# Numerical Investigation of the Temporal Contrast in ps-OPCPA with Compact Double BBO Arrangement
## 2 Numerical Method  
PSF originates from a quantum fluctuation effect first predicted by [PERSON] with the quantum mechanical model [22]. It can be understood as the spontaneous splitting of pump photons to signal and idler photons and then being amplified along the nonlinear crystal together with seed pulses. As one main type of noise in OPCPA, PSF delivered to the target before the arrival of the main pulse may radically change the conditions of the interactions, so it has gained much attention in past decades [23; 24; 25; 26; 27; 28]. In some work [28; 29], PSF is assumed to propagate along the direction of the smallest phase mismatch because in this case it is amplified most efficiently and becomes considerable noise in OPCPA. Compared to the common non-collinear OPCPA, which eliminates the group velocity dispersion only at the specific wavelength--as depicted in Figure 1a--PSF under the perfect phase-matching assumption (\(\triangle k=0\)) has the feature of angular dispersion--as depicted in Figure 1b. Specifically, the frequency-dependent angle \(\alpha_{f}\) between signal fluorescence and pump pulses, and the angle \(\Omega_{f}\) between signal fluorescence and idler fluorescence can be determined by solving the momentum conservation relationships in non-collinear OPCPA geometry [30]:  
\[\alpha_{f}(\omega_{s},\omega_{i})=\arccos\left(\frac{k_{p}^{2}+k_{s}^{2}-k_{i}^{2} }{2k_{p}k_{s}}\right), \tag{1}\]  
\[\Omega_{f}(\omega_{s},\omega_{i})=\pi-\arccos\left(\frac{k_{i}^{2}+k_{s}^{2}-k_ {p}^{2}}{2k_{i}k_{s}}\right), \tag{2}\]  
where \(k_{p}\), \(k_{i}\), \(k_{s}\) are the wave numbers of the pump, signal and idler, respectively. In addition, as shown in Figure 1b, \(\alpha_{f}^{\prime}\) is the angle between signal fluorescence and signal pulses and \(\Omega_{f}^{\prime}\) is the angle between idler fluorescence and signal pulses, which can be denoted by:  
\[\alpha_{f}^{\prime}(\omega_{s},\omega_{i})=\alpha-\alpha_{f}(\omega_{s}, \omega_{i}), \tag{3}\]  
\[\Omega_{f}^{\prime}(\omega_{s},\omega_{i})=\Omega_{f}(\omega_{s},\omega_{i}) +\alpha_{f}^{\prime}(\omega_{s},\omega_{i}). \tag{4}\]  
It is worth mentioning that actually PSF may have more complicated spatial structures as a result of its spontaneous generation. The assumptions that PSF has other phase-matching conditions could be investigated by choosing the proper \(\alpha_{f}^{\prime}\) and \(\Omega_{f}^{\prime}\). For example, when \(\alpha_{f}^{\prime}\) equals to zero and \(\Omega_{f}^{\prime}\) equals to \(\Omega\), it corresponds to the case that PSF has the same phase mismatch with the signal. This condition and the perfect phase-matching condition of PSF will both be considered in Section 3.1. This angle information is necessary for simulating broadband parametric amplification in the frequency domain where the full dispersion is taken into account [31]. Besides, based on the model in [31], we develop the five-wave-coupled equations to include the amplification of PSF:  
\[ \begin{cases}\frac{\partial\widetilde{E}_{p}}{\partial z}+ik_{p}\cos(\alpha) \widetilde{E}_{p}=-i\frac{\chi^{(2)}\omega_{p}}{n_{p}c\cos(\alpha)}\frac{1}{ \cos^{2}(\rho-\alpha)}\mathcal{F}\{E_{i}(t)E_{s}(t)\},\\ \frac{\partial\widetilde{E}_{s}}{\partial z}+ik_{s}\widetilde{E}_{s}=-i\frac{ \chi^{(2)}\omega_{s}}{n_{s}c}\mathcal{F}\{E_{p}(t)E_{i}^{*}(t)\},\\ \frac{\partial\widetilde{E}_{i}}{\partial z}+ik_{i}\cos[\Omega(\omega_{s}, \omega_{i})]\widetilde{E}_{i}=-i\frac{\chi^{(2)}\omega_{i}}{n_{i}c\cos[\Omega (\omega_{s},\omega_{i})]}\mathcal{F}\{E_{p}(t)E_{s}^{*}(t)\},\\ \frac{\partial\widetilde{E}_{sf}}{\partial z}+ik_{s}\cos[\alpha_{f}^{\prime}( \omega_{s},\omega_{i})]\widetilde{E}_{sf}=-i\frac{\chi^{(2)}\omega_{s}}{n_{s} c\cos[\alpha_{f}^{\prime}(\omega_{s},\omega_{i})]}\mathcal{F}\{E_{p}(t)E_{sf}^{*}(t)\},\\ \frac{\partial\widetilde{E}_{if}}{\partial z}+ik_{i}\cos[\Omega_{f}^{\prime}( \omega_{s},\omega_{i})]\widetilde{E}_{if}=-i\frac{\chi^{(2)}\omega_{i}}{n_{i} c\cos[\Omega_{f}^{\prime}(\omega_{s},\omega_{i})]}\mathcal{F}\{E_{p}(t)E_{sf}^{*}(t)\}, \end{cases}