## 2 Single-Pass and Multi-Pass Properties of Binary Amplitude Masks
### Absorbing Ring: Single-Pass Properties
The same reasoning can be applied to the \(LG_{20}\) mode by considering the variations in the loss ratios \(L_{0}/L_{2}\), \(L_{1}/L_{2}\), and \(L_{3}/L_{2}\), as functions of \(Y_{A}\), and the normalised ring radius. Similarly to the \(LG_{20}\) mode, it can be that the loss ratios \(L_{0}/L_{2}\), \(L_{1}/L_{2}\), and \(L_{3}/L_{2}\) display a pair of peaks when \(Y_{A}\) is varied, and results are shown in Figure 5b. These two peaks are centred on the two \(LG_{20}\) nodes, i.e., \(Y_{A}=0.54\) and \(Y_{A}=1.3\). Forcing a laser cavity to oscillate on a single-high-order transverse mode clearly results in the difficulty of averting the risk of the oscillation of the Gaussian \(LG_{00}\) mode. In terms of this, when enabling the \(LG_{20}\) mode, the plots in Figure 5b suggest that the ideal position of the opaque ring is on the first node, i.e., \(Y_{A}=0.54\). When considering the oscillation of the \(LG_{10}\) mode, however, the situation is quite the opposite, since setting the ring on the second node allows a better discrimination against the \(LG_{10}\) mode. Let us now perform the same analysis concerning the \(LG_{30}\) mode. Figure 5c represents the variations in the loss ratios \(L_{0}/L_{2}\), \(L_{1}/L_{2}\), and \(L_{3}/L_{2}\) and logically presents three peaks centred on the three nodes of the desired \(LG_{30}\) mode, i.e., \(Y_{A}=0.45\), \(1.07\), and \(1.77\). It can be seen that setting the ring on the first (third) node, results in the bad discrimination of the \(LG_{20}\) (\(LG_{00}\)) mode, since the loss ratio \(L_{2}/L_{3}\) (\(L_{0}/L_{3}\)) is low. Finally, in order to favour the \(LG_{30}\) mode, the second node (\(Y_{A}=1.07\)) seems to be the best position for setting the opaque ring, since the undesired modes (\(p=0\), \(1\), and \(2\)) are well discriminated against. Figure 5: Variations in the loss ratios as a function of \(Y_{A}\), the normalised ring radius. Now, it is important to check whether there is the possibility of improving the discrimination of undesired transverse modes by introducing a second opaque ring in order to favour the \(LG_{20}\) mode. The first step of the study is the calculation of the losses introduced by the two rings shown in Figure 6. Figure 7 shows the variations in the losses \(L_{p}\) introduced by the twin absorbing ring as a function of \(Y_{A}\) while maintaining the ratio \(Y_{C}/Y_{A}\) at a value \(2.41\), which represents the ratio of the position of the two \(LG_{20}\) node positions, i.e., \(2.41=1.3065/0.5411\). In Figure 7, it can be seen that the \(LG_{10}\) beam losses are significantly increased compared to the losses shown in Figure 4, and then an improvement of the discrimination is expected. To confirm this, the loss ratio variations in Figure 8 are investigated. The first peak corresponds to the twin ring positioned on the two nodes of the \(LG_{20}\) beam. It is clear that compared to the plots in Figure 5b, the presence of the second absorbing ring greatly improved the discrimination against the \(LG_{10}\) and \(LG_{30}\) beams. However, nothing changed for the second peak centred on \(Y_{A}=1.30\) since the second absorbing ring positioned at \(Y_{C}=1.30\times 2.41=3.13\) is outside the lateral extent of the \(LG_{20}\) beam. Figure 7: Variations in the losses \(L_{p}=(1-T_{p})\) introduced by two absorbing rings versus the normalised radius \(Y_{A}=\rho_{A}/W\). The incident light is a symmetrical Laguerre–Gauss \(LG_{p0}\) beam of order \(p\). The lowest order \(LG_{00}\) beam has a width \(W=1\) mm.