## 2 Single-Pass and Multi-Pass Properties of Binary Amplitude Masks
### Multi-Pass Properties of Absorbing Rings  
The resonator considered here is plano-concave cavity of length \(L\) and is made up of a plane mirror and a concave mirror with a radius of the curvature as \(R\). Figures A1 and A2 (see Appendix A) show the cavity, including an absorbing ring or a CPP or BAPP on the concave mirror side, and a circular diaphragm on the plane mirror. The method used for the determination of the resonant field is based on its decomposition via Laguerre-Gauss functions (see Appendix A). Moreover, the diaphragm and the absorbing ring or the binary phase mask can be switched. Both situations will be addressed below. On one hand, the sensitivity of the beam width on mirrors with thermal lensing is different, and consequently, so is the \"detuning\" of the targeted zero of the expected \(LG_{p0}\) mode. On the other hand, this provides a verification for the method which forces the fundamental mode to be a \(LG_{p0}\) mode whatever the relative position of the selecting diaphragm and the diffracting object (ring or phase mask). Before proceeding, it is useful to note two geometric parameters of interest which characterise the Gaussian mode of the bare cavity, i.e., without the diaphragm or other diffracting object, namely, the Gaussian mode radius \(W_{p}\) (\(W_{c}\)) on the plane (concave) mirror:  
\[W_{p}^{2}=\frac{\lambda L}{\pi}\bigg{(}\frac{g}{1-g}\bigg{)}^{1/2} \tag{14}\]  
\[W_{c}^{2}=W_{p}^{2}/g \tag{15}\]  
where the cavity geometric parameter \(g=(1-L/R)\) is kept in the stable region (\(0<g<1\)) by varying \(R\), the radius of the curvature of the concave mirror. In the case where the amplifying medium is set in front of the concave mirror, the effect of the thermal lensing can be accounted for through a variable effective radius of curvature. The variations in \(W_{p}\) and \(W_{c}\) versus parameter \(g\) are shown in Figure 9a, and their derivatives are shown in Figure 9b. Note that \(g=0.5\) is a remarkable value since the derivative \(dW_{c}/dg\) is null, and the consequence is that the best position of the diffracting object (ring or phase mask) should be near the concave mirror so as to avoid the phenomena of the transverse mode flip due to a variable thermal lens. Figure 8: Variations in the loss ratios \(L_{0}/L_{2}\), \(L_{1}/L_{2}\), and \(L_{3}/L_{2}\) as a function of the normalised first ring radius. The width of the each ring is \(\Delta=50\) μm. Three quantities (see Appendix A) are calculated in order to characterise the fundamental mode of the cavity, including an absorbing ring and a diaphragm: \(L_{FM}\) is the fundamental mode losses, \(F_{c}\) is the transverse mode discrimination, and \(M^{2}\) is the propagation factor of the output beam. The latter allows us to identify the type of \(LG_{p0}\) beam, which is susceptible to oscillate, as shown in equation \(M^{2}=(2p+1)\), corresponding to a pure \(LG_{p0}\) beam. Since the absorbing ring may only impose one single zero, we are interested in a fundamental mode; specifically, a \(LG_{10}\) mode since it has one node. It will be shown later that the absorbing ring inside the resonator can impose a fundamental mode, which can be a higher-mode order (\(p\geq 2\)). Let us first examine an important point concerning the positioning of the ring relative to the dark ring of the desired \(LG_{10}\) mode. Indeed, when the ring has a certain width \(\Delta\), it is crucial to pay attention to the overlap of the target node which has to be at its middle. In the next section, we will use the ratio \(Y_{R}=(Y_{A}+Y_{B})/2\), which is associated with the normalised radius of the central part of the absorbing ring. In the following text, we will confirm whether a single absorbing ring is able to impose a \(LG_{p0}\) mode as the fundamental mode of a cavity of length \(L\). An absorbing ring with a fixed radius \(\rho_{R}=(\rho_{A}+\rho_{B})/2=180\)\(\upmu\)m is inserted inside the cavity, as shown in Figure A2.