## 2 Material and Theories  
KTP isomorphs are usually expressed by the formula MTiOXO\({}_{4}\), where M can be potassium (K), rubidium (Rb), or cesium (Cs), and X can be arsenic (As) or phosphorus (P) [12]. Here, combinations other than KTP, RTP, KTA, RTA, and CTA are not commonly used because, except for these five, the growth of large and homogeneous bulk crystals is very difficult [13]. The KTP isomorphs have point symmetry of orthorhombic mm2 and exhibit biaxial birefringence. Their crystallographic axes \(a\), \(b\), and \(c\) are all perpendicular to each other and correspond to the optical axes \(x\), \(y\), and \(z\), respectively, as shown in Figure 1a. In this definition, the order of refractive-index magnitudes is given by \(n_{x}<n_{y}<n_{z}\)[14]. A wave vector representing the direction of beam propagation is defined in the spherical coordinate shown in Figure 1a, as follows:  
\[\overset{\rightarrow}{k}=(\sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta). \tag{1}\]  
Here, \(\varphi\) and \(\theta\) are the azimuthal and polar angles, respectively. For Type II SPDC, an input single photon (pump) with a frequency of \(2\omega\) generates a pair of photons (signal and idler) with the same frequency of \(\omega\) that are orthogonally polarized to each other, as illustrated in Figure 1b. When the light is incident into the crystal, the refractive indices (RIs) of two eigen polarization modes for the given \(\overset{\rightarrow}{k}\)-direction can be obtained by solving the Fresnel equation of the wave normal as follows [15],  
\[n_{q\omega}^{(p)}=\sqrt{\frac{2}{-B_{q}\pm\sqrt{B_{q}^{2}-4C_{q}}}}, \tag{2}\]where the definitions of parameters are:  
\[B_{q}=-k_{x}^{2}(b_{q}+c_{q})-k_{y}^{2}(a_{q}+c_{q})-k_{z}^{2}(b_{q}+a_{q}), \tag{3}\]  
\[C_{q}=k_{x}^{2}b_{q}c_{q}+k_{y}^{2}a_{q}c_{q}+b_{q}a_{q}k_{z}^{2}, \tag{4}\]  
\[a_{q}=n_{q\omega,x},\,b_{q}=n_{q\omega,y},\,c_{q}=n_{q\omega,z}. \tag{5}\]  
Here, \(q\) can be 1 or 2, then in the frequency-degenerate SPDC process, \(n_{2\omega}\) and \(n_{\omega}\) represent the RIs of pump photon with the frequency \(2\omega\) and signal (or idler) photon with the identical frequency \(\omega\), respectively. \(k_{x}\), \(k_{y}\), and \(k_{z}\) in Equations (3) and (4) are defined in Equation (1). \(p\) in Equation (2) can be \(h\) or \(l\), representing high or low RI. Each of these corresponds to when taking minus and plus in the \(\pm\) symbol of the denominator in Equation (2). The RIs of all 5 KTP isomorphs (i.e., \(n_{x}\), \(n_{y}\), and \(n_{z}\)) in Equation (5) are found in [16, 17, 18, 19]. When photons of different frequencies pass through the crystal, a temporal walk-off (\(\Delta T\)) between the interacting photons occurs due to their difference in GV. This GV mismatch is defined as \(\Delta T\) per unit crystal length as follows,  
\[\frac{\Delta T}{L}=\frac{\Delta n_{g}}{c}, \tag{6}\]  
where \(L\), \(c\), and \(\Delta n_{g}\) represent the crystal length, the speed of light in vacuum, and the group index difference between interacting photons, respectively. Therefore, when \(\Delta n_{g}=0\), the GV matching is satisfied. Since the phase-matching (PM) condition between the pump, signal, and idler photons is given by \(k_{p}=k_{s}+k_{i}\), the PM and GV matching conditions for Type II SPDC can be simplified as:  
\[2n_{2\omega}^{(l)}=n_{\omega}^{(h)}+n_{\omega}^{(l)}, \tag{7}\]  
\[2n_{g,2\omega}^{(l)}=n_{g,2\omega}^{(h)}+n_{g,2\omega}^{(l)}. \tag{8}\]  
Now Type II EPM conditions are defined as when Equations (7) and (8) are satisfied simultaneously. Each of the equations is a 3-variable function for pump wavelength (\(\lambda_{p}\)), \(\theta\), and \(\varphi\), thus by solving a system of equations (i.e., Equations (7) and (8)) for a given \(\lambda_{p}\), we can find a set of \(\theta\) and \(\varphi\), i.e., the direction of \(\overset{\rightarrow}{k}\).