## Reference Pulse  
An important step in the SRSI method is the generation of a reference pulse from the incident pulse using a frequency-conserved nonlinear process. Here, we consider how an appropriate reference pulse can be obtained. XPW, SD, and TG are all frequency-conserved four-wave-mixing processes that are used to improve the temporal contrast of ultrahigh-peak-power femtosecond laser systems [51; 55; 56]. The performances of the generated XPW wave, SD signal, and TG signal show that these processes satisfy the requirement as a reference pulse for the SRSI method. The principles of the XPW generation, SD, and TG processes are shown in Figure 3.  
Figure 3a presents the principle of XPW. A linear polarized pulse is focused into a BaF\({}_{2}\) crystal, generating an XPW wave with polarization perpendicular to that of the incident pulse. The generated XPW wave is separated from the incident pulse by placing a polarizer after the BaF\({}_{2}\) crystal. XPW is a self-phase-matched, polarization-nondegenerate, frequency-degenerate four-wave-mixing process. Figure 3b shows that for the SD effect, two incident laser pulses from a single pulse are focused into a transparent medium with a small crossing angle, generating SD signals in addition to the two incident beams on the other side of the medium. Higher-order cascaded SD signals are also generated when the incident pulse energy is sufficiently high, because SD is a frequency-degenerated cascaded four-wave-mixing process [57]. Figure 3c presents the principle of the TG process, which requires three input laser beams, making the TG setup complicated. However, TG is a self-phase-matched process because a BOXCARS beam geometry is used [58]. Moreover, in comparison to the XPW and SD processes, TG is notably more sensitive to the incident pulse energy and thus can be extended to measure weak pulses [31]. XPW, SD, and TG are all third-order nonlinear processes. Therefore, the spectra and temporal profiles of the generated pulses are defined by the following equations:  
\[I_{\mathrm{r}}(t)\propto{I_{\mathrm{u}}}^{2}(t)I_{\mathrm{u}}(t-\tau) \tag{5}\]  
\[I_{\mathrm{r}}(\omega_{\mathrm{r}})\propto\Bigg{|} \begin{array}{c}\iint d \omega_{1}d\omega_{-1}\chi^{(3)}\widetilde{E}_{1}^{*}(z,\omega_{1})\widetilde {E}_{-1}(z,\omega_{-1})\\ \times\widetilde{E}_{1}(z,\omega_{\mathrm{r}}-\omega_{-1}+\omega_{1})\sin c [\Delta_{k}(\omega_{\mathrm{r}},\omega_{1},\omega_{-1})L/2]\end{array} \Bigg{|} ^{2}. \tag{6}\]  
The temporal profile of the generated signal is cleaned because its intensity is proportional to the cube of that of the incident pulse. In these very fast parametric processes, which have a response time of a few hundreds of femtoseconds, the temporal profile is cleaned within picoseconds. The spectrum and spectral phase of this kind of cleaned signal pulse are smoothed. Equation (6) shows that the spectrum of the generated reference pulse is smoothed and broadened because it is an integral (i.e., an average contribution) of the spectra of the incident pulses. Figure 4a shows the typical improvement in the temporal contrast of the reference pulse compared to that of the incident pulse during the SD process. The temporal contrast of the SD signal was improved by four orders of magnitude. Figure 4b shows that the typical spectrum of the reference pulse (thick red dotted line) is broadened and smoothed in comparison to that of the incident pulse (thick black solid line). In addition, the spectral phase of the reference pulse (thin red dotted line) is smoothed and flattened in comparison to that of the incident pulse (thin black solid line). Figure 3: Principles of (**a**) cross-polarized wave generation (XPW), where a polarizer is used to separate the generated XPW. (**b**) Self-diffraction (SD) process, where \(k_{\mathrm{sd1}}\) and \(k_{\mathrm{sd2}}\) are the first- and second-order SD signals, respectively. (**c**) Transient-grating (TG) process, where \(K_{TG}\) is the generated TG signal.