## 3 Results and Discussion  
In the reflection scheme, by recording the spectrum as a function of time delay, we obtained an interferogram based on coherent Raman scattering. In this paper, we describe our procedure to retrieve the spectral phases from the interferogram by using a theoretical simulation. The experimental details were presented in Reference [21] and all the data presented in this paper is generated from the same experimental setup. The spectra of Raman sidebands were recorded by spectrometer (HR 4000, Ocean Optics). The pulse durations of pump and Stokes were around 50 fs. The pulse durations of the sidebands were a little longer than 50 fs due to the dispersion of the crystal [17,18]. The experimental interferograms are shown in Figure 2a (The range for the spectrometer is from 200 nm-1100 nm and in Figure 2a, we only show the range from 500 nm-700 nm). The interferogram is recorded with an integration time (100 ms) of the spectrometer. Therefore, the spectrogram did not display the shot-to-shot robustness of the sideband generation. The spectrogram shows intensity oscillations due to Raman scattering. A cross-section of the spectrogram at 562.57 nm, which is the central wavelength of AS4, is displayed in Figure 2b. The frequency of the intensity oscillations is roughly equal to the differences between the central frequencies of two sidebands. The peak intensity of the pump and the Stokes beam at the focal point were about 4 \(\times\) 10\({}^{11}\) W/cm\({}^{2}\). These beams will induce the cross phase modulation for the sidebands. Theoretically, the phase shift at 562.57 nm is 7.99 (it is evaluated using the parameter from Reference [24]). During the scanning, when the pump and Stokes beams encountered the sidebands, at the first moment, the intensity of the driving beams increased. The changing of the refractive index is proportional to the intensity, and the spectral phase will change faster when the beam approaches the maximum intensity. This will increase the intensity oscillation frequency on the interferogram. Moreover, when the intensity is decreasing, the intensity oscillation frequency will decrease. In our experiments, we have tried our best to limit the cross phase modulation. By analyzing the spectrogram, we did not see significant variation of the intensity oscillation frequency. However, we could still see that the central frequency of the sidebands is slightly shifted due to the cross-phase modulation induced by pump and Stokes. Another nonlinear effect which plays an important role is self-phase modulation. The high intensity of the pump and the Stokes beams will induce self-phase modulation. It would distort their spectral phases and also broaden their spectrum. All of these would affect the energy and spectrum of the Raman sideband. The goal of our simulation is to retrieve the relative spectral phases of the sidebands from the interferogram. We emphasize that, in the experiment, the spectrogram was recorded with an integration time 100 ms, which resulted in averaging out the fluctuation of the phase and the energy of the sideband generation (according to Reference [23], the deviation of the sideband's energy is around 15% due to the shot-to-shot fluctuation). Our method is to retrieve the spectral phase for the particular spectrogram and so far we do not consider the shot-to-shot fluctuation in the simulation. Therefore, when judging our retrieval algorithm, the root-mean-square (rms) deviation of the retrieved _vs._ original spectrogram is important. In the experiment, the spectral phase distortion was a combined effect resulting from the dispersion of the crystal and nonlinearities, such as cross phase modulation [17,18,21]. The spherical aberration of the spherical mirror also contributed to the phase distortion. To find the best value for the spectral phases, we compared the theoretical simulation with the experimental interferogram. Therefore, the experimental data was processed for the convenience of comparison with the theoretical simulation.