# Flexible Construction of a Partially Coherent Optical Array
## 3 Results and Discussion
Through our method, various optical coherent arrays can be obtained. The density plots of the normalized intensity and its square modulus of the CDOC of the SCLs after propagation through a thin lens with a focal length of \(f_{1}=400\) mm are shown in Figure 3. The figure shows that the SCLs gradually evolve from a Gaussian distribution to an SCL structure during propagation (see Figure 3(a1-d1,a3-d3)). The evolution of the intensity distribution of the SCLs is similar to the exponential phase beams, with the intensity distribution changing from a symmetric distribution to an asymmetric distribution gradually during propagation. The physical mechanism can be explained by the symmetry-breaking structure of the phase distribution, i.e., the original axial symmetry of the beam in the focal plane is disrupted by the introduced nonuniform spiral phase. Not only does the center of gravity of the intensity shift, but the energy flux distribution in the focal plane also changes significantly [56; 57]. The SCLs have a similar phase distribution to the exponential phase beams, with the phase of the CSD function consisting of the offset superposition phase terms (see Equation (9)). Therefore, discrete (or uniform) and continuous (or nonuniform) exponential phase-like beams can be achieved by modulating the SCLs. The corresponding square modulus of the CDOC is plotted in Figure 3(a2-d2,a4-d4). We found that the square modulus of the CDOC of USCLs in the source plane has a shape similar to a windmill, and  
Figure 2: Density plots of (**a1–d1**) the normalized intensity distribution \(\mathrm{I}(\mathbf{v})/\mathrm{I}(\mathbf{v})_{\mathrm{max}}\) of the incoherent light for generating NUSCLs, (**a2–d2**) the square modulus of the CDOC, and (**a3–d3**) its phase distribution of the CDOC of the NUSCLs with \(\delta_{0}=1\) mm for different spot numbers M.  
the square modulus of the CDOC in windmill shape rotates during propagation, which is interesting. Compared with the USCLs, the square modulus of the CDOC of NUSCLs shows fewer sidelobes, although the square modulus of the CDOC will rotate during propagation, too. The distribution of the square modulus of the CDOC of SCLs evolves from a spiral shape to a Gaussian distribution, and rotation occurs during propagation, with the sidelobes gradually degrading. Compared to the USCLs, the square modulus of the CDOC of an NUSCLs has a pure background, which is helpful when used as the freedom to load information. Two types of spiral coherent lattices have been studied, which included USCLs and NUSCLs, and their arrangement is the same, but there is a difference in the size of the light spot. It should be noted that the SCLs are only an example proposed for our method, and various coherent lattice structures can be generated by this flexible and programmable method. The density plots of the normalized intensity distribution of the SCLs with different transverse coherence widths in the focal plane after passing through a thin lens with a focal length of \(f_{1}=400\) mm are shown in Figure 4. Uniform (see Figure 4(a1-d1)) and nonuniform (see Figure 4(a2-d2)) lattice distributions in the focal plane can be observed gradually with the increase in the transverse coherence width. Additionally, the focal plane intensity distribution of the NUSCLs is more uniform than that of the USCLs with a lower transverse coherence width (see Figure 4(a1,a2)). These phenomena can be explained based on Equation (9). The distribution of the CDOC is determined by the Gaussian term and the Schell-mode term, in contrast to that of the USCLs, and the NUSCLs with the Schell-model terms are modulated by the parameter \(m\), i.e., Schell-model terms decrease with the increase in \(m\), and the phase term is the same as that of the USCLs. The influence of the Gaussian amplitude of the CDOC is small when utilizing the modulation parameter \(m\) with a lower transverse coherence width.