# Flexible Construction of a Partially Coherent Optical Array
## 3 Results and Discussion  
In this section, we not only theoretically simulated the evolutionary characteristics of normalized intensity distributions and the square modulus of CDOC distributions of SCLs during propagation using the proposed method in the article, but also verified the feasibility of our method in experiments. Figure 1 shows the normalized intensity distribution \(\mathrm{I}(\mathbf{v})/\mathrm{I}(\mathbf{v})_{\mathrm{max}}\) of the incoherent light, the square modulus, and its phase of the CDOC of USCLs in the source plane with \(\delta_{0}=1\) mm, for different spot numbers M. The incoherent spiral intensity from a discrete distribution gradually turns into a continuous structure, with the increase in the total spots of number M (see Figure 1(a1-d1)). Amazingly, the square modulus of the CDOC distribution in the source plane also shows a spiral shape, and the spiral shape of the CDOC distribution in the source plane becomes continuous as the number of spots M increases for USCLs (see Figure 1(a2-d2)), with its phase consisting of singularity arrays (see Figure 1(a3-d3)) and gradually turning into a continuous shape as M increases. The generation of phase singularities in this article is caused by correlated singularities which are caused by the discontinuity of the light field distribution. This can be seen from Equation (8) in the article. Equation (8) for CDOC distribution is a \(\sum\) sum form, resulting in discontinuity. This is different from the fact that the phase singularity of a typical vortex beam is caused by the generation of the vortex phase. From the figure, we can see that the phase distribution of CDOC of USCLs gradually becomes clear with M increases and we can manipulate the number of phase singularities and distributions by adjusting the total number of M spots. Figure 2 show the normalized intensity distribution \(\mathrm{I}(\mathbf{v})/\mathrm{I}(\mathbf{v})_{\mathrm{max}}\) of the incoherent light, the square modulus, and its phase of the CDOC of NUSCLs in the source plane with \(\delta_{0}=1\) mm, for different spot numbers M; a much more continuous shape can be achieved by manipulating the beam waist size \(\omega_{m}\), showing a great difference between the USCLs and NUSCLs. The incoherent spiral intensity from a discrete distributi  
Figure 1: 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 USCLs, (**a2–d2**) the square modulus of the CDOC, and (**a3–d3**) its phase distribution of the CDOC of the USCLs with \(\delta_{0}=1\) mm, for different spot numbers M.  
continuous structure by manipulating the beam waist size \(\omega_{m}\) and the total spots of number M (see Figure 1(a1-d1) and Figure 2(a1-d1)). The square modulus of the CDOC distribution in the source plane also shows a spiral shape in the source plane which is similar to the USCLs; the spiral shape of the CDOC distribution in the source plane becomes continuous as the number of spots M increases for NUSCLs, compared with the USCLs, and the square modulus of the CDOC distribution of NUSCLs has fewer side lobes (see Figure 1(a2-d2) and Figure 2(a2-d2)), with its phase consisting of singularity arrays, and gradually turns into a continuous shape as M increases. The phase singularities show a clearer distribution than USCLs (see Figure 1(a3-d3) and Figure 2(a3-d3)). The phase distribution of CDOC of NUSCLs gradually becomes clear with M increases, which is similar to USCLs, but phase distributions of the CDOC of NUSCLs are clearer and more compact compared with the USCLs. Additionally, one can not only introduce phase singularities through a vortex phase [54; 55] but also through constructing coherent lattices appropriately, and its phase distribution can be manipulated by the beam parameters, e.g., the spots number M and the beam waist size \(\omega_{m}\), flexibly. The spiral coherent lattices are only an example of our flexible construction of the partially coherent optical array.