## 3 Results and Discussion  
### Residual TOD and FOD and Influence on the Multi-PW Design  
As previously mentioned, in the process of chirped pulse amplification, the stretcher, compressor, transmission materials, and amplifier media can introduce different magnitude levels of GVD, TOD, and FOD. While the pulse can be compressed to the near Fourier transform limit (FTL) after GVD is completely compensated, residual high-order dispersions are still present. As shown in Figure 2, the existing SG PW laser system [18] can afford a compressed pulse duration of 1 ps with an energy level of 1 kJ and a bandwidth of approximately 3 nm, which is mainly caused by the gain narrowing effect and system B integral of 1.78. A multi-PW with a 150 fs level and energy of over 0.75 kJ can be achieved by changing the oscillator to a pulse duration of 80 fs, updating the stretcher and compressor, together with a bandwidth extension achieved by spectral shaping after OPCPA with an energy of 1 J, which is similar to [28]. The bandwidth can be extended after compensation for the gain-narrowing effect, which can also decrease the B integral to less than 1 and improve the amplified pulse energy to over 0.75 kJ simultaneously. However, as the pulse duration becomes shorter, the residual high-order dispersions of the TOD and FOD may significantly influence the contrast and pulse shape. For the existing SG PW laser system, a 180 fs hyperbolic secant pulse shape centered at 1053 nm from the oscillator is stretched to 3.3 ns by an eight-pass single-grating Offner stretcher to suit the 6 ns pulse duration of the OPCPA pump. It is then amplified to 2 kJ by the Nd:glass amplifier chain and finally compressed by a four-grating single-pass compressor. In addition, the total length of the Nd:glass and BK7 lenses in the system are approximately 3.35 m and 0.36 m [6]. The GVD can be adjusted using an adjustment compressor set after the OPCPA, which can control the pulse duration from the FTL to 30 ps. To confirm the criterion for high-order dispersion compensation, the residual TOD and FOD for the multi-PW design are first analyzed, and then the influences of the dispersions on the compressed pulse are illustrated. As can be seen in Figure 3, the absolute values of the GVD, TOD, and FOD for pulse durations of 180 fs (Figure 3a-c) and 80 fs (Figure 3d-f) are simulated using the Offner stretcher and compressor design [29; 30; 31] without considering the dispersions of transmission materials and amplifier mediums, together with the differences in dispersion between the stretcher and compressor. The pulse is stretched to 3.3 ns and compressed to the FTL in the simulation. The stretcher affords a positive dispersion of the GVD, and the compressor affords negative values. As illustrated in Figure 3, the GVD introduced by the stretcher and compressor coincide and can be canceled at a central wavelength of 1053 nm, but the residual TOD and FOD are \(6\times 10^{3}\) fs\({}^{3}\) and \(-3.6\times 10^{8}\) fs\({}^{4}\) for a pulse duration of 180 fs, and \(1.2\times 10^{2}\) fs\({}^{3}\) and \(-8\times 10^{6}\) fs\({}^{4}\) for a pulse duration of 80 fs. These are mainly caused by the pulse with a longer duration according to a narrower bandwidth, which demands a larger GVD to achieve the same stretched pulse duration of 3.3 ns, and needs to pass through a longer optical path in the stretcher and compressor, thereby introducing more aberrations for the TOD and FOD. The analysis dispersion results are consistent with implementing a stretcher and compressor for the SG PW laser  
Figure 2: Block diagram of the existing SG PW laser system. with a seed pulse duration of 180 fs. They can also provide a design basis for upgrading to multi-PW. Compared with the pulse duration of 180 fs, the residual TOD and FOD are small in the multi-PW design with a seed pulse duration of 80 fs.