## 4 Experimental Results
With the maximum widths of fringe patterns, \(a\), the six separated circular fringe patterns can be reconstructed and shown in Figure 7a-f. Figure 8a-h shows the eight separated circular fringe patterns from a more complicated mixed signal shown in Figure 3b, based on the similar process shown in Figures 4-6. Finally, the results are shown in Tables 1-3. Figure 7a-f shows the separated fringe patterns, which are reconstructed by using the determined parameters shown in Tables 1-3. In Table 3, we can find that the CC values for most reconstructed signals are higher than 0.746. However, the CC value \(r\) of the reconstructed signal in Figure 7f is only 0.431, which is much lower than the others and could be caused by the higher variability on the determined position and width. Tables 4 and 5 show the center positions and the maximal widths, respectively, of the eight circular fringe patterns separated from Figure 3b. As shown in Table 4, the differences between the original and determined center positions are small and the range is similar to that shown in Table 2. Regarding the difference of the maximal width of the circular fringe patterns shown in Table 5, the error percentages are all less than 3%. The CC values between the eight reconstructed and the original circular fringe patterns are also shown in Table 5, which are, on average, lower than that shown in Table 3. The possible reasons for this are the increased number of mixed circular fringe patterns and the partial overlapping among these patterns. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Result** \\ \hline \(\alpha_{x}\) & (0.14, 0.16, 0.18, 0.20, 0.22, 0.25, 0.37, 0.44, 0.51, 0.53, 0.73, 0.76, 0.76, 0.81) \\ \hline \(\alpha_{y}\) & (0.73, 0.76, 0.79, 0.82) \\ \hline \(\alpha_{\text{best}}\) & (0.73, 0.76, 0.79, 0.82) \\ \hline \(a\) & (45.39, 39.80, 34.42, 29.21) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The determined best orders \(\alpha_{\text{best}}\) and width \(a\) of fringe patterns. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Signal** & **Position of Original** & **Position of Reconstructed** & **Difference** \\  & **Signals** & **Signals** & **Signals** \\ \hline (a) & (0.1, \(-\)0.2) & (0.095, \(-\)0.209) & (0.005, 0.009) \\ \hline (b) & (1.6, \(-\)0.2) & (1.595, \(-\)0.209) & (0.005, 0.009) \\ \hline (c) & (0.6, \(-\)1.6) & (0.594, \(-\)1.613) & (0.006, 0.013) \\ \hline (d) & (1.2,1.2) & (1.206, 1.206) & (0.006, 0.006) \\ \hline (e) & (\(-\)1.5, \(-\)0.8) & (\(-\)1.512, \(-\)0.812) & (0.012, 0.012) \\ \hline (f) & (\(-\)1.3,1.0) & (\(-\)1.344, 1.008) & (0.044, 0.008) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Center positions (\(x_{0}\), \(y_{0}\)) of the fringe patterns separated from Figure 3a. Figure 6: Normalized fractional Fourier spectra \(F_{r}(u,v)\) (left column) and the corresponding filtered results \(G(u,v)\) (right column) with the determined best fractional orders: (**a,b**) \(a_{\rm best}\) = (0.73, 0.73); (**c,d**) \(a_{\rm best}\) = (0.76, 0.76); (**e,f**) \(a_{\rm best}\) = (0.79, 0.79); (**g,h**) \(a_{\rm best}\) = (0.82, 0.82). Figure 9a shows a multiple-fringe pattern, which is the recorded intensity of light field diffracted from three pinholes illuminated by a plane wave in an optical experiment. The wavelength of blue laser illumination is 473 nm. Figure 9b shows the grayscale image of Figure 9a. By using the proposed method, three fringe patterns can be separated from the mixed signal. Figure 10a-c shows the separated circular fringes by using the proposed method, while Figure 10d shows the mixed fringe pattern reconstructed by combining the three separated ones. This example shows that the proposed method can be applied to the separation problem for the real fringe patterns. Figure 8: The eight separated circular fringe patterns (**a–h**) from the mixture signal shown in Figure 3b. Figure 7: The separated and reconstructed fringe patterns (**a–f**) from the mixture signal shown in Figure 3a.