## 2 Single-Pass and Multi-Pass Properties of Binary Amplitude Masks
### Absorbing Ring: Single-Pass Properties
In order to determine whether the absorbing ring is able to change the divergence hierarchy, we will calculate the angular divergence \(\theta_{p}\) of the diffracted \(LG_{p0}\) beam upon the ring as follows:  
\[\theta_{p}=\frac{W_{e}}{D} \tag{10}\]  
where the effective width \(W_{e}\), determined in the far-field at a distance \(D\) = 30 m, is based on the second moment [43] of the diffracted intensity distribution \(I_{d}(r,D)\) in plane \(z=D\):  
\[W_{e}^{2}=\frac{2\int\limits_{0}^{\infty}I_{d}(r,D)r^{3}dr}{\int\limits_{0}^{ \infty}I_{d}(r,D)rdr} \tag{11}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Zero\#** & **1** & **2** & **3** \\ \hline (\(L_{1}\))\({}_{\text{min}}\) in \% & 0.002 & & \\ (\(L_{2}\))\({}_{\text{min}}\) in \% & \(7.5\times 10^{-4}\) & 6.2 \(\times\) 10\({}^{-4}\) & \\ (\(L_{3}\))\({}_{\text{min}}\) in \% & 0.0016 & 9.4 \(\times\) 10\({}^{-4}\) & 6.7 \(\times\) 10\({}^{-4}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Minimum value of losses \(L_{p}\) when a ring of width \(\Delta\) = 20 \(\upmu\)m is set on the successive nodes. The transverse intensity distribution \(I_{d}(r,D)=\left|E_{d}(r,z=D)\right|^{2}\) is obtained from the diffracted electric field \(E_{d}(r,z)\), expressed by the well-known Fresnel-Kirchhoff integral:  
\[E_{d}(r,z)=\frac{2\pi}{\lambda z}\int\limits_{0}^{\infty}\tau_{R}(\rho)E_{p}( \rho)\exp\left[\frac{-i\pi\rho^{2}}{z\lambda}\right]J_{0}\left[\frac{2\pi}{ \lambda z}r.\rho\right]\rho d\rho \tag{12}\]  
where \(J_{0}\) is the zero-order Bessel function of first order. \(r\) (\(\rho\)) is the radial coordinate ob the plane \(z\) (ring). The calculation of Equation (12) is carried out with the help of a FORTRAN code based on the numerical integrator _adapag_ from the International Mathematics and Statistical Library (IMSL). The quantity \(\tau_{R}(\rho)\) defines the opaque ring transmission:  
\[\tau_{R}(\rho)= \begin{cases}0&\text{for }\rho_{A}\leq\rho\leq\rho_{B}\\ +1&\text{elsewhere}\end{cases}  \tag{13}\]  
Figure 4a shows that the ring with a width \(\Delta=20\)\(\upmu\)m does not change the hierarchy of divergence expressed by Equation (8), but, as shown in the plot in Figure 4b, when \(\Delta=150\)\(\upmu\)m, the beam divergence is highly impacted so that, for instance, \(\theta_{0}\) and \(\theta_{1}\) move very close to, i.e., the node of \(LG_{10}\). As a consequence of the fundamental mode of the cavity, including an opaque ring (\(Y_{A}=0.707\)), initially desired to be a \(LG_{10}\) mode, could potentially be a \(LG_{00}\) mode. Consequently, it would be wise to use a narrow width opaque ring unless a cavity without a selecting diaphragm is envisaged. The single-pass properties of the opaque ring suggest that positioning the ring inside a cavity on a node of a given \(LG_{p0}\) mode would enable the laser oscillation on the chosen \(LG_{p0}\) mode. However, this could occur only if the other modes are subject to higher losses, thus preventing their oscillation. As it will be shown later, this issue of the oscillation locking onto a single high-order transverse mode can be addressed by considering the transverse mode discrimination expressed as the ratio of losses associated with two neighbouring modes in terms of losses. Here, we can extent this concept to the single-pass properties of the ring by considering the ratio of losses, such as \(L_{0}/L_{1}\), \(L_{2}/L_{1}\), and \(L_{3}/L_{1}\) in a situation when, for instance, the desired mode is \(LG_{10}\). The plots in Figure 5a show \(Y_{A}=0.707\) (\(LG_{10}\) node), where the opaque ring efficiently identifies the undesired modes, i.e., the \(LG_{00}\), \(LG_{20}\), and \(LG_{30}\) modes. Consequently, the ring set inside a laser cavity could force the laser oscillation onto the \(LG_{10}\), since the losses for the other modes are greater than those of \(L_{1}\). Figure 4: Variations in the \(LG_{p0}\) beam divergence diffracted by an opaque ring of width (**a**) \(\Delta=20\)\(\upmu\)m and (**b**) \(\Delta=150\)\(\upmu\)m versus its normalised radius \(Y_{A}=\rho_{A}/W\).