## 3 Results and Discussion
### Residual TOD and FOD and Influence on the Multi-PW Design
However, to compensate for the dispersion caused by the transmission materials and amplifier mediums [6], in which the GVD, TOD, and FOD are approximately \(8.8\times 10^{4}\) fs\({}^{2}\), \(1.54\times 10^{5}\) fs\({}^{3}\), and \(-1.69\times 10^{5}\) fs\({}^{4}\). An adjustment compressor is used to compensate for the above GVD; however, this process can introduce extra TOD and FOD simultaneously. The change in the distance for adjusting the compressor gratings, GVD, TOD, and FOD for pulse durations of 180 fs (dotted line in blue) and 80 fs (solid in black line) with adjustable pulse durations are shown in Figure 4a-d. To achieve an FTL compressed pulse, the adjusted compressor needs to afford a GVD of \(-8.8\times 10^{4}\) fs\({}^{2}\) and introduce an additional but the same TOD of \(1.3\times 10^{6}\) fs\({}^{3}\) and FOD of \(-3\times 10^{7}\) fs\({}^{4}\) simultaneously. After compensating for the GVD of the transmission materials, the final TOD and FOD for the existing SG PW laser system are \(1.3\times 10^{6}\) fs\({}^{3}\) and \(-3.9\times 10^{8}\) fs\({}^{4}\) for a pulse duration of 180 fs and \(1.3\times 10^{6}\) fs\({}^{3}\) and \(-3.8\times 10^{7}\) fs\({}^{4}\) for a pulse duration of 80 fs. The values agree with the implementation of pulse duration controlled by the adjustment compressor in the SG PW laser and can provide a basis for the multi-PW design. Figure 3: Absolute value of the GVD, TOD, and FOD for pulse durations of 180 fs and 80 fs, which are stretched to 3.3 ns and compressed to FTL, together with the differences of dispersion between stretcher and compressor. (**a**) The GVD and difference for the pulse duration of 180 fs; (**b**) the TOD and difference for the pulse duration of the pulse duration of 180 fs; (**c**) the FOD and difference for the pulse duration of 180 fs; (**d–f**) are similar as above for the pulse duration of 80 fs. To illustrate the influence of the residual high-order dispersions of the TOD and FOD on the contrast and pulse shapes, Figure 5a shows the intensity contrast for bandwidths of 3 nm, 5 nm, and 7 nm with the TOD of \(1.3\times 10^{6}\) fs\({}^{3}\) and FOD of \(-3.9\times 10^{8}\) fs\({}^{4}\), according to the gain narrowing effect and compressed FTL pulse durations of 388 fs, 233 fs, and 166 fs, respectively. A comparison of the contrast with the compensation of TOD and the residual FOD is shown in Figure 5b; the B integral is neglected in the simulation. Similarly, Figure 5c shows the intensity contrast for bandwidths of 7 nm, 11 nm, and 15 nm, according to the compressed FTL pulse durations of 388 fs, 105 fs, and 77 fs, respectively, and includes the TOD of \(1.3\times 10^{6}\) fs\({}^{3}\) and FOD of \(-3.8\times 10^{7}\) fs\({}^{4}\); only the residual FOD is shown in Figure 5d. From the comparison results of Figure 5, we can see that the residual TOD and FOD had no significant effect on the contrast for the existing SG PW laser with a bandwidth of 3 nm after gain narrowing [18], but high-order dispersion should be considered if the pulse bandwidth is broadened and compressed to the FTL. The simulated result for the bandwidth of 3 nm can afford the measured contrast of \(10^{-8}\), by which there is no additional TOD dispersion adjustment in the SG PW laser. However, when the pulse bandwidth is broadened to over 7 nm, and the duration is shorter than 150 fs in the multi-PW design, the TOD can significantly distort the pulse shape and the pulse is no longer symmetric, which can cause the peak power of the pulse to decrease. Therefore, the TOD must be compensated, but the FOD is negligible. Furthermore, when the pulse duration is less than 100 fs, the FOD should be considered to further improve the intensity contrast, which is consistent with the results obtained in [32]. Figure 4: Distance of adjustment compressor gratings, the GVD, TOD, and FOD change with pulse durations by the adjustable compressor for the pulse durations of 180 fs (solid black lines) and 80 fs (dotted blue lines), respectively. (**a**) Distance; (**b**) GVD; (**c**) TOD; (**d**) FOD.