## 3 Simulations and Discussion  
Figure 2 shows the numerical simulation results of the PM and GV matching for Type II EPM: Figure 2a-e correspond to KTP, RTP, RTA, KTA, and CTA in turn. In each graph, the blue and cyan surfaces represent the PM and GV matching calculated using Equations (7) and (8), respectively. The crossing line of two surfaces means the range of \(\overrightarrow{k}\)-direction (i.e., \(\theta\) and \(\varphi\)) satisfying the EPM, which spans a specific wavelength region as expected in Section 2. Figure 3 plots \(\theta\) and \(\varphi\) as a function of the resonant \(\lambda_{p}\), corresponding to the crossing lines in Figure 2. Figure 3a-e correspond to KTP, RTP, RTA, KTA, and CTA, respectively, as in Figure 2. For each KTP isomorph, Type II EPM is possible for all azimuthal angles (\(\varphi\) = 0-90\({}^{\circ}\)), whereas for polar angles (\(\theta\)) only in a specific angular region as plotted in Figure 3. Type II EPM properties calculated for all five crystals are summarized in Table 2 with the references to Sellmeier equations used for the simulations [16; 17; 18; 19]. The spectral ranges of the entangled photon pairs calculated for all five crystals span 1883.8-2068.1 nm, which is still within the silica fiber transparency (e.g., SM2000, 1700 nm to 2300 nm, Thorlabs) [26]. Therefore, quantum light sources based on KTP isomorphs have good potential for quantum communication in the sense that the existing optical communication infrastructure can be used almost as it is. In addition, since absorption lines of various gas molecules (e.g., hydrogen chloride (HCl), ammonium (NH\({}_{3}\)), water vapor (H\({}_{2}\)O), carbon dioxide (CO\({}_{2}\)), and nitrous oxide (N\({}_{2}\)O)) exist in this spectral region, the KTP isomorph-based quantum light sources can be exploited in sensing applications [27]. The spectral ranges of the photon pairs, which can be tuned by sweeping the direction of \(k_{p}\), reach tens of nm as listed in Table 2, which provides flexibility in choosing the wavelength of the quantum light source while maintaining the high spectral purity. Figure 2: Numerical simulation results of the phase matching (PM) and group velocity (GV) matching for Type II extended phase matching (EPM): (**a**) KTP, (**b**) RTP, (**c**) RTA, (**d**) KTA, and (**e**) CTA. In each graph, the blue and cyan surfaces represent the PM and GV matching, respectively. The crossing line of two surfaces means the range of \(k_{p}\)-direction satisfying the EPM. Since the SPDC efficiency is proportional to the square of \(d_{\mathit{eff}}\), it is critical to estimate the effective nonlinearity for the given \(\overset{\rightarrow}{k}\)-direction that satisfies the EPM. Figure 4 shows the \(d_{\mathit{eff}}\) values plotted as a function of \(\lambda_{p}\), which are calculated numerically using Equation (9). The \(\lambda_{p}\) values in the horizontal axes in Figure 4 correspond to the \(\lambda_{p}\) range of Type II EPM listed in Table 2. For each case, the \(d_{\mathit{eff}}\) value is larger at longer wavelengths under Type II EPM. The maximum \(d_{\mathit{eff}}\) values are obtained at \(\varphi=0\) for all five crystals (i.e., the case that \(k_{p}\) lies on the \(x\)-\(z\) plane shown in Figure 1a). These \(d_{\mathit{eff}}\) values at \(\varphi=0\) and their corresponding polar angles \(\theta\) and resonant \(\lambda_{p}\)'s are listed in Table 3. In the PPKTP case, the nonlinear optic coefficient (\(d_{\mathit{CPM}}\)) is given as \((2/\pi)d_{24}\) for the first-order QPM. The simulation results show that the \(d_{\mathit{eff}}\) values are larger for non-poled crystals (2.71-3.18) than for PPKTP (2.39). The ratios of \(d_{\mathit{CPM}}\) to the \(d_{\mathit{eff}}\) values of the non-poled KTP isomorphs are summarized in Table 3. Since the SPDC efficiency is proportional to the square of \(d_{\mathit{eff}}\), \((d_{\mathit{eff}}/d_{\mathit{CPM}})^{2}\) represents the ratio of efficiency that can be increased for each case compared to the case of PPKTP.