## 2 WTIPM Solutions for LiGaS\({}_{2}\)-Based OPA
### Noncollinear WIPM Calculations  
Here, we tried to find possible WTIPM solutions in a noncollinear OPA. In the collinear mid-IR OPA discussed in Section 2.2, the mid-IR light could either be seeded as a signal or generated as an idler because the two geometries were equivalent for the PM calculation. However, the two geometries were different in noncollinear OPA. If the mid-IR light was the idler of noncollinear OPA, it would be angularly dispersed. To avoid this, we only considered the noncollinear OPA geometry with the mid-IR light seeded as the signal. According to Equations (3) and (5), the noncollinear angle \(\alpha\) could be used to achieve WIPM and TIPM, respectively; however, the two noncollinear angles are usually not equal. The WTIPM solution can only exist under some specific pump and signal wavelengths. Taking Type-I OPA in the XZ plane with \(\lambda_{\text{p}}\) = 2.06 \(\upmu\)m as an example, we calculated the noncollinear angles for WIPM and TIPM, respectively, as shown in Figure 3a. A WTIPM solution was found at \(\lambda_{\mathrm{s}}\) = 5 \(\upmu\)m, where WIPM and TIPM corresponded to the same noncollinear angle of \(\sim\)2.9\({}^{\circ}\). For other signal wavelengths, however, the WIPM and TIPM could not be fulfilled simultaneously under the same noncollinear angle. For the strict WTIPM solution, the OPA could provide a broad gain bandwidth from 4.6 to 5.6 \(\upmu\)m, and could almost maintain the same gain, even when the temperature varied from 0 to 200 \({}^{\circ}\)C (Figure 3b). Such strict noncollinear WTIPM solutions exist in other pump wavelengths, as shown in Figure 3c,d. The WTIPM signal wavelength can be tuned from 2.16 to 2.35 \(\upmu\)m (Figure 3c) and from 4.73 to 5.07 \(\upmu\)m (Figure 3d) when the pump wavelength varies from 1.17 to 2.4 \(\upmu\)m. Note that there is an end to the WTIPM solution at around \(\lambda_{\mathrm{p}}\) = 1.53 \(\upmu\)m, with the noncollinear angle \(\alpha\) decreasing to 0 at this point. As \(\alpha\) < 0 is nonsense, there was not any solution in the short-wavelength range when \(\lambda_{\mathrm{p}}\) > 1.53 \(\upmu\)m (Figure 3c) or in the long-wavelength range when \(\lambda_{\mathrm{p}}\) < 1.53 \(\upmu\)m (Figure 3d). As a result of their large temperature bandwidth, these WTIPM solutions were not sensitive to changes in the preset temperature, as shown in Figure 3c,d. The calculations in Figure 3 show the potential of noncollinear configuration in the design of high-power ultrafast mid-IR OPA and OPCPA. According to Equation (2), the wavelength bandwidth may be further expanded if \(\partial^{2}\Delta k/\partial\lambda_{\mathrm{s}}{{}^{2}}\) = 0 is also fulfilled on the basis of \(\partial\Delta k/\partial\lambda_{\mathrm{s}}\) = 0. We termed this PM second-order as WIPM. Usually, it is more difficult to achieve TIPM and second-order WIPM simultaneously, so we first explored the solution of second-order WIPM and then evaluated  
Figure 3: (**a**) Dispersion of a noncollinear angle corresponding to WIPM (blue curves) and TIPM (red curves) in a LiGaS\({}_{2}\)-based OPA with \(\lambda_{\mathrm{p}}\) = 2.06 \(\upmu\)m, \(T_{0}\) = 323 K. The intersection point (\(\lambda_{\mathrm{s}}\) = 5 \(\upmu\)m, \(\alpha\) = 2.9\({}^{\circ}\), \(\theta\) = 50.9\({}^{\circ}\), \(\varphi\) = 0, and \(d_{\mathrm{eff}}\) = 4.5 pm\(\cdot\)V\({}^{-1}\)) represents the WTIPM solution. (**b**) Small-signal gain as a function of signal wavelength and temperature around the WTIPM point (**a**) with crystal length \(L\) = 10 mm and peak pump intensity \(I_{\mathrm{p}}\) = 5 GW\(\cdot\)cm\({}^{-2}\). (**c**) Signal wavelength solution for WTIPM when the pump wavelength varies from 1.17–1.53 \(\upmu\)m. (**d**) Signal wavelength solution for WTIPM when the pump wavelength varies from 1.53–2.4 \(\upmu\)m. Blue, red, and green curves represent the results at the preset temperatures of 293 K, 323 K, and 353 K, respectively. its temperature bandwidth.