## 3 Results
### Length of Linear Dispersive Medium  
The change of \(L_{\text{SMF}}\) causes dramatic effects on the NLI patterns. In this test, \(\lambda_{p}\) is fixed (1550.92 nm), and \(L_{\text{DSF}}\) is 100 m. Figure 4a-d show the measured JSI for various SMF lengths. The spectra right and above the JSI in Figure 4a indicate the transmittance spectra of idler and signal filters, respectively. The filter type used in this study is the DWDM filters whose center frequencies are detuned by 400 GHz from the pump frequency, and its 3-dB bandwidth is about 0.6 nm (\(\sim\)75 GHz). Insets in Figure 4a-d show the JSI islands after passing through the signal/idler filters for \(m=1\), 2, 3, and 4, respectively. Note that the \(m=0\) islands in Figure 4b-d is not shown as the \(m=0\) islands end before 200 GHz. Figure 4e shows the differences between the pump center frequency and the peak frequency for \(m=1\), 2, 3, and 4 islands against the SMF length. The maximum island frequency gets closer as the SMF length becomes longer. The solid curved lines are the theoretical predictions using the identical parameters as in Section 3.1 except the DSF lengths and SMF lengths with the phase shift induced in DSF. The frequency difference decreases rapidly as increasing \(L_{\text{SMF}}\). The measured results match well with the theoretical predictions. This fast frequency shift yields the coarse-tuning ability of the center frequency of islands.  
Figure 3: The measured JSI for various dispersion-shifted fiber (DSF) lengths, (**a**) 100 m, (**b**) 150 m, and (**c**) 200 m. (**d**) The plot of the diagonal lines. Dotted lines are measured single stage (\(N=1\), non-interference) JSI case with same length. (**e**) Peak frequency against the DSF length for \(m=1\), 2, and 3. Lines are theoretical predictions extracted from the calculated JSI by using Equation (1) with (solid lines) and without (dashed lines) the phase shift induced in DSF (\(\Delta k_{\text{DSF}}L_{\text{DSF}}\)).  
The maximum island frequency can be centered at the center of the 100-GHz DWDM filter frequency detuned by 400 GHz from the pump frequency if the condition, \(\Delta k_{\text{DSF}}L_{\text{DSF}}+\Delta k_{\text{SMF}}L_{\text{SMF}}=2\) m\(\pi\), can be satisfied [18]. The first, second, third, and fourth islands can be within the signal/idler filter transmittance bandwidth if the SMF length is about 50 m, 90 m, 135 m, and 180 m, respectively. The \(m=1\) island in Figure 4a is elongated diagonally. The JSI bandwidth is wider than the filter bandwidth so that the brightness and collection efficiency, i.e., heralding efficiency, will be degraded. The \(m=4\) island in Figure 4d, however, is almost round, indicating high spectral purity. The spectral purity (\(P\)) can be extracted from the measured data using the Schmidt decomposition method [22]. The extracted \(p\) values from the data are \(p=\) 0.74, 0.85, 0.95, and 0.91 for inset of (a), (b), (c), and (d), respectively. Therefore, we could obtain the photon pairs with high spectral purity by choosing the proper length of SMF.  
Figure 4: The measured JSI for various SMF lengths, (**a**) 50 m, (**b**) 94 m, (**c**) 140 m, and (**d**) 185 m. Spectra right side and above JSI in (**a**) are signal/idler filters (100-GHz DWDM filter (3 dB-bandwidth \(\sim\)80 GHz)). Inset in each JSI indicates the filtered JSI after the signal/idler filters within filter bandwidth. (**e**) Frequency difference against the SMF length for \(m=1\), 2, 3, and 4. Lines are theoretical predictions extracted from the calculated JSI by using Equation (1).