## 2 Material and Theories
For Type II EPM, the low-RI signal (or idler) beam (\(\rho_{1}^{(l)}\)) is always placed between the low-RI pump beam (\(\rho_{2}^{(l)}\)) and the high-RI idler (or signal) beam (\(\rho_{1}^{(h)}\)), as can be seen from Equation (7) [24]. In \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Crystals** & \(\boldsymbol{d_{15}}\) & \(\boldsymbol{d_{31}}\) & \(\boldsymbol{d_{24}}\) & \(\boldsymbol{d_{32}}\) & \(\boldsymbol{d_{33}}\) \\ \hline KTP [21] & 2.02 \(\pm\) 0.15 & 2.01 \(\pm\) 0.07 & 3.75 \(\pm\) 0.07 & 3.75 \(\pm\) 0.07 & 15.4 \(\pm\) 0.2 \\ RTP [21] & 1.98 \(\pm\) 0.34 & 2.05 \(\pm\) 0.07 & 3.98 \(\pm\) 0.39 & 3.82 \(\pm\) 0.1 & 15.6 \(\pm\) 0.3 \\ RTA [21] & 2.17 \(\pm\) 0.20 & 2.25 \(\pm\) 0.07 & 3.92 \(\pm\) 0.15 & 3.89 \(\pm\) 0.08 & 15.9 \(\pm\) 0.3 \\ KTA [21] & 2.30 \(\pm\) 0.05 & 2.30 \(\pm\) 0.05 & 3.64 \(\pm\) 0.34 & 3.66 \(\pm\) 0.08 & 15.5 \(\pm\) 0.3 \\ CTA [22] & 2.1 \(\pm\) 0.41 & 2.1 \(\pm\) 0.4 & 3.4 \(\pm\) 0.71 & 3.4 \(\pm\) 0.7 & 18.1 \(\pm\) 1.8 \\ \hline \hline \end{tabular}  
* Reasonably, assuming that Kleinman symmetry is valid, \(d_{24}=d_{32}\) and \(d_{15}=d_{31}\).  
\end{table} 
Table 1: Non-zero nonlinear optic coefficients of 5 KTP isomorphs at 1064 nm (in pm/V). this case, the walk-off angle \(w\) can be defined as the largest angle between the Poynting vectors of the interacting photons:  
\[\cos w=\cos\rho_{1}^{(h)}\cos\rho_{2}^{(l)}. \tag{13}\]  
As expected in Equation (13), \(w\) is also the 3-variable function for \(\lambda_{p}\), \(\theta\), and \(\varphi\), which can be obtained by substituting the set of solutions of \(\lambda_{p}\), \(\theta\), and \(\varphi\) that satisfies Type II EPM. Then the maximum deviation between the interacting beams after passing through a crystal of length \(L\) can be expressed as:  
\[\Delta=L\tan w. \tag{14}\]  
The construction of the signal-idler joint spectral amplitude (JSA) and the calculation of the purity of biphoton state via Schmidt decomposition is a good way to quantify the heralded-state spectral purity of SPDC output. The biphoton state \(|\psi\rangle\) generated from SPDC can be expressed as,  
\[|\psi\rangle=\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{s}d\omega_{i}f(\omega_ {s},\omega_{i})\hat{a}_{s}^{\dagger}(\omega_{s})\hat{a}_{i}^{\dagger}(\omega_ {i})|0\rangle|0\rangle, \tag{15}\]  
where \(\hat{a}_{s}^{\dagger}\) and \(\hat{a}_{i}^{\dagger}\) denote the creation operators of the signal and idler photons, respectively, and \(\omega_{s}\) and \(\omega_{i}\) are the corresponding frequencies [25]. Here, the normalized correlation function \(f(\omega_{s},\omega_{i})\), representing the biphoton JSA, is expressed as the product of pump envelop function and the PM function as follows:  
\[f(\omega_{s},\omega_{i})=\alpha(\omega_{s},\omega_{i})\varphi(\omega_{s}, \omega_{i}). \tag{16}\]  
Assuming a pump with Gaussian spectral shape, the pump envelope (PE) function can be written as,  
\[\alpha(\omega_{s},\omega_{i})\propto\exp\left[-\frac{(\omega_{s}+\omega_{i}- \omega_{p})^{2}}{\sigma_{p}^{2}}\right], \tag{17}\]  
where \(\omega_{p}\) and \(\sigma_{p}\) represent the center frequency and bandwidth of the pump, respectively. The PM function is given in the form of a sinc function as shown below,  
\[\varphi(\omega_{s},\omega_{i})\propto\text{sinc}\left(\frac{\Delta kL}{2} \right), \tag{18}\]  
where \(\Delta k\) represents the phase mismatch defined as \(\Delta k=|\,k_{p}-k_{s}-k_{i}|\). For the given \(k_{p}\)-direction (i.e., \(\theta\) and \(\varphi\)) that satisfies the EPM, Equation (16) is given as a function of the signal and idler wavelengths (or \(\lambda_{s}\) and \(\lambda_{i}\)), thus the JSA can be plotted on a two-dimensional plane as a function of \(\lambda_{s}\) and \(\lambda_{i}\).