## 2 Numerical Method  
A schematic for when a birefringent plate is used in dispersion compensation, which is similar to spectral modulation, is shown in Figure 1. \(\theta\) is the angle between the incident ray and the crystal surface, which denotes the inclination angle, and \(\phi\) is the angle between the crystal optical axis and the incident surface which denotes the in-plane rotation angle.  
Figure 1: Schematic of the dispersion compensation based on the birefringent plate. P1, P2: Polarizers; HWP: Half-wave plate.  
The electric field of the pulsed laser after the polarizer P2 can be described by \(E_{Out}=E_{In}m_{22}\), where \(m_{22}\) is the Jones matrix element of the birefringent plate [25; 26], which can be expressed as follows:  
\[\begin{split}&\text{M}_{\text{b}}=\left(1-\frac{\cos^{2}\Phi\sin^{2} \theta}{\omega_{0}^{2}}\right)\\ &\times\left[\begin{array}{cc}\left(1-\frac{\sin^{2}\theta}{ \omega_{0}^{2}}\right)\cos^{2}\Phi\exp(i\delta_{e})+\sin^{2}\Phi\exp(i\delta_{ e})&\left(1-\frac{\sin^{2}\theta}{\omega_{0}^{2}}\right)^{1}\sin\Phi\cos\Phi[\exp(i \delta_{e})-\exp(i\delta_{e})]\\ \left(1-\frac{\sin^{2}\theta}{\omega_{0}^{2}}\right)^{1}\sin\Phi\cos\Phi[\exp( i\delta_{e})-\exp(i\delta_{e})]&\left(1-\frac{\sin^{2}\theta}{\omega_{0}^{2}} \right)\cos^{2}\Phi\exp(i\delta_{e})+\sin^{2}\Phi\exp(i\delta_{e})\end{array} \right]\end{split} \tag{1}\]  
where  
\[q=1-\left(\frac{1}{n_{o}{}^{2}}-\frac{1}{n_{e}{}^{2}}\right)\sin^{2}\theta_{ i}\cos^{2}\phi \tag{2}\]  
\[\delta_{o}=\frac{2\pi}{\lambda}d\frac{n_{o}}{\left(1-\frac{\sin^{2}\theta_{i} }{n_{o}^{2}}\right)^{1/2}} \tag{3}\]  
and  
\[\begin{split}\delta_{e}=\frac{2\pi}{\lambda}d\left[n_{e}\left(1 -\frac{\sin^{2}\theta_{i}\sin^{2}\phi}{n_{e}^{2}}-\frac{\sin^{2}\theta_{i}\cos ^{2}\phi}{n_{o}^{2}}\right)^{1/2}-n_{o}\left(1-\frac{\sin^{2}\theta_{i}}{n_{o }^{2}}\right)+\frac{n_{o}}{\left(1-\frac{\sin^{2}\theta_{i}}{n_{o}^{2}}\right) ^{1/2}}\right]\end{split} \tag{4}\]  
The thickness \(d\) is related to temperature according to  
\[d(\text{T})=d_{0}[1+\alpha(\text{T}-\text{T}_{0})] \tag{5}\]  
A quartz birefringent crystal was used in this study. The expansion coefficient of quartz was \(a=13.3667\times 10^{-6}\) at 18 \({}^{\circ}\)C [27]. The refractive indices depend on both wavelength and temperature. For quartz, the following empirical formula was used for the principal refractive indices:  
\[n_{i}=[t_{i}+s_{i}(\lambda-\lambda_{2i})](\text{T}_{0}-\text{T})+\sqrt{m_{i}+ m_{1i}\lambda^{2}/\left(\lambda^{2}-\lambda_{1i}^{2}\right)-k_{i}\lambda^{2}} \tag{6}\]  
where the subscript \(i\) corresponds to o and e, for ordinary and extraordinary, respectively, \(\lambda\) is the wavelength, T is the temperature, and \(t_{i}\), \(s_{i}\), \(m_{i}\), and \(k_{i}\) are parameters that are evaluated from the literature [27].  
The intensity and phase of the output field were obtained using \(I_{Out}=E_{Out}E_{Out}^{*}\) and \(\varphi(\omega){=\tan^{-1}[\text{Im}(E_{Out})/\text{Re}(E_{Out})]}\). The dispersions introduced by the birefringent plate can then be described as  
\[\varphi(\omega)=\varphi_{0}+\varphi_{1}(\omega-\omega_{0})+\frac{1}{2}\varphi_ {2}(\omega-\omega_{0})^{2}+\frac{1}{6}\varphi_{3}(\omega-\omega_{0})^{3}+\ldots \tag{7}\]  
where \(\omega\) and \(\omega_{0}\) are the laser and center frequencies, respectively. \(\varphi_{0}\) is the absolute phase, \(\varphi_{1}\) is the group delay (GD), and \(\varphi_{2}\) and \(\varphi_{3}\) are GVD and TOD, respectively. The dispersion of each order introduced by the crystal can be controlled by the thickness d, inclination angle \(\theta\), in-plane rotation angle \(\phi\), and temperature T.  
According to Equations (1)-(4), the center wavelength of transmission and dispersions can be adjusted by a combination of length, angle, and temperature, which is a periodic function and can be precisely controlled by tuning the temperature of the crystal, especially when the length is fixed after processing.