## 3 Simulation Results
The relevant parameters are set as \(n=2\), \(m=0\), \(\omega_{0}=1\)mm, and \(\lambda=532\)nm. If the source beam is assumed to be a quasi-homogeneous planar source [1; 35; 36], then, Figure 2d can be treated as the modulus of the square of the DOC in the far field. Before presenting the simulation results, we want to determine the relationship between the initial coherence width and the size of the soccer pattern. As stated in [1; 10; 43], the source transverse coherence width is inversely proportional to the size of the function of \(p\). Here, we assume that the transverse coherence width is \(\delta_{0}=1/l\) where \(l\) denotes the size of the distribution of \(p(\mathbf{v})\) (radius of the soccer). Figure 3 presents the simulation results of the DOC of the GPCB with different initial coherence width at several propagation distances. The parameters used in the simulation are \(n=2\), \(m=0\), \(\omega_{0}=1\) mm, and \(\lambda=532\) nm. One can see that the DOC pattern can keep the profile similar to that in the source plane (see in Figure 2c), in a range from \(z=0\) to certain propagation distances, but this range is closely dependent on the initial coherence width. When the coherence width is large (\(\delta_{0}=0.64\) nm), the DOC can keep its profile invariant at about 0.7 m (see Figure 3a3-c3). In the case of \(\delta_{0}=0.32\) nm, the profile of the DOC changes much faster compared to that in the case for \(\delta_{0}=0.64\) nm and evolves into a shape similar to that in the far field at distance \(z=1.0\) m (see Figures 2d and 3d1). Figure 2: (**a**) Soccer pattern as a function of \(p(\mathbf{v})\); (**b**) intensity distribution \(I_{0}=|\tau_{0}|^{2}\) of the Hermite–Gaussian beam with \(n=2\), \(m=0\) in the source plane; (**c**) modulus of the square of the source degree of coherence, (**d**) modulus of the square of the Fourier transform of \(I_{0}\). The Fourier transform was performed by FFT in MATLAB. Let us now turn our attention on the influence of the initial coherence width on the DOC pattern of such GPCB in the far field. Figure 4a-c shows our numerical simulations of the DOC distributions at propagation distance \(z=1000\) m (far field) with different values of coherence width. The simulation parameters are the same as those used in Figure 3. For the case of \(\delta_{0}=0.32\) nm (see Figure 4a), the DOC pattern is similar to that shown in Figure 2d, implying that the beam can be be considered as a quasi-homogeneous beam whose DOC in the far field is the Fourier transform of the source intensity distribution, independent of the DOC in the source plane. As the initial coherence width increases, the effects of the source DOC gradually appear. In this situation, the DOC pattern is determined by both the intensity distribution and the DOC function in the source plane. As the coherence width increases, the area of the side lobes becomes large. The enlargement of the side lobes in Figure 4b,c can be explained by the fact that, as the coherence width increases, the globe coherence in our calculation area, which is \(2.0\) mm \(\times\)\(2.0\) mm, increases. As a result, the side lobes are enlarged in the high-coherence case (see in Figure 4b,c) compared with the low-coherence case (see in Figure 4a). It is also found that the distributions of the DOC in Figure 4 are left-right asymmetric, which may be caused by the fact that the \(p\) function (related to the DOC) has a slight left-right asymmetry, as shown in Figure 2a. To confirm our results, we choose as a circular function a p function, which is exactly circularly symmetric. The expression for a circular function is given by  
\[p(\mathbf{v})=\left\{ \begin{array}{l}1,|\mathbf{v}|\leq l\\ 0,|\mathbf{v}|>l\end{array} \right. \tag{15}\]  
where \(l\) is the radius of soccer defined previously. Figure 4d-f presents the corresponding numerical simulations of the DOC distributions at distance \(z=1000\) with the corresponding coherence width. The \(p\) function is shown in Equation (15).