# Investigation of Spatial Chirp Induced by Misalignments in a Parallel Grating Pair Pulse Stretcher
## 3 Spatial Chirp Induced by the Misalignments and its Adjustment
### Spatial Chirp Analysis with the Ray Tracing Simulation Method  
The spatial chirp caused by each one of the above five misalignments can be theoretical calculated [16; 17; 19; 21; 23]. However, due to the complexity of the theoretical equations, the interactions among different misalignments are unknown and are difficult to be analyzed; thus, the overall effects cannot be evaluated when two or more misalignments exist simultaneously. In this work, we use the Zemax software and the ray tracing simulation method to analyze the spatial chirp of the stretched pulses more visually. In the Zemax software, we constructed an ideal parallel grating pair stretcher model, in which parameters are set the same as the designed values in our home-made stretcher described in Section 2. To simulate the spatial chirp induced by different misalignments, we just need to intentionally set the corresponding grating or mirror at a deviation from their designed value. Then, we use the ray tracing method to simulate the off-axis deviations (OFD) and the line dispersion (LD) induced by the misalignments on a view screen that is far away (3000 mm) from the first grating G1. Three different wavelengths \(\lambda_{1}\) = 775 nm, \(\lambda_{2}\) = 800 nm, and \(\lambda_{3}\) = 825 nm are selected to carry out the simulation, which covers the major wavelengths of our fs pulses. First, we investigated the spatial chirp characteristics when each one misalignment exists individually. Figure 3 is the line dispersion between \(\lambda_{1}\) and \(\lambda_{3}\) induced by each misalignment, from which we can conclude that \(\theta_{\rm h}\) and \(\varphi_{\rm h}\) can only cause dispersion in the horizontal direction, \(\theta_{\rm r}\) and \(\varphi_{\rm v}\) can cause mainly dispersion in vertical direction and negligible dispersion in the horizontal direction, while \(\theta_{\rm v}\) cause very small dispersion in both the horizontal direction and vertical direction. It also can be seen that all the dispersion induced by each misalignment is changed approximately linear against the alignment error. Except line dispersion, misalignments can also lead to off-axis deviation. Table 1 lists the full simulation results with one set of misalignments, from which we can conclude that \(\theta_{\rm h}\) and \(\varphi_{\rm h}\) can cause the off-axis deviation in the horizontal direction, while \(\theta_{\rm v}\), \(\theta_{\rm r}\), and \(\varphi_{\rm v}\) can only cause the off-axis deviation in the vertical direction. It can also be read from Table 1 that \(\varphi_{\rm h}\) contributes the most to the line dispersion in the horizontal direction, while \(\theta_{\rm r}\) contributes the most to the line dispersion in the vertical direction. When the five misalignments exist simultaneously, the overall off-axis deviation and the line dispersion approximately equal to the linear superposition of the effects due to these misalignments when they exist individually. Simulation with several sets of misalignments have been carried out, and each set of data can achieve the same conclusions as above, indicating little coupling effects exist among different misalignments. Figure 2: The schematic diagram of the five misalignments of the grating and mirror in the stretcher. G2, the second grating in grating pair stretcher; \(\mathbf{n}\), the normal direction of G2; \(\theta_{\rm h}\), the tilt angle error of grating G2 in the horizontal direction; \(\theta_{\rm v}\), the tilt angle error of grating G2 in the vertical direction; \(\theta_{\rm r}\), the rotation angle error of grating G2 that can lead to unparallel grooves of the grating pair; \(\varphi_{\rm h}\), the tilt angle error of mirror M in the horizontal direction; \(\varphi_{\rm v}\), the tilt angle error of mirror M in the horizontal direction. The arrows denote the positive direction. The values of \(\theta_{\rm h}\), \(\theta_{\rm v}\), and \(\theta_{\rm r}\) in Table 1 are set arbitrarily. In this situation, the off-axis deviation and the line dispersion will exist simultaneously.