## 3 Results and Discussion
The boundary conditions are assumed to be the same as the experimental conditions. For example, in Figure 2, in addition to AS3 and AS4, the amplitudes \(A_{\mathrm{q}}(\omega)\) of all the other sidebands, AS2, AS5, AS6, AS7, and AS8, are zero in the initial condition. We solved the equations using the Runge-Kutta method. In order to find the right spectral phases, we attributed the phase distortion to the dispersion of the crystal (1 mm thick synthetic single-crystal diamond in our experiment) and the retardation among the sidebands. By comparing the theoretical results (Figure 2d) with the experimental results (Figure 2c), the spectral phases were optimized by changing time delay between sidebands and the thickness of the crystal and obtained a best fit with experimental results for a thickness of 200 \(\upmu\)m. The thickness of the crystal was about 1 mm, which was much larger than the 200 \(\upmu\)m in the simulation. In other words, the spectral phases included the effect resulting from the dispersion of the crystal and cross phase modulation. For the results presented in the paper, we did not consider the effect of cross phase modulation separately. The reason was that when adding the cross-phase modulation induced by pump and Stokes fields into the simulation, we did not see significant improvement in the simulation. The theoretical results were also normalized in order to compare with the experimental data. Figure 2e,f displays the cross section of the interferogram at the central wavelength of AS3 at 597.7 nm AS3 and AS4 at 562.57 nm, which showed that our theoretical results fit the experimental results well. However, the results were not perfect, as we did not separately consider the phase distortions due to the nonlinear effects, such as self/cross phase modulation and spherical aberration of the mirrors. The spectral phases retrieved in our simulation were relative. Thus, we took the spectral phase of AS3 as the reference. After finding the proper phase for AS4 using the interferogram of AS3 and AS4,we continued to retrieve the spectral phases of AS5 from the interferogram between AS4 and AS5. With the same method, the spectral phases of AS6 and AS7 were retrieved. Then we put multi-sidebands with the retrieved spectral phases together and reconstructed the corresponding interferogram to compare with the experimental result. The theoretical interferogram and experimental interferogram of multi-sidebands are shown in Figure 3. Figure 3a,b show the numerical results and experimental results with sidebands AS3, AS4, and AS5. Figure 3c,d show the numerical results and experimental results with sidebands AS3, AS4, AS5, and AS6. Figure 3e,f show the numerical results and experimental results with sidebands AS3, AS4, AS5, AS6 and AS7. After we normalized both the retrieved and the original spectrograms, we calculated the total rms deviation between simulation and experimental spectrogram as: \(\mathrm{rms}=\sqrt{\frac{1}{Nt\times N_{\omega}}}\sum(I_{\mathrm{simulation}}( \omega,t)-I_{\mathrm{experiment}}(\omega,t))^{2}\) to be 0.0294 for our retrieval (here \(\omega\) is the frequency, \(t\) is the time delay, \(Nt\) is the total scanning steps in time domain an, \(N\omega\) is the total number in frequency domain, and \(I_{simulation}(\omega,t)\) and \(I_{experiment}(\omega,t)\) are the normalized intensities of simulation and experiment). In Figure 4a, we show the rms deviation at a variable \(\omega\) (\(\mathrm{rms}=\sqrt{\frac{1}{Nt}\sum(I_{\mathrm{simulation}}(\omega,t)-I_{ \mathrm{experiment}}(\omega,t))^{2}}\), which is a sum over scanning time steps) together with the spectral phases obtained from the retrieval. In Figure 4, we convert \(\omega\) to the wavelength following \(\lambda=2\pi c/\omega\), here, \(c\) is the speed of light. The retrieval method we used did not include the iteration routine. Including the iteration in our algorithm is our future plan. The ultrafast waveform retrieved from Figure 3e,f could be found in Reference [21].