## 3 Numerical Simulation and Discussion
### Results and Discussion  
In our simulations, the signal and pump were assumed to have the same repetition rates and their pulse energies were fixed at 0.15 mJ and 15 mJ, respectively. The heat load depended on the repetition rate. Due to the lack of experimental data on the allowable tensile stress or fracture temperature for the LiGaS\({}_{2}\) crystal, here, we controlled the temperature difference within the crystal with a maximum of 100 K. Therefore, the pulse repetition rate was set at 5 kHz, corresponding to a pump power of 75 W. In the following, we conducted the simulation by assuming a constant temperature of 293 K in the lateral face of crystal (equivalently, \(h_{1}\) is infinite), and also presented the efficiency evolutions along the crystal under \(h_{1}=400\) W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\) for comparison. The strict WTIPM solution at \(\lambda_{\text{p}}=2.06\) m and \(\lambda_{\text{s}}=5\) m, as shown in Figure 3a, was simulated first. A signal with a very broad signal spectrum around 5 m was seeded (black curve in Figure 5c). Figure 5a shows the temperature distribution within the crystal. The maximum temperature appeared at the front of the crystal because the absorption of the 2.06 m pump light was larger than for the signal and idler (Figure 1a). With the depletion of the pump light, the temperature in the crystal center decreased from 390 K (front surface) to 357 K (back surface). Figure 5b shows the evolution of the conversion efficiency within the crystal. The signal efficiency grew at first and then dropped after reaching a peak of \(\sim\)12% at \(L=10.5\) mm (red curve in Figure 5b). The conversion efficiency was lower than the quantum-limited efficiency of \(\sim\)40% due to the Gaussian pulse profiles and the inherent back conversion effect [36]. In order to judge the influence of the thermal effect on the conversion efficiency, we also simulated the same OPCPA process under \(h_{1}=400\) W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\) (pink curve in Figure 5b) and without considering the crystal absorption (blue curve in Figure 5b). The three efficiency curves overlapped quite well, suggesting that the temperature increase in the crystal had almost no effect on the conversion efficiency. Such behavior can be expected from the large temperature bandwidth of WTIPM solutions. The simulated signal spectra with and without considering the thermal effect were also identical and close to the calculated small-signal gain profile, as shown in Figure 5c. Such a spectral bandwidth was comparable or even larger than that reported in 5 m OPCPA with ZnGeP\({}_{2}\) crystals [16; 17]. After compression, the amplified mid-IR spectrum supported a pulse duration of 59 fs (Figure 5d), corresponding to about 3.5 cycles at 5 m.  
The simulation results achieved for OPCPA with the third-order WIPM solution of \(\lambda_{\text{p}}=1.65\) m and \(\lambda_{\text{s}}=3.3\) m are shown in Figure 6a-d. Due to the similar absorption of the pump, signal, and idler, the temperature distribution within the crystal (Figure 6a) resembled that shown in Figure 5a. The peak conversion efficiency in this case (Figure 6b) was larger than that in Figure 5b due to the larger quantum efficiency. As the third-order WIPM solution is not a strict WTIPM solution, the thermal effect had some influence on the conversion efficiency, especially in the case of \(h_{1}=400\) W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\) (Figure 6b). The narrowing of the amplified signal spectrum due to the thermal effect was minor and insignificant (Figure 6c). The large amplification bandwidth from the third-order WIPM supported a shorter pulse of \(\sim\)16 fs (Figure 6d), corresponding to sub-two cycles at 3.3 \(\mu\)m.  
The amplified mid-IR few-cycle pulses in the above two cases reached \(\
eg 9\) W in average power. Due to the adoption of WTIPM, the thermal dephasing effect was suppressed significantly, as shown in Figures 5 and 6.