## 2 Single-Pass and Multi-Pass Properties of Binary Amplitude Masks
### Absorbing Ring: Single-Pass Properties  
The geometry of the considered absorbing ring of interest is shown in Figure 2. The first property of the opaque ring illuminated by a \(LG_{p0}\) beam to be considered is its transmission, labelled as \(T_{p}\), defined as the ratio of transmitted and incident powers:  
\[T_{p}=\frac{\int\limits_{0}^{\rho_{A}}I_{p}(\rho)\rho d\rho+\int\limits_{\rho_ {B}}^{\infty}I_{p}(\rho)\rho d\rho}{\int\limits_{0}^{\infty}I_{p}(\rho)\rho d\rho} \tag{9}\]  
where \(I_{p}(\rho)=\left|E_{p}(\rho)\right|^{2}\) is the intensity distribution of the incident \(LG_{p0}\) beam. The losses introduced by the ring are equal to \(L_{p}=(1-T_{p})\) and are shown in Figure 3a as a function of the normalised ring radius at \(\Delta\) = 20 \(\mu\)m.  
In Figure 3a, it can be seen that the hierarchy formed by the set of losses (\(L_{0},L_{1},L_{2}\), and \(L_{3}\)) changes as the ring diameter changes. In particular, one can observe that the losses introduced by the ring have a minimum value very close to zero whenever the ring is positioned on a node of intensity while the other modes suffer higher losses. This results in interesting modal properties when the absorbing ring is inserted inside a cavity as will be discussed in the next section. Figure 3: Variations in the losses \(L_{p}=(1-T_{p})\) introduced by an absorbing ring versus the normalised radius \(Y_{A}=\rho_{A}/W\). The incident light is a symmetrical Laguerre–Gauss \(LG_{p0}\) beam of the order \(p\). The lowest-order \(LG_{00}\) beam has a width \(W=1\) mm. The ring width is (**a**) 20 \(\upmu\)m, (**b**) 50 \(\upmu\)m, (**c**) 150 \(\upmu\)m.  
Figure 2: Absorbing ring of internal (external) radius \(\rho_{A}(\rho_{B})\) and of width \(\Delta\). discussed later. In order demonstrate that a thin absorbing ring positioned on a \(LG_{p0}\) node results in low losses, in Table 2, we outlined the minima of \(L_{p}\). Usually, the fundamental mode \(TEM_{00}\) (transverse electromagnetic) of the cavity is a Laguerre-Gauss \(LG_{00}\) mode (cylindrical symmetry) or a Hermite-Gauss \(HG_{00}\) (rectangular symmetry), i.e., a Gaussian beam. However, if an opaque ring is introduced inside the resonator, then the fundamental mode \(TEM_{00}\) of the cavity could be a high-order transverse \(LG_{p0}\) mode, since it is the mode with the lowest losses. This is why it is pertinent to talk about a fundamental mode, which is a high-order transverse mode depending on the inserted filter or mask, introducing high losses to all transverse modes except the desired \(LG_{p0}\) mode, which suffers the lowest losses. Since the hierarchy of divergence given in Equation (8) plays an important role in the transverse mode discrimination, we need to verify if the ring is able or unable to change the hierarchy of divergence. However, we will begin by considering the influence of the ring width \(\Delta\) on the loss hierarchy. The plots in Figure 3b,c of losses \(L_{p}\) for \(\Delta\) = 50 \(\upmu\)m (\(\Delta\) = 150 \(\upmu\)m) versus the normalised ring radius shows that the curves exhibit similar profiles except that one obtains higher losses when increasing the ring width \(\Delta\). Hereafter, we will determine the fundamental mode, i.e., the mode with the lowest losses, of a plano-concave cavity, including a circular diaphragm and an opaque ring. It is clear that, if we aim for instance for a \(LG_{10}\) mode, then it would be judicious to set the size of the ring so that \(Y_{A}=0.707\), which corresponds to the \(LG_{10}\) node, as shown by the arrow in Figure 3a-c. However, the ring will have succeeded in overcoming the other transverse modes, but given the losses due to the diaphragm this will not favour them. This is why it is important to examine the variations in the far-field angular divergence \(\theta_{p}\) of the \(LG_{p0}\) beam diffracted by the amplitude ring. The results are shown in Figure 4a,b for \(\Delta\) = 20 \(\upmu\)m (\(\Delta\) = 150 \(\upmu\)m).