## 4 Discussion  
In this paper, we considered the use of binary amplitude or phase masks as intra-cavity filters for forcing the fundamental mode of a laser to be a single high-order radial Laguerre-Gauss \(LG_{p0}\) mode. The intra-cavity binary mask aims to impose the zeros of intensity of the oscillating laser beam. As soon as the nodes of the oscillating mode are imposed by the  
Figure 19: Intensity profile of the focused beam emerging from Out#1 and Out#2 of the cavity, including a BAPP#3. The focused beam from output Out#1 (Out#2) is an \(LG_{30}\) (quasi-Gaussian distribution). Figure 20: Variations in the \(M^{2}\) factor of \(LG_{p0}\) beams (\(p=0\), \(1\), \(2\), and \(3\)) passing through a CPP versus \(Y_{PI}\). mask, according to the positions of the zeros of the Laguerre polynomial, then the oscillating mode undergoes the lowest loss: this is a \(LG_{p0}\) mode given by Equation (1). For convenience and clarity, we limited ourselves to \(p=3\), but it may be perfectly feasible to obtain laser oscillation on higher-order transverse modes forced by a binary amplitude or phase mask. It is important to note that the actions of the phase and amplitude masks are very different. The binary amplitude masks made up of absorbing rings induces high losses for all modes except the specific desired \(LG_{p0}\) mode. The cavity does not necessitate any supplementary clipping induced by a circular diaphragm. In this case, it was found that the transverse mode discrimination is high, provided that the rings thickness is small (10-20 \(\upmu\)m). The phase mask is made of a transparent material, glass for instance, on which is etched a relief which results in annular zones that introduce a phase shift equal to 0 or \(\pi\), giving rise to a transmittance equal to +1 or \(-1\). The radius of phase discontinuities (0\(\rightarrow\)\(\pi\)) and (\(\pi\)\(\rightarrow\)0) coincide with the zeros of intensity of the desired \(LG_{p0}\) mode. The action of the binary phase mask is the change in the divergence hierarchy of the \(LG_{p0}\) basis so that the fundamental mode selected by a diaphragm set on the other side of cavity is a single high-order \(LG_{p0}\) mode. It was found that the transverse mode discrimination obtained using the binary phase mask was lower than that obtained using a binary amplitude mask. It could be added that the fabrication of a binary phase mask is more difficult than a binary amplitude mask because of the need to control the etched quantity of glass which defines the phase shift discontinuity (0\(\rightarrow\)\(\pi\)). As was mentioned in the introduction, the \(LG_{p0}\) beams and the redressed \(LG_{p0}\) beams are characterised by a beam propagation factor \(M^{2}=(2p+1)\), which disqualifies them for laser applications needing a high brightness. However, such laser beams have certain qualities which the Gaussian beam does not have [16]. In particular, a rectified \(LG_{p0}\) beam is particularly useful for improving the spatial resolution in 3D-laser prototyping [61] or the longitudinal force of optical tweezers [62]. In this context, it may be noted that the setup shown in Figure 11 has the advantage of enabling the perfect self-alignment of the binary phase mask to achieve beam rectification. This would not be the case if the beam rectification was achieved outside the laser cavity. By contrast, it is useful to note that the output Out#2 through the opaque rings (Figure 11) is a undeformed \(LG_{p0}\) beam if the intracavity beam is a \(LG_{p0}\) beam because we are dealing with zero-field occluding or non-diffracting occluding since the absorbing rings are positioned on nodes. Since, even today, most commercial lasers utilise a Gaussian beam, there is no alternative but to build our own home laser able to deliver a \(LG_{p0}\) beam. For that, there is an important feature to consider which concerns the lateral size of the gain region in the laser medium.