of ratio \(\rho_{i}/W\) such that \(L_{p}(2\rho_{i}^{2}/W^{2})=0\). aperture is based on the hierarchy of beam divergence of the \(LG_{p0}\) modes expressed, in accordance with Equation (6), by the inequalities  
\[\theta_{0}<\theta_{1}<\theta_{2}<\theta_{3}\ldots \tag{8}\]  
By adjusting the diaphragm opening, it is possible to select the mode with the smallest divergence. In Section 3, it will be shown that the binary phase mask known as the circular phase plate (CPP) is able to change the hierarchy divergence so that the fundamental mode selected by the internal aperture is a high-order \(LG_{p0}\) mode. There are some experimental works on lasers oscillating in a high-order transverse mode selected with the help of a binary phase mask [36; 37; 38].