## 3 Results and Discussion
### Experiment  
Figure 7 shows the schematic diagram of the principle-proof experiment for TOD control using a quartz birefringent crystal, according to the arrangement in Figure 1. The spectral phase of short pulse is generally measured by commercial equipment of Frequency-resolved optical gating or Wizzler. However, to measure the GVD and TOD directly with high precision, we used the Wizzler (Wizzler-1030) in the experiment. Considering the Wizzler needs the energy of several microjoules and a pulse duration of nearly FTL, the oscillator is not suitable anymore. Therefore, we used a laser source (Origami-10 XP) with a pulse duration of 400 fs (FTL = 280 fs), 100 kHz, and an energy of 30 uJ at 1039 nm to satisfy the energy and pulse duration demand for the dispersion measurement. The light from the laser passes through the quartz birefringent crystal and polarizers of P1 and P2 using the reflecting mirrors M1 and M2 and is finally directed into the Wizzler. The energy  
Figure 6: (**a**) Maximum amplitude of TOD and FSR changing with the length of the birefringent plate; (**b**) The GVD and TOD change with \(\phi\), in which the length of the plate is 6.5 mm; (**c**) The transmission curve for TOD compensation; (**d**) The GVD and TOD curves changing with wavelength for the TOD compensation parameters.  
of input Wizzler can be adjusted by half waveplate (HWP) and P1. It is worth noting that the laser has inherent GVD and TOD caused by a non-FTL compressed pulse and the seed of a fiber oscillator. To distinguish the GVD and TOD clearly in the experiment, the quartz of length 11.98 mm was used with an environmental temperature of 25 \({}^{\circ}\)C, the angle \(\theta\) was set to 86\({}^{\circ}\) to reduce the effect of reflected light on the laser, and the angle \(\phi\) can be varied in plane by a rotation adjustment mechanism.  
The changes in GVD and TOD with \(\phi\) were measured with Wizzler, and the results are shown in Figure 8a,b, in which the measured GVD and TOD introduced by the quartz crystal are obtained using the difference between the measured values and the inherent dispersion of the laser. The GVD and TOD periodically change with the angle \(\phi\), and the non-symmetrical amplitudes of the peaks and valleys are caused by \(\theta\) not being equal to 90\({}^{\circ}\). To the best of our knowledge, the experimental results for the GVD and TOD caused by birefringent crystals are first demonstrated here. The differences between the experiments and simulations mainly caused the residual GVD of \(-6\times 10^{4}\) fs\({}^{2}\) and the TOD of \(5\times 10^{6}\) fs\({}^{3}\) from the laser, resulting in more deviation, especially when measuring the peaks and valleys for TOD. Figure 8c shows the spectrum and phase with a GVD of \(-6\times 10^{4}\) fs\({}^{2}\) and TOD of \(5\times 10^{6}\) fs\({}^{3}\) from the initial laser without inserting quartz, and Figure 8d shows the GVD of \(-8\times 10^{4}\) fs\({}^{2}\) and TOD of \(-1.3\times 10^{7}\) fs\({}^{3}\) when using the birefringent quartz crystal, according to the valleys in Figure 8b. In addition, the spectrum in Figure 8d is modulated mainly by the longer crystal, which can limit the FSR.  
Furthermore, Compared to the case of spectral intensity shaping by using the birefringent crystal, the initial position of \(\phi=0^{\circ}\) should be confirmed when using the crystal for dispersion compensation. The thickness and temperature should also be precisely processed and controlled to fix the maximum TOD simultaneously. The results suggest that the birefringent crystal can independently and continuously control the TOD through \(\phi\) and can be used to compensate for the TOD in the multi-PW design.  
Figure 7: Schematic diagram of principle–proof experiment for dispersion control, which uses a quartz birefringent crystal and measured by Wizzler; HWP is the half waveplate; M1 and M2 are reflector mirrors; P1 and P2 are polarizers.