## 2 WTIPM Solutions for LiGaS\({}_{2}\)-Based OPA  
### PM Design and Manipulation  
In OPA, a low-frequency signal light (angular frequency \(\omega_{\rm s}\)) can be amplified by a high-frequency pump light (angular frequency \(\omega_{\rm p}\), and \(\omega_{\rm p}>\omega_{\rm s}\)). Governed by the energy conservation law, an idler light is also generated in this process, with an angular frequency of \(\omega_{\rm i}=\omega_{\rm p}-\omega_{\rm s}\). Under the assumption that the pump cannot be depleted by conversion to the signal and idler, the small-signal gain that the signal light experiences is as follows [21]:  
\[\rm G=1+\frac{\Gamma^{2}}{g^{2}}\sinh^{2}(gL), \tag{1}\]  
where \(g=\sqrt{\Gamma^{2}-\left(\frac{\Delta k}{2}\right)^{2}}\), \(\Gamma^{2}=\frac{2\omega_{\rm i}\omega_{\rm j}\omega_{\rm e}^{2}l_{\rm e}}{n \mu_{\rm i}n_{\rm p}l_{\rm e}\omega_{\rm s}^{3}}\), \(\Delta k=k_{\rm p}-k_{\rm s}-k_{\rm i}\) is the wave\(-\)vector mismatch and the \(k_{\rm i}\) depends on the refractive index \(n_{\rm j}\) by \(k_{\rm j}=n_{\rm j}\omega_{\rm j}/c\) (\(c\) is the light speed in vacuum), \(\epsilon_{0}\) is the permittivity of the vacuum, \(l_{\rm p}\) is the pump intensity, and \(d_{\rm eff}\) is the effective nonlinear coefficient. From Equation (1), the small-signal gain is maximized under a perfect PM condition (or momentum conservation condition), where \(\Delta k=k_{\rm p}-k_{\rm s}-k_{\rm i}=0\). If the phase mismatch is large, the small-signal gain will decrease significantly. Hereinafter, the gain bandwidth (\(\Delta\lambda\) denotes wavelength bandwidth, \(\Delta T\) denotes temperature bandwidth) is defined as the full width at half maximum (FWHM) of the gain profile. The necessary PM environment can be created in a nonlinear crystal with the aid of birefringence. In a broadband OPA, however, it is difficult to achieve the perfect PM condition for all the wavelength components because of the dispersion of the refractive index. Such a wavelength-dependent phase mismatch can be expressed by the Taylor expansion of \(\Delta k\) against the signal wavelength \(\lambda_{\mathrm{s}}\),  
\[\Delta k=\Delta k_{0}+\left.\frac{\partial\Delta k}{\partial\lambda_{\mathrm{s}} }\right|_{\lambda_{\mathrm{s}}=\lambda_{\mathrm{s}0}}\Delta\lambda_{\mathrm{s} }+\left.\frac{1}{2}\frac{\partial^{2}\Delta k}{\partial\lambda_{\mathrm{s}}^{2 }}\right|_{\lambda_{\mathrm{s}}=\lambda_{\mathrm{s}0}}(\Delta\lambda_{\mathrm{ s}})^{2}+\frac{1}{6}\left.\frac{\partial^{3}\Delta k}{\partial\lambda_{\mathrm{s}}^{3}} \right|_{\lambda_{\mathrm{s}}=\lambda_{\mathrm{s}0}}(\Delta\lambda_{\mathrm{s}} )^{3}+O\Big{[}(\Delta\lambda_{\mathrm{s}})^{4}\Big{]}, \tag{2}\]  
where \(\Delta k_{0}\) represents the phase mismatch among the central wavelength of three waves, and \(\Delta\lambda_{\mathrm{s}}=\lambda_{\mathrm{s}}-\lambda_{\mathrm{s}0}\) is the wavelength deviation from the central wavelength \(\lambda_{\mathrm{s}0}\). Normally, the condition of \(\Delta k_{0}=0\) can be easily fulfilled by tuning the PM angle. To enlarge the wavelength bandwidth, the first-order derivative term in Equation (2) should be made zero,  
\[\left.\frac{\partial\Delta k}{\partial\lambda_{\mathrm{s}}}\right|_{\lambda= \lambda_{\mathrm{s}0}}=0. \tag{3}\]  
This corresponds to the first-order WIPM condition. In this case, the wavelength bandwidth of PM is determined by the second-order derivative term in Equation (2). The first-order WIPM solutions can be found in collinear configuration and are only limited to several specific central wavelengths [12]. Instead, such first-order WIPM solutions are widespread under a noncollinear configuration (Figure 1b), because the noncollinear interaction introduces more degrees of freedom in the PM manipulation. Therefore, the noncollinear configuration has been widely adopted in broadband OPA and OPCPA systems [22; 23; 24; 25].