## 2 WTIPM Solutions for LiGaS\({}_{2}\)-Based OPA
### Collinear WTIPM Calculations  
In the following PM calculations, the adopted Sellmeier equations and thermo-optic dispersion for the negative biaxial LiGaS\({}_{2}\) crystal were taken from [29]. According to these equations, we plotted the dependence of both the refractive index \(n\) and its first-order derivative against temperature \(\partial n/\partial T\) on the wavelength from 1 to 10 \(\upmu\)m, as shown in Figure 1c. As far as we know, all current nonlinear frequency conversions with LiGaS\({}_{2}\) crystal adopt a collinear PM (i.e., \(\alpha=\Omega=0\) in Figure 1b); thus, the collinear configuration was firstly evaluated for possible TWIPM solutions. We calculated the dispersions of both \(\partial\Delta k/\partial\lambda_{\mathrm{s}}\) and \(\partial\Delta k/\partial T\) for a LiGaS\({}_{2}\)-based OPA under both Type-I PM in the XZ plane and Type-II PM in the XY plane. The signal wavelengths were scanned from 3 to 10 \(\upmu\)m. Two pump wavelengths of 1030 nm (Figure 2a) and 2060 nm (Figure 2b) were adopted, corresponding to the Yb-based and Ho-based high-power lasers, respectively. To show the wavelength and temperature tolerance, we also calculated the small-signal gain according to Equation (1) by tuning the wavelength and temperature, as shown in Figure 2c-f. In the common pump case of \(\lambda_{\mathrm{p}}=\) 1030 nm (Figure 2a), two WIPM solutions (\(\partial\Delta k/\partial\lambda_{\mathrm{s}}=0\)) were found at \(\lambda_{\mathrm{s}}=\) 7.8 \(\upmu\)m for Type-I PM and 8.6 \(\upmu\)m for Type-II PM under the preset temperature of 323 K (50 \({}^{\circ}\)C). However, there was no TIPM solution (\(\partial\Delta k/\partial T=0\)) in the studied signal wavelength range from 3 to 10 \(\upmu\)m. Therefore, although large wavelength bandwidths of 1.25 \(\upmu\)m (\(\lambda_{\mathrm{s}}=\) 7.8 \(\upmu\)m) and 1.2 \(\upmu\)m (\(\lambda_{\mathrm{s}}=\) 8.6 \(\upmu\)m) could be achieved with the two WIPM solutions, their temperature bandwidths were limited to 72 K and 100 K, respectively (Figure 2c,d). As \(\partial\Delta k/\partial T\
eq 0\), the PM wavelength would change once the temperatures deviated from the preset temperatures. For the type-I PM in the XZ plane (Figure 2c), the PM wavelengths would shift from the WIPM solutions toward short-wavelength and long-wavelength directions with the increase in temperature, while the PM condition was no longer satisfied when the temperature was below the preset temperature. The type-II PM curve in the XY plane evolved in the opposite direction from the type-I PM in the XZ plane (Figure 2d), which originated from the opposite sign of \(\partial\Delta k/\partial T\) for the two WIPM solutions. These calculations suggest that the current 1 \(\upmu\)m-pumped OPA and OPCPA systems with the collinear WIPM solutions are not suitable in high average-power applications. Figure 1: (**a**) Transmission spectrum of a 10 mm-thick LiGaS\({}_{2}\) crystal cited from the free SNLO software [8; 30], which eliminated the influence from the Fresnel reflection. (**b**) Schematic for non-collinear PM in LiGaS\({}_{2}\) crystal. The \(k_{\mathrm{p}},k_{\mathrm{s}}\), and \(k_{\mathrm{i}}\) are the wave-vectors of the pump, signal, and idler, respectively. The PM angle \(\theta\) (\(\varphi\)) is defined by the angle between \(k_{\mathrm{p}}\) and the crystal \(Z\) (X) axis in the XZ (XY) plane. The noncollinear angle \(\alpha\) (\(\Omega\)) refers to the angle between \(k_{\mathrm{s}}\) and \(k_{\mathrm{p}}\) (\(k_{\mathrm{i}}\)). For collinear PM, \(\alpha=\Omega=0\). (**c**) Dispersion of the principle refractive index (solid) and the first-order derivative of principle refractive index against the temperature (dashed) of the LiGaS\({}_{2}\) crystal. Blue, red, and green curves refer to the parameters in the principle axes of X, Y, and \(Z\), respectively.