## 2 Single-Pass and Multi-Pass Properties of Binary Amplitude Masks
### Multi-Pass Properties of Absorbing Rings
The ratio \(Y_{R}=(Y_{A}+Y_{B})/2=\rho_{R}/W_{c}\) is varied by changing the geometrical parameter \(g=(1-L/R)\) by adjusting the cavity length. The ratio \(Y_{R}\) is set as equal to the first zero of the three first \(LG_{p0}\) modes (\(p=1\), \(2\), and \(3\)) and the normalised diaphragm radius noted \(Y_{c}\) is adjusted so that the fundamental mode loss \(L_{FM}\) is the same for the different values of \(Y_{R}\), insofar as is possible. The results are shown in Table 3, which displays the cavity length \(L\), \(Y_{R}\), \(Y_{c}\), \(L_{FM}\), the transverse mode discrimination \(F_{c}\), and the \(M^{2}\) factor of the beam emerging from the plane mirror. All these parameters are defined in the Appendix A. The transverse intensity profiles of the cavity fundamental mode, selected with the parameters given in Table 3, in the far-field region (\(z=10\) m) are shown in Figure 10 (blue curves). The plots in red colour represent the intensity distributions of the pure \(LG_{p0}\) beam. Figure 9: (**a**) Gaussian mode radius \(W_{p}\) (\(W_{c}\)) on the plane (concave) mirror versus the geometric parameter \(g\). (**b**) Gaussian mode derivative radius \(dW_{p}/dg\) (\(dW_{c}/dg\)) versus the geometric parameter \(g\). One can note that the fundamental mode in Figure 10a is very close to the perfect \(LG_{10}\) mode, while it slightly moves away from a pure \(LG_{p0}\) beam as \(p\) increases, although not very much. At this point, the opaque ring is set against the plane mirror and the diaphragm is set close to the concave mirror, playing the role of output (Figure 11). This configuration of cavity is very close to an experiment setup used to control the transverse mode of a solid-state laser using an amplitude mask made up of concentric absorbing rings [27]. The characteristics of the cavity are as follows: \(L=260\) mm; \(R=300\) mm; \(\lambda=1064\) nm; \(g=0.133\); and \(W_{p}=186\)\(\mu\)m.  
Figure 11: Scheme of the plano-concave cavity, including an amplitude mask on the plane mirror and a circular diaphragm on the output concave mirror of radius of curvature, \(R=300\) mm. The cavity length  
Figure 10: The plots in blue colour represent the transverse intensity distribution of the cavity fundamental mode. The calculations have been made up for the parameters of Table 3 in the far-field region at a distance \(z=10\) m from the cavity of Figure A2 (out#1). The plots in red colour correspond to a pure \(LG_{p0}\) mode. (**a**) \(L=93\) mm, (**b**) \(L=124\) mm, and (**c**) \(L=137\) mm. is constant and equal to \(L=277\) mm. This cavity configuration is close to that used in the experiment [27], and the parameters are as follows: \(L=260\) mm; \(R=300\) mm; \(\lambda=1064\) nm; \(g=0.133\); \(W_{p}=186\) um. Before considering the cavity shown in Figure 11, which contains an amplitude mask set against the plane mirror and is made up of one, two, or three absorbing rings, we will examine the role of the presence or the absence of the diaphragm on the selection of a \(LG_{p0}\) mode with one absorbing ring. For this study, we considered a cavity with different parameters: \(L=600\) mm; \(g=0.5\); \(\lambda=1064\) nm; and \(W_{\rm p}=0.45\) mm. Initially, we chose this half-confocal configuration (\(g=0.5\)) because it corresponds to the maximum of the transverse mode discrimination [44]. In order to force the laser cavity to exhibit the fundamental mode, i.e., an \(LG_{p0}\) mode, it is not sufficient to set the opaque ring on one of the zeros of the polynomial \(L_{p}\). Indeed, doing that would in principle introduce more losses to the other modes than the desired \(LG_{p0}\) mode, as shown in Figure 3. However, as is shown in Figure 4a, the hierarchy of mode divergence is not disturbed by the presence of the absorbing thin ring. Consequently, the losses introduced by the diaphragm could be in competition with the action of the ring so that the fundamental mode could remain the usual Gaussian beam with \(M^{2}=1\).