## 3 Single-Pass and Multi-Pass Properties of a Binary Phase Mask  
### Single-Pass Properties of a Binary Phase Mask  
As mentioned previously, it is possible to force the laser cavity into the fundamental mode, which is an \(LG_{p0}\) mode, by imposing the zeros of intensity by inserting a binary phase mask made up of a transparent material on which is etched a relied, as this gives rise to annular zones which introduce a phase shift equal to 0 or \(\pi\) and create a transmittance equal to +1 or \(-1\). The radii of phase discontinuities (0\(\rightarrow\)\(\pi\)) and (\(\pi\)\(\rightarrow\)0) correspond to the node radii of the desired \(LG_{p0}\) mode, as shown in Table 1. At the position of these phase discontinuities, the electric field passes from a positive to a negative value, and thus, it is necessarily zero at that position. Note that the binary diffractive optical elements constitute a family of diffractive devices, which are the simplest diffractive components to fabricate since they require only one level of etching. Binary diffractive optics have been used very extensively for beam shaping in various wavelength fields (terahertz, infrared, visible, and X-rays) for a very long time [51; 52; 53; 54; 55; 56; 57; 58]. Following a methodology similar to the one used for the amplitude mask, it will be shown below that the mechanism of transverse mode selection differs from that mentioned above for the binary amplitude mask. The annular phase mask (BAPP) described above is made up of several \"elementary bricks\". This elementary brick consists of a CPP made up of a circular phase discontinuity of radius \(R_{PI}\) described by Equation (A14). Obviously, the CPP does not attenuate the incident \(LG_{p0}\) beam since it is transparent. However, the hierarchy of divergence associated with each \(LG_{p0}\) beam is no longer described by the inequality given by Equation (8). In fact, we will use the same method implemented for the absorbing ring when calculating the angular divergence of the beam passing through the CPP not once, but twice, as shown in Figure 14 The reasons justifying this double-pass are given in the Appendix A. Figure 15 shows the set-up considered for determining the variations of the \(LG_{p0}\) beam passing twice through the CPP. The results are shown in Figure 16, which illustrates the variations in the divergence of the \(LG_{p0}\) versus the normalised CPP radius \(Y_{PI}=R_{PI}/W_{0}\)  
where \(W_{0}\) is the width of the incident Gaussian term. It can be seen in Figure 16 that the beam has the smallest divergence (indicated by arrows \(A\), \(B\), and \(C\)) and can be either the \(LG_{30}\), \(LG_{20}\), or \(LG_{10}\) beam, depending on \(R_{PI}\) the CPP radius. A closer look shows that these minima occur when \(R_{PI}\) corresponds to the first zero of intensity of the incident \(LG_{p0}\) beam. More generally, the divergence is also minimum when the CPP radius corresponds to any zero of the \(LG_{p0}\) beam. If we combine the CPP with a diaphragm set in its far-field, as shown in Figure 15, then it would be possible to install a hierarchy of losses susceptible to favouring the transmission of any \(LG_{p0}\) beam for adequate \(R_{PI}\) and diaphragm radius \(R_{D}\). For convenience, we introduce the normalised diaphragm radius \(Y_{D}=R_{D}/W_{z}\), where \(W_{z}\) would be the width of the \(LG_{00}\) beam in the diaphragm plane, i.e., at a distance of \(z=30\) m. The variations in the losses introduced by the diaphragm versus \(Y_{PI}\) were determined for two values of the normalised diaphragm radius, \(Y_{D}=3.5\) (Figure 17a) and \(Y_{D}=6.5\) (Figure 17b).