## 3 Single-Pass and Multi-Pass Properties of a Binary Phase Mask
### Multi-Pass Properties of a Binary Phase Mask  
First, we will consider the fundamental mode of the cavity previously shown in Figure 11, with the parameters given in Table 3, in which a CPP with its phase discontinuity \begin{table}
\begin{tabular}{c c c c c c} \hline
**BAPP** & **Loss for \(p=0\)** & **Loss for \(p=1\)** & **Loss for \(p=2\)** & **Loss for \(p=3\)** & \(F_{c}\) \\ \hline \#1 & 7.7\% & 0.19\% & 2.13\% & 3.95\% & 11.2 \\ \#2 & 10.67\% & 10.74\% & 0.48\% & 4.92\% & 10.25 \\ \#3 & 15.2\% & 10.98\% & 11\% & 0.65\% & 16.9 \\ \hline \end{tabular}
\end{table} 
Table 10: Variations in losses induced by the diaphragm of normalised radius \(Y_{D}=6.5\) set at a distance \(z=30\) m from the cascade of the two BAPP. \(F_{c}\) is the discrimination factor defined as the inverse ratio of the two first losses. The minimum in the loss value is indicated in red. Figure 18: Transmittance profile (field ratio) of the BAPP#\(p\) intended to select a \(LG_{p0}\) when inserted in the cavity shown in Figure 11. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Y\({}_{\text{PI}}\)** & **Loss for \(p=0\)** & **Loss for \(p=1\)** & **Loss for \(p=2\)** & **Loss for \(p=3\)** & \(F_{c}\) \\ \hline
0.707 & 7.7\% & 0.19\% & 2.13\% & 3.95\% & 11.2 \\
0.541 & 9.38\% & 1.64\% & 0.26\% & 1.2\% & 4.6 \\
0.455 & 8.9\% & 3.33\% & 0.95\% & 0.35\% & 2.7 \\ \hline \end{tabular}
\end{table} 
Table 9: Variations in losses induced by the diaphragm of normalised radius \(Y_{D}=6.5\) set at a distance \(z=30\) m from the cascade of the two CPP. \(F_{c}\) is the discrimination factor defined as the inverse ratio of the two first losses. The minimum in the loss value is indicated in red. positioned on the first node of \(LG_{p0}\) modes (\(p\) = 1, 2, and 3) is inserted. The results are summarised in Table 11. The fundamental mode can be either a \(LG_{00}\), \(LG_{10}\), or \(LG_{20}\) mode, but with a low transverse mode discrimination. These results are in a good agreement with a previous experiment [59]. It is worth mentioning that the opening or closing of the diaphragm does not allow any transverse mode jumps, as shown above in Figures 12 and 13. Indeed, whatever the diaphragm opening, it is the mode with the smallest divergence which undergoes the lowest losses. The hierarchy of losses is imposed by the hierarchy of divergences, as shown in Figure 16, and remains independent from the diaphragm opening. In order to illustrate this property, we extracted the values of \(LG_{p0}\) divergences when \(Y_{PI}\) is equal to (0.707, 0.541, and 0.455) from Figure 16, and the results are given in Table 8. The values in red colour in Table 8 represent the minimum values of \(LG_{p0}\) beam divergence, and the minimum of losses are shown in Table 9. It now remains to be seen whether the characteristics of the fundamental mode of a plano-concave cavity, including a binary annular phase plate, simultaneously impose the \(p\) nodes of the fundamental mode, thus taking the form of \(LG_{p0}\) mode. The key parameter will mainly be the transverse mode discrimination, TMD, which is expected to be improved in comparison with the TMD obtained using a CPP (Table 11). Now, let us consider the cavity shown in Figure 11 with the same parameters used in Section 2.2 when considering the multi-pass properties of absorbing rings: \(L\) = 260 mm; \(R\) = 300 mm; \(\lambda\) = 1064 nm; \(g\) = 0.133; \(W_{p}\) = 186 \(\upmu\)m; \(W_{c}\) = 260 \(\upmu\)m. The results are shown in Table 12. It can be seen that the selection of a \(LG_{p0}\) mode, as the fundamental mode with low losses, \(L_{FM}\), is effectively carried out by inserting a BAPP#\(p\) since the \(M^{2}\) factor is very close to (\(2p+1\)). It cannot be said that using a BAPP in place of a CPP inside a laser cavity significantly improves the transverse mode discrimination \(F_{c}\). It is important to point out that the far-field of the two cavity outputs shown in Figure 11 are characterised by distinct intensity profiles.