## 3 Simulations and Discussion
As shown in Table 3, the SPDC efficiency is expected to be improved as much as from 29% to 77% in a quantum light source using non-poled KTP isomorphs. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Crystals** & \(\varphi\)[\({}^{\circ}\)] & \(\theta\)[\({}^{\circ}\)] & \(\lambda_{p}\) [nm] & \(\lambda_{s}\) **(or \(\lambda_{j}\))** [nm] & \(\Delta\lambda_{s}\) **(or \(\Delta\lambda_{j}\))** \\  & & & & & **[nm]** \\ \hline KTP [16] & 0–90 & 40.3–49.2 & 941.9–952.2 & 1883.8–1904.4 & 20.6 \\ RTP [19] & 0–90 & 42.7–52.8 & 950.3–965.0 & 1900.5–1930.0 & 29.4 \\ RTA [18] & 0–90 & 45.1–53.9 & 1023.4–1035.3 & 2046.8–2070.5 & 23.7 \\ KTA [17] & 0–90 & 42.3–48.1 & 1032.0–1051.4 & 2064.0–2102.8 & 38.8 \\ CTA [17] & 0–90 & 47.3–69.4 & 1026.1–1034.0 & 2052.2–2068.1 & 15.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: \(\overset{\rightarrow}{k}\)-direction and wavelength ranges of the interacting photons satisfying Type II EPM. Figure 3: \(k\)-directions (\(\varphi\) and \(\theta\)) for Type II EPM and the corresponding resonant pump wavelength (\(\lambda_{p}\)). The EPM is possible for all azimuthal angles (\(\varphi=0\)–90\({}^{\circ}\)), whereas for polar angles (\(\theta\)), only in a specific angular region: (**a**) KTP, (**b**) RTP, (**c**) RTA, (**d**) KTA, and (**e**) CTA. Figure 5 shows the walk-off angles (\(w\)) calculated using Equation (13) under Type II EPM: (a) KTP and RTP and (b) RTA, KTA, and CTA plotted as a function of the EPM \(\lambda_{p}\) values. Over the whole range of \(\lambda_{p}\), the calculated \(w\) spans 1.6-2.8\({}^{\circ}\) for the five crystals. The corresponding beam deviation (\(\Delta\)) per unit millimeter crystal length calculated using Equation (14) spans 27.9-48.9 \(\upmu\)m/mm, which can be sufficiently overcome by a large beam window in thick crystals. In each crystal, the calculated values of \(w\) and \(\Delta\) for the case with the maximum \(d_{\mathit{eff}}\) value are also summarized in Table 3.  
Figure 4: Effective nonlinear optic coefficients (\(d_{\mathit{eff}}\)) calculated numerically under Type II EPM: (a) KTP and RTP; (b) RTA, KTA, and CTA. Figure 5: Walk-off angle (\(w\)) between the interacting beams numerically calculated under Type II EPM: (a) KTP and RTP; (b) RTA, KTA, and CTA. In order to quantify the heralded-state spectral purity of SPDC output, we analyzed the signal-idler JSA properties and calculated the purity of the biphoton state. Figure 6 shows the joint spectral properties of Type II SPDC in the non-poled KTP with the largest \(d_{\mathit{eff}}\) shown in Table 3. Note that in this case, \(k_{\mathit{p}}\) lies on the \(x\)-\(z\) plane (i.e., \(\varphi\) = 0 in Figure 1a), thus the optical alignment is relatively easy compared to the other cases for \(\varphi\
eq\) 0 (Table 2). The density plot of the PM function (Figure 6b) shows typical EPM characteristics as in the PPKTP case [10]. The joint spectral intensity (JSI) calculated for the 3.3-nm bandwidth PE function (Figure 6a) is plotted in Figure 6c. Here, the crystal length of 10 mm is used for calculations, and the JSI is defined as \(\mid\)JSA \(\mid^{2}\). The solid and dashed lines in Figure 6d represent the contour lines for JSI = 0.5 and a circle with a diameter of 11 nm, respectively. The contour line shows a circular shape, and the purity calculated using Schmidt decomposition in Equation (20) is as high as 0.996. Figure 7 shows the JSIs of Type II SPDC in non-poled RTP, RTA, KTA, and CTA with maximum \(d_{\mathit{eff}}\) values (Table 3), where the PE functions of 2.2-nm, 4.05-nm, 4.05-nm, and 3.0-nm bandwidth, respectively, were considered for higher purities. The contours of JSI = 0.5 (blue solid line) for each case shown in Figure 7 are plotted in Figure 8. The red dashed lines shown in Figure 8 represent the circles with diameters of 7.35 nm, 13.6 nm, 13.6 nm, and 10 nm, respectively. All calculation results in Figures 7 and 8 were performed for 10-mm crystal length.