## 3 Results and Discussion
At first, to remove the experimental noise due to laser fluctuations, the data was smoothed. Next, the interferogram was recorded by the spectrometer with a resolution of 0.75 nm; the sampling points were not equally distributed in the frequency domain. Thus, we used 2D bilinear interpolation to reconstruct an interferogram from the experimental data. The reconstructed spectrogram had data points uniformly sampled in the frequency domain with spectral range cut down to 400 nm-700 nm, which covered the spectrum of the sidebands used in our simulation. Between 400 nm and 700 nm, there were 250 sampling points. The reconstructed interferogram is shown in Figure 2c. The vertical axis in the interferogram gives wavlenegth in nanometers, which is converted from \(\omega\) following \(\lambda=2\pi\)c/\(\omega\), here, c is the speed of light. The step of interpolations is 1 fs in the other dimension of the interferogram (which is the time delay between the two spherical mirrors). Figure 2: **(a)** Interferogram of sidebands AS3 and AS4 recorded by spectrometer; **(b)** The cross-section of the interferogram at 562.57 nm, which is the central wavelength of AS 4, shows the intensity oscillation; **(c)** Reconstructed interferogram of AS3 and AS4 with the 2D bilinear interpolation; **(d)** Theoretical simulation of the interferogram using AS3 and AS4; **(e)** Cross-section of the interferogram from simulation (blue) and experiments (red) at the central wavelength of AS 3 (597.7 nm); **(f)** Cross-section of the interferogram from simulation (blue) and experimental data (red) at 562.57 nm. As a demonstration of our phase retrieval algorithm, the total time delay span in our theoretical calculation was 100 fs, which suppressed the effect of phase distortion from spherical aberration induced by the displacement of translation stage. In the simulation, we solved the partial differential equations that described the coherent Raman interaction between the Raman sidebands. We took all the sidebands as being far from the resonance as in Reference [21]. The equations can be written as Equation (1):  
\[\frac{\partial\mathrm{E_{q}}(\omega,\mathrm{z})}{\partial\mathrm{z}}=-\mathrm{ j}\frac{\mathrm{b_{0}}}{\mathrm{a_{0}}}\mathrm{\beta_{q}}(\int\limits_{0}^{ \infty}\chi(\Omega)\rho_{ab}(\Omega)\mathrm{E_{q-1}}(\Omega-\omega)\mathrm{d \Omega}+\int\limits_{0}^{\infty}\chi^{\star}(\Omega)\rho_{ab}^{\star}(\Omega) \mathrm{E_{q+1}}(\Omega+\omega)\mathrm{d\Omega}) \tag{1}\]  
where \(\rho_{ab}=\frac{1}{2}\sin(\theta)\exp(j\varphi)\) and \(\tan\theta=\frac{2|B|}{2\Delta\omega-D+A}\). \(B=|B|\exp(j\varphi)\). The line profile is \(\chi(\Omega)=\frac{\dot{a}}{\alpha-(\omega_{p}-\omega_{s})-i\gamma}\) where \(\alpha\) is a constant related to the spontaneous Raman cross-section, \(\omega_{s}\) and \(\omega_{p}\) are the angular frequencies of the pump and Stokes beams, and \(\gamma\) is the half-width of the spontaneous Raman line. The propagation constant is defined as \(\beta_{q}(\omega)=\eta\hbar\omega_{q}Na_{0}\) for different wavelength and the definition of \(a_{0}\) and \(b_{0}\) can be found in References [8,21]. \(E_{\mathrm{q}}(\omega)=A_{\mathrm{q}}(\omega)\times\exp(j(\varphi_{\mathrm{q} }(\omega)+\omega_{0}t_{0}))\). For the amplitude \(A_{\mathrm{q}}(\omega)\), we used the value measured by the spectrometer. We applied the same interpolation as that in the interferogram to reconstruct \(A_{\mathrm{q}}(\omega)\), whose sampling points were uniformly distributed in the frequency domain. Spectral phase \(\varphi_{\mathrm{q}}(\omega)\) and time delay \(t_{\mathrm{q}}\) of the sidebands are the fitting parameters we need to determine through simulation. The other parameters are determined by the theoretical evaluation and optimized by the calculation. In principle, there are infinite equations as n goes to \(\pm\infty\). However, since our sidebands were AS3-AS5, we only considered the relevant equations, including these sidebands, within the spectral range 400 nm-700 nm.