## 2 Single-Pass and Multi-Pass Properties of Binary Amplitude Masks
### Multi-Pass Properties of Absorbing Rings

Table 6: Amplitude mask with two absorbing rings, intended to yield the \(LG_{20}\) mode with a propagation factor \(M^{2}=5\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(p\) & \multicolumn{3}{c}{**Radius of the Rings in \(\mu\)m**} \\ \hline
1 & 131 & \\
2 & 100 & 243 & \\
3 & 85 & 199 & 330 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Sizes of the absorbing rings, which correlate to the amplitude masks with one (\(p=1\)), two (\(p=2\)), or three rings (\(p=3\)). \begin{table}
\begin{tabular}{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 & 1.016 & 17\% & 1.23 \\
10 & 3.5 & 1.78 & 7 & 0.4\% & 23.6 \\
15 & 2 & 1.02 & 1.016 & 24\% & 1.12 \\
15 & 3.5 & 1.78 & 7 & 1.5\% & 10 \\
20 & 2 & 1.02 & 1.016 & 31\% & 1.08 \\
20 & 3.5 & 1.78 & 7 & 3.2\% & 5.6 \\
25 & 2 & 1.02 & 1.016 & 37.5\% & 1.05 \\
25 & 3.5 & 1.78 & 7 & 5.7\% & 1.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Amplitude mask with three absorbing rings, intended to yield the \(LG_{30}\) mode with a propagation factor \(M^{2}=7\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\Delta\) & \(Y_{c}\) & \(\rho_{c}\) (mm) & \(M^{2}\) & \(L_{\text{FM}}\) & \(F_{c}\) \\ \hline
10 & 1.65 & 0.84 & 1.08 & 11.5\% & 1.63 \\
10 & 2.5 & 1.3 & 3.018 & 0.25\% & 53 \\
15 & 1.65 & 0.84 & 1.09 & 16.25\% & 1.42 \\
15 & 2.5 & 1.3 & 3.02 & 0.4\% & 23.6 \\
20 & 1.65 & 0.84 & 1.07 & 20.7\% & 1.34 \\
20 & 2.5 & 1.3 & 3.008 & 0.87\% & 11.35 \\
25 & 1.65 & 0.84 & 1.1 & 25\% & 1.14 \\
25 & 2.5 & 1.3 & 3.008 & 1.7\% & 9.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Amplitude mask with one absorbing ring, intended to yield the \(LG_{10}\) mode with a propagation factor \(M^{2}=3\). The following aspects can be extracted from Tables 5-7:
* The selection of a pure high-order transverse \(LG_{p0}\) mode by an amplitude mask made up of \(p\) absorbing rings positioned on the nodes of the desired mode works well provided that the diaphragm is sufficiently open. Otherwise, the fundamental mode becomes the Gaussian mode with \(M^{2}=1\). * The losses of the fundamental mode \(LG_{p0}\) increase with the width \(\Delta\) of the absorbing rings. * The transverse mode discrimination of the cavity decreases with the increase in \(\Delta\). The main conclusion is that it is preferable to use absorbing rings with a small width in order to ensure low losses and a high transverse mode discrimination and to prevent the overlapping of several nodes, leading to a mixing of high-order transverse modes. Note that the intra-cavity generation of selected Laguerre-Gaussian modes of variable radial order, from 0 to 5, has been demonstrated experimentally [27]. The major disadvantage of this method of transverse mode selection is that the selecting mask is enables loss and is questionable for use in a high-power laser, even though the rings are set in beam regions where the intensity is relatively weak. As a consequence, in the next section, a binary phase mask, which is fundamentally transparent and does not possess Fresnel reflections, is used to reach the same objectives. For high-power solid-state laser systems operating on a fundamental mode, a high-order transverse mode imposed by an intra-cavity amplitude or a phase mask, one of the major problems is the thermal effects, which could introduce a certain instability in the mode order through a change in the beam size with pumping power. However, there are several solutions for compensating the thermally induced lens, and these are described in detail in the literature [45; 46; 47; 48; 49; 50].