## 3 Single-Pass and Multi-Pass Properties of a Binary Phase Mask
### Single-Pass Properties of a Binary Phase Mask
It can be seen in Figure 17a that a diaphragm that is not open enough does not allow the instigation of the fundamental mode in a cavity, as shown in Figure A2, but could result  
Figure 16: Variations in the divergence \(\theta_{e}=W_{e}/z\), calculated by using Equations (11) and (12), of the diffracted \(LG_{00}\), \(LG_{10}\), \(LG_{20}\), and \(LG_{30}\) beams passing twice through the CPPs versus \(Y_{PI}\), the normalised CPP radius. The arrows A, B, and C indicate the smallest divergence of the \(LG_{10}\), \(LG_{20}\), \(LG_{30}\), respectively. Figure 15: Apparatus allowing to determine the diffracting properties of the cascade of two circular phase plates (CPP) separated by a distance \(2D=10\) mm. The diffracted field is determined in the far-field at a distance \(z=30\) m by using Equation (12). in a high-order transverse mode by setting the phase discontinuity of the CPP on the first node of the desired \(LG_{p0}\) mode. However, for a diaphragm that is wide open (\(Y_{D}=6.5\)), as is seen in Figure 17b, a hierarchy of losses takes place which is similar to that shown in Figure 16. The latter allows the possible oscillation of a \(LG_{p0}\) mode, provided the value of \(Y_{PI}\) is adjusted for minimizing the losses. It has been demonstrated experimentally [59] that a CPP set inside a solid-state laser cavity is able to impose the fundamental mode, which is an \(LG_{p0}\) mode (\(p=1,2,3\)). The discrimination property of the CPP is related to the value of the divergence of the other modes rather than the one targeted by the phase circle discontinuity of the CPP. Table 8 shows that the greater the divergence difference, the more discrimination is expected. It is also interesting to estimate the expected transverse mode discrimination by considering the losses introduced by a diaphragm with a normalised radius equal to \(Y_{D}=6.5\) and located at a distance \(z=30\)m from the cascade of the two CPP. The results are shown in Table 9, which also contains the discrimination factor \(F_{c}\), defined as the inverse ratio of the two first losses. In the next section, we will confirm whether the use of a BAPP, setting a phase circle discontinuity on all nodes of the desired \(LG_{p0}\) mode, improves or does not improve the transverse mode discrimination factor \(F_{c}\). Now, it remains for us to study the single-pass properties of the binary annular phase plate (BAPP) with geometry described by Equation (A15) and by Figure 18, which represents the transmittance profile \(\tau_{DOE}(\rho)\) of the considered BAPP. The phase discontinuities circles of the BAPP correspond to the zeros of intensity given in Table 1.  
Figure 17: Variations in the losses of the diaphragm set at a distance of \(z=30\) m versus \(Y_{PI}=R_{PI}/W\) for (**a**) \(Y_{D}=3.5\) and (**b**) \(Y_{D}=6.5\). The beam incident on the cascade of the CPP is a \(LG_{p0}\) beam whose Gaussian term has a width of \(W_{0}=1\) mm. \begin{table}
\begin{tabular}{c c c c c} \hline \(Y_{PI}\) & \(\theta_{0}\) (rad) & \(\theta_{1}\) (rad) & \(\theta_{2}\) (rad) & \(\theta_{3}\) (rad) \\ \hline
0.707 & \(22.6\times 10^{-4}\) & \(6.1\times 10^{-4}\) & \(12.4\times 10^{-4}\) & \(16.4\times 10^{-4}\) \\
0.541 & \(23.5\times 10^{-4}\) & \(10.9\times 10^{-4}\) & \(7.5\times 10^{-4}\) & \(11\times 10^{-4}\) \\
0.455 & \(23\times 10^{-4}\) & \(14.6\times 10^{-4}\) & \(10\times 10^{-4}\) & \(8.9\times 10^{-4}\) \\ \hline \end{tabular}
\end{table} 
Table 8: Particular values of the divergence of diffracted \(LG_{p0}\) (\(p=0,1,2\), and \(3\)) beams passing twice through the CPP shown in Figure 15. The circle phase discontinuity is set on the first node of the incident \(LG_{p0}\) beam. The minimum divergence is indicated in red.