## 2 Single-Pass and Multi-Pass Properties of Binary Amplitude Masks
### Multi-Pass Properties of Absorbing Rings
This is confirmed by the plot in Figure 12, which displays a flip in the \(M^{2}\) factor value when the diaphragm radius is changed. Consequently, positioning the absorbing ring on a node is not sufficient for forcing the fundamental mode to be a high-order transverse \(LG_{p0}\) mode. We must be aware of any clipping effect that could favour the Gaussian \(LG_{00}\) mode. On the other hand, in the absence of a diaphragm, the selection of a particular \(LG_{p0}\) mode can be compromised by the phenomenon of transverse mode jump. Indeed, in order to be concrete, let us consider an absorbing ring of width \(\Delta=20\) um set against the plane mirror (Figure 11), intended to impose the first dark ring of the \(LG_{20}\) mode. The cavity has a beam waist radius of \(W_{p}=0.45\) mm. As a consequence, the normalised radius of the ring is characterised by \(Y_{R}=0.54\), \(Y_{A}=Y_{R}-\Delta/(2W_{p})=0.51\), and \(Y_{A}=Y_{R}+\Delta/(2W_{p})=0.56\). In Figure 13, we observe, effectively, a fundamental mode, i.e., an \(LG_{20}\) mode, confirmed by the propagation factor \(M^{2}\approx 5\) and characterised by very a low loss \(L_{FM}=0.1\%\). However, if the beam waist radius \(W_{p}\) suffers a variation of about 6.6% for any reason (thermal lensing, cavity length change), such that the normalised ring radius becomes \(Y_{R}=0.57\) (\(Y_{A}=0.546\), \(Y_{B}=0.593\)), then the selected mode is a high-order transverse mode with \(p=11\), since its second zero is \((\rho/W_{p})=0.57\). Thus, to avoid the phenomenon of transverse mode jump when selecting a high-order transverse \(LG_{p0}\) mode, two precautions must be taken, i.e., a sufficiently small ring width \(\Delta\) and an intra-cavity diaphragm for reducing high losses to the undesired higher-order transverse modes. Instead of the intra-cavity diaphragm, the same result can be obtained by limiting the radial size of the pumped region in the amplifying medium. Another possibility for avoiding the transverse mode jumping is to use an amplitude mask made up of concentric absorbing rings positioned on each node of the desired \(LG_{p0}\) mode, as shown in Figure 14.  
Now, let us consider the performance of the cavity shown in Figure 11 with the parameters \(L=260\) mm; \(R=300\) mm; \(\lambda=1064\) nm; \(g=0.133\); and \(W_{p}=186\) um, as well as an amplitude mask, as shown in Figure 14, made up of one, two, or three absorbing rings that coincide, respectively, with the nodes of the modes \(LG_{10}\), \(LG_{20}\), and \(LG_{30}\). The radii of the rings are given in Table 4.  
Figure 14: Amplitude mask made up of opaque rings coinciding with the zeros of intensity of the desired \(LG_{p0}\) mode, for instance, an \(LG_{30}\) mode. \[W_{c}^{2}=W_{p}^{2}/g \tag{15}\]0 The cavity configuration is shown in Figure 10c The plot shows the variation in the output \(M^{2}\) factor versus \(Y_{c}\), the normalised diaphragm radius, when the opaque ring of width \(\Delta\) = 20 \(\upmu\)m is set on the third zero (\(Y_{R}=1.77\)) of the \(LG_{30}\) mode. \[W_{c}^{2}=W_{p}^{2}/g \tag{15}\]1 Far-field pattern of the fundamental mode of the cavity, including the plane mirror, an absorbing ring characterised by a normalised radius, \(Y_{R}\). In the following section, we will provide the parameters that characterise the transverse mode selection (\(L_{FM}\), \(M^{2}\), and \(F_{c}\)) for a selecting mask made up of one, two, or three absorbing rings with a variable width \(\Delta\). The results are given in Table 5 (one ring), Table 6 (two rings), and Table 7 (three rings). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\Delta\) & \(Y_{c}\) & \(\rho_{c}\) (mm) & \(M^{2}\) & \(L_{\text{FM}}\) & \(F_{c}\) \\ \hline
10 & 2 & 1.02 & 1.03 & 14.5\% & 1.9 \\
10 & 3 & 1.53 & 5 & 0.3\% & 28.4 \\
15 & 2 & 1.02 & 1.03 & 20.5\% & 1.15 \\
15 & 3 & 1.53 & 5 & 0.8\% & 19 \\
20 & 2 & 1.02 & 1.012 & 26.2\% & 1.05 \\
20 & 3 & 1.53 & 5.01 & 1.9\% & 9.2 \\
25 & 2 & 1.03 & 1.03 & 31.7\% & 1.15 \\
25 & 3 & 1.53 & 5 & 3.4\% & 1.28 \\ \hline \hline \end{tabular}
\end{table}