## Appendix A
The boundary conditions at the DOE and aperture planes impose a relation between the different coefficients, and their determination involves the matrix \(M\), known as the round-trip operator and expresses the change in the forward coefficients after a round-trip in the apertured cavity:  
\[f^{\prime}_{jp}=\sum_{m}M_{pm}f_{jm} \tag{10}\]  
The matrix \(M\) is defined by its typical element \(M_{1q}\) given by  
\[M_{1q}=(R_{1}R_{2})^{1/2}\exp[-i2\phi_{D}]\sum_{p}C^{A}_{qp}\sum_{n}C^{D}_{l}c^{ A}_{np}\exp[-i4p(\Phi_{D}-\Phi_{A})]\exp[-i(n+q)\Phi_{A}] \tag{11}\]  
where  
\[\Phi_{D}=Arc\tan\left(\frac{\lambda L}{\pi W_{0}^{2}}\right) \tag{12}\]  
\[\Phi_{A}=Arc\tan\left(\frac{\lambda z_{DOE}}{\pi W_{0}^{2}}\right) \tag{13}\]  
\[C^{A}_{pm}=\int\limits_{0}^{\infty}\tau_{DOE}(X)\exp(-X)L_{p}(X)L_{m}(X)dX \tag{14}\]  
\[C^{D}_{pn}=\int\limits_{0}^{2\gamma_{C}^{2}}\exp(-Y)L_{p}(Y)L_{m}(Y)dY \tag{15}\]  
with  
\[E_{bj}(\rho,z)=\exp\{i[k(2L-z)-\omega t]\}\underset{p}{\sum}b_{jp}G_{bp}(\rho, z). \tag{13}\]0  
The complex transmittance \(\tau_{DOE}(\rho)\) associated with the binary circular phase plate, annular phase plate, and spherical aberration takes the following form:  
For the circular phase plate of the radius:  
\[R_{PI}:\rightarrow\;\tau_{DOE}(\rho)= \begin{cases}-1&\text{for}\rho\leq R_{ PI}\\ +1&\text{for}\rho>R_{PI}\end{cases}  \tag{17}\]  
For the binary annular phase plate:  
The binary annular DOE is made up of annular zones introducing a phase shift equal to 0 or \(\pi\), giving rise to a transmittance given by Equation (16):  
\[\tau_{DOE}(\rho)= \begin{cases}-1&\text{for}\;0<\rho\leq\rho_{1}\\ (-1)^{i+1}&\text{for}\;\rho_{i}<\rho\leq\rho_{i+1}\;\text{and}\;(i+1)<p\\ (-1)^{p+1}&\text{for}\;\rho>\rho_{p}\end{cases}  \tag{18}\]  
The position of the phase jumps from 0 to \(\pi\) or \(\pi\) to 0, exactly following the zeros of the Laguerre polynomial \(L_{\text{p}}\) given in Table 1, where \(\rho_{i}\)/\(s\) are the \(p\) radial positions for which the intensity of the desired \(LG_{p0}\) mode is zero. It is important to note that the round-trip operator \(M\) contains the information about reflection at the mirrors and about amplitude (phase) clipping at the edge of the diaphragm (DOE). The resonance condition is expressed by the relation \(f^{\prime}_{jp}=\Gamma f_{jp}\) which holds for all \(p\) after a round-trip. This allows us to consider the eigenmodes of the phase/amplitude apertured cavity as the eigenvectors \(u\) of the matrix \(M\), defined as  
\[Mu=\Gamma u. \tag{10}\]  
where \(\Gamma\) is the complex eigenvalue associated with the eigenvector \(u\). A remarkable eigenvector is that which has the largest eigenvalue, labelled \(|\Gamma_{FM}|\), thus defining the fundamental mode of the cavity whose power loss per round-trip is given by  
\[L_{FM}=1-|\Gamma_{FM}|^{2} \tag{11}\]  
The second eigenvector of \(M\) has the second greatest eigenvalue, labelled as \(|\Gamma_{SEC}|\), and is the second transverse mode whose power loss per round-trip is given by  
\[L_{SEC}=1-|\Gamma_{SEC}|^{2} \tag{12}\]  
From a practical point of view, the cavity with an amplifying medium constitutes a laser which, at the threshold, will oscillate first on the above-mentioned fundamental mode since it has the lowest losses. From this step, increasing the pumping action will lead to the emergence of a second transverse mode. In order for the laser to be able to oscillate on the desired fundamental mode, which could be a high-order transverse \(LG_{p0}\) mode before the second transverse mode begins the oscillation, the difference between \(L_{FM}\) and \(L_{SEC}\) must be as large as possible. This possibility of a cavity to contain a second transverse mode can be described by a transverse mode discrimination factor, defined as  
\[F_{c}=L_{SEC}/L_{FM} \tag{13}\]  
(b) Cavity with a diaphragm and an absorbing ring (Figure 10):  
In this case, as shown in Figure 10, the determination of the resonant field requires knowledge of only two coefficients, \(f_{p}\)/s and \(b_{p}\)/s.