# Investigation of Spatial Chirp Induced by Misalignments in a Parallel Grating Pair Pulse Stretcher
## 3 Spatial Chirp Induced by the Misalignments and its Adjustment
### Adjustments Method to Calibrate the Misalignments  
Since no significant coupling effects on the spatial chirp among different misalignments have been observed, the calibration of different misalignments can be carried out individually. According to the sensitivity of the spatial chirp to each misalignment, we have concluded a set of adjustment procedures, as described in Figure 5. The adjustment procedure can be roughly divided into two steps, the coarse adjustment and the fine adjustment. After assembling each element of the stretcher in their appropriate positions as designed, the first step is making sure the incident beams are in the horizontal direction before being projected on grating G1 in the coarse adjustment. Then, we need to adjust grating G2 and mirror M to make the output beam coaxial with the input beam. At this moment, the off-axis deviation of the output pulses can be regarded as zero in both the horizontal and vertical directions. The fine adjustment consists of two sub-steps. The first sub-step is adjusting the vertical chirp of the output pulse. Since the vertical tilt angle \(\varphi_{\mathrm{v}}\) of M can be well adjusted individually, and it contributes little to the vertical chirp compared to \(\theta_{\mathrm{r}}\) according to Table 1, the vertical chirp can be regarded as being induced by the unparallel grooves of the grating pair only. Thus, we can just adjust \(\theta_{\mathrm{r}}\) to make the vertical chirp zero. Then, adjust the vertical tilt \(\theta_{\mathrm{v}}\) of G2 to make the off-axis deviation zero again. The second sub-step of the fine adjustment is adjusting the horizontal chirp. Since the horizontal tilt \(\varphi_{\mathrm{h}}\) of M contributes to the horizontal chirp much more than the horizontal tilt \(\theta_{\mathrm{h}}\) of G2 does when the off-axis deviation is zero, we could adjust \(\varphi_{\mathrm{h}}\) first to zero the horizontal chirp, and then adjust \(\theta_{\mathrm{h}}\) to make the off-axis deviation zero again. Repeat the above adjustments of \(\varphi_{\mathrm{h}}\) and \(\theta_{\mathrm{h}}\) until  
Figure 4: The spatial chirp of the output pulses induced by the stretcher misalignments. (**a**) The horizontal dispersion when only \(\theta_{\mathrm{h}}\) and \(\varphi_{\mathrm{h}}\) exist, (**b**) the vertical dispersion when only \(\theta_{\mathrm{v}}\), \(\theta_{\mathrm{r}}\), and \(\varphi_{\mathrm{v}}\) exist, (**c**) the dispersion when the five misalignments exist simultaneously. The simulation parameters used in (**a**)–(**c**) are identical to those in Table 1. (**d,e**) are the spot profiles of a stretched Gaussian pulse with a diameter of 2 mm when the view screens are 0.5 and 3.0 m away from the first grating G1, respectively. (**f,g**) are the same as (**d,e**), except the diameter of the pulse is 5 mm. VS1 and VS2 denote the position of the view screen is 0.5 and 3.0 m away from the first grating, respectively. The dimensions of (**a**)–(**c**) are \(2\times 2\) mm, while the dimensions of (**e**)–(**g**) are \(10\times 10\) mm. both the spatial chirp and the off-axis deviation are zeros. Then, the off-axis deviation and the spatial chirp are zeros in both the vertical and horizontal directions, and the stretcher can be regarded as being well adjusted. Since the spatial chirp for the beam with a large diameter is difficult to be distinguished by the naked eye, an imaging spectrometer can be used for more precise real-time detection in the fine adjustment phase, which will be shown in Section 4.1. For our home-made single grating stretcher shown in Figure 1b, each of the two roof mirrors consists of two plane mirrors. The non-perpendicularity between the two mirrors in HRM and VRM can be equivalent to the horizontal tilt of G2 and the vertical tilt of M in Figure 1a, respectively. The vertical tilt of HRM can be equivalent to the vertical tilt of G2 in Figure 1a, while the horizontal tilt of VRM can be equivalent to the horizontal tilt of M in Figure 1a. The grating rotation error for the two structures in Figure 1 are the same.