# Investigation of Spatial Chirp Induced by Misalignments in a Parallel Grating Pair Pulse Stretcher
## 2 Design of Chirped Pulse Stretcher  
To design a satisfactory pulse stretcher, we should first review the dispersion theory of the parallel grating pair. Figure 1a is the typical structure of a parallel grating pair stretcher. When an fs pulse propagates through the stretcher, the optical path of BCD can be expressed as:  
\[L=\text{BC}+\text{CD}=L_{0}\bigg{(}\frac{1}{\cos\beta}+\cos\alpha+\sin\alpha \cdot\tan\beta\bigg{)}, \tag{1}\]  
where \(L_{0}\) is the vertical distance between the grating pair, \(a\) and \(\beta\) are the incident angle and diffraction angle of the incident pulse, respectively. Figure 1: (**a**) The typical structure of a parallel grating pair chirped pulse stretcher, (**b**) the optical structure of our home-made stretcher with a single grating. BS, beam splitter; G, grating; M, mirror; \(a\), incident angle; \(\beta\), diffraction angle; \(L_{0}\), the vertical distance between the grating pair, HRM, horizontal roof mirror; VRM, vertical roof mirror. The arrows and numbers in (**b**) denote the light propagation order. The grating diffraction equation is:  
\[\sin\alpha+\sin\beta=10^{-6}md\lambda, \tag{2}\]  
where \(m\) is the diffraction order, \(d\) is the groove density (lines/mm), and \(\lambda\) is the wavelength (nm). Combining Equation (2) and (1) can be rewritten as follows (for \(m=1\)):  
\[L=L_{0}\left(\frac{1+\sin\alpha\left(10^{-6}d\lambda-\sin\alpha\right)}{\sqrt{ 1-\left(10^{-6}d\lambda-\sin\alpha\right)^{2}}}+\cos\alpha\right). \tag{3}\]  
Then, the chromatic dispersion of the parallel grating pair can be expressed as the derivative of \(L\) against the wavelength \(\lambda\):  
\[\frac{\mathrm{d}L}{\mathrm{d}\lambda}=\frac{L_{0}\left(10^{-6}d\lambda-\sin \alpha\right)^{2}\lambda}{\left[1-\left(10^{-6}d\lambda-\sin\alpha\right)^{2} \right]^{3/2}}. \tag{4}\]  
According to Equation (4), we can find the dispersion of the grating pair is proportional to \(L_{0}\). For a certain \(a\) and \(d\), we can adjust its dispersion by only change the vertical distance \(L_{0}\) between the grating pair. Since the pulses will pass through the stretcher twice, the pulse width \(\tau_{\mathrm{c}}\) of the stretched pulse can be expressed as:  
\[\tau_{\mathrm{c}}=\frac{2\mathrm{d}L}{c}\approx\frac{2\cdot L_{0}\left(10^{-6 }d\right)^{2}\lambda_{0}\cdot\Delta\lambda}{c\cdot\left[1-\left(10^{-6}d \lambda_{0}-\sin\alpha\right)^{2}\right]^{3/2}}, \tag{5}\]  
where \(\lambda_{0}\) is the center wavelength of the pulse, and \(\Delta\lambda\) is the band width. To achieve the dynamic parameters in a single shot in a shock compression experiment, a chirped pulse with good linearity and a relatively long duration is needed. Here, we choose the stretch principle of the parallel grating pair to build our own chirped pulse stretcher. The Ti:sapphire laser (Newport Corporation, SOL-ACE35F1K-HP) used in the experiment can output Gaussian limit pulses with about 7 mJ single pulse energy, which have a pulse width of about 35 fs and a band width of about 42 nm, while the center wavelength is about 800 nm. In our home-made stretcher, we use a horizontal roof mirror (HRM) to replace the second grating G2; thus, only one grating is needed, and the stretcher is more compact, as shown in Figure 1b. The vertical roof mirror (VRM) in the stretcher can make the input fs pulse and the output ps pulses propagate in two different heights so that the ps pulses can be directly outputted with a mirror instead of a beam splitter. A gold-coated blazed grating is employed in the stretcher, in which the groove density is 1200 lines/mm and its Littrow angle is 28.7\({}^{\circ}\). To guarantee the diffraction efficiency of the gratings, the incident angle \(a\) is set as 35\({}^{\circ}\). In the single grating structure, the incident pulses will hit the grating four times, and the optical path between two hits is equivalent to \(L_{0}\), as shown in Figure 1a, which is set as 910 mm.