## 4 Results and Discussion  
Figure 3 shows the experimental results which have one-to-one correspondence with the numerical results shown in Figure 1. It can be seen that all experimental patterns agree very well with the numerical results. Such excellent agreement, not only confirms the validity of the present analysis, but also presents a distinctive approach for the generation of structured laser beams with novel amplitude and phase. Moreover, the comprehensive agreement ensures that the derived formula can be exploited to investigate the phase structures of those patterns. The phase structures of the optical patterns can be evaluated from \(\Theta(\rho,\phi;\theta_{n})=tan^{-1}\{Im[\Phi_{Q}(\rho,\phi;\theta_{n})]/\ Re[ \Phi_{Q}(\rho,\phi;\theta_{n})]\}\), where \(Im[\Phi_{Q}(\rho,\phi;\theta_{n})]\) and \(Re[\Phi_{Q}(\rho,\phi;\theta_{n})]\) are the imaginary and real parts of the wave function \(\Phi_{Q}(\rho,\phi;\theta_{n})\). Figure 4 shows the experimental results for the propagation invariant region for the case with \(Q=6\). The second row of Figure 4 displays the \(\Theta(\rho,\phi;\theta_{n})\) for the boxed region shown in the first row of Figure 4. For the cases of \(n=0\) and \(n=3\), there were no obvious phase singularities in the phase distribution. These optical fields do not carry a topological charge. On the other hand, the phase distribution displays phase singularities with a clockwise increasing phase for the cases of \(n=1\) and \(n=2\), corresponding to the vortices with a topological charge \(1\) and \(2\). Although the intensity distribution for \(n=4\) and \(n=5\) are equal to the cases of \(n=2\) and \(n=1\), the phase singularities show a counterclockwise increasing phase, corresponding to the vortices of topological charge \(-1\) and \(-2\). With the wave field \(\Phi_{Q}(\rho,\phi;\theta_{n})\), the current density \(\overset{\rightarrow}{J}\propto\left(\Phi_{Q}{}^{*}\
abla\Phi_{Q}-\Phi_{Q} \
abla\Phi_{Q}{}^{*}\right)\) can be calculated to obtain physical aspects of the vortices. The third row of Figure 4 displays the probability current density for the boxed region which is shown in the first row of Figure 4. Again, the current density does not exhibit any vortex structures for \(n=1\) and \(n=3\), but exhibits various features of vortex lattices for the other cases.  
Figure 2: (**a**) Experimental setup for generating crystalline and kaleidoscope light beams with vortex–lattice structures. (**b**) The charge map and corresponding vortex geometry of the spiral phase plate. (**c**) The photo of steel mask for \(Q=12\) and \(Q=16\).  
Figure 3: Experimental interference patterns that correspond one-to-one with the theoretical results shown in Figure 1.  
Figure 5 shows the result for the case of \(Q=21\). The intensity distribution reveals the characteristics of 21-symmetry kaleidoscopic wave patterns. Once again, the complicated structure of phase singularities can be resolved. As \(n\) increases, the central area exhibits phase singularities with corresponding topological charges and ends up with the eighth order. The experimental results verify that the numerous structures of vortex lattice can be generated in the presented method.  
Figure 4: Experimental crystalline patterns (the first row) for the case of \(Q=6\). The contour plots of the phase fields \(\Theta(\rho,\varphi)\partial_{n})\) (the second row) for the boxed region in the first row. The last row: the probability current density corresponds to the second row.  
Figure 5: Experimental interference patterns and the corresponding phase fields for the case of \(Q=21\) with various relative phases.