## 2 Theoretical Analysis
### CFWM Process  
The theoretical analysis of CFWM processes for multicolored laser pulse generation is given in [60]. The schematic of CFWM processes is shown in Figure 1a. Two vectors, \(k_{1}\) and \(k_{2}\), correspond to the two input beams with frequencies of \(\omega_{1}\) and \(\omega_{2}\) (\(\omega_{1}>\omega_{2}\)) respectively. The \(m\)th-order anti-Stokes (spectrally blue-shifted) and Stokes (spectrally red-shifted) sidebands are marked as ASm and Sm (m = 1, 2, 3...). Figure 1b-e show the phase-matching geometries for generating the first three anti-Stokes sidebands (AS1, AS2, AS3) and the first Stokes sideband (S1). Based on these phase-matching geometries, the phase-matching condition for the \(m\)th-order anti-Stokes sideband can be written as: \(k_{\mathbf{ASm}}=k_{\mathbf{AS(m-1)}}+k^{\mathbf{(m)}}{}_{1}-k^{\mathbf{(m)} }{}_{2}\approx(\mathrm{m+1})k^{\mathbf{(1)}}{}_{1}-\mathrm{m}k^{\mathbf{(1)}}{ }_{2}\), \(\omega_{\mathrm{ASm}}\approx(\mathrm{m+1})\omega_{1}{}^{(1)}-\mathrm{m}\omega _{2}{}^{(1)}\). Since the two input beams are never single-frequency lasers, \(k^{\mathbf{(m)}}{}_{1}\) and \(k^{\mathbf{(m)}}{}_{2}\) are used instead of \(k_{1}\) and \(k_{2}\). The values of \(\omega_{1}{}^{(\mathrm{m})}\), \(\omega_{2}{}^{(\mathrm{m})}\), \(|k^{\mathbf{(m)}}{}_{1}|\), and \(|k^{\mathbf{(m)}}{}_{2}|\) may be different for every step of the m FWM processes. Similarly, with \(k^{\mathbf{(m)}}{}_{1}\) and \(k^{\mathbf{(m)}}{}_{2}\) used instead of \(k_{1}\) and \(k_{2}\), the \(m\)th-order Stokes sideband will have the following phase-matching condition: \(k_{\mathbf{Sm}}=k_{\mathbf{S(m-1)}}+k^{\mathbf{(m)}}{}_{2}-k^{\mathbf{(m)}}{} _{1}\approx(\mathrm{m+1})k^{\mathbf{(-1)}}{}_{2}-\mathrm{m}k^{\mathbf{(-1)}}{} _{1}\),\(\omega_{\rm Sm}\approx(\rm m+1)\omega_{2}{}^{(-1)}-\rm m\omega_{1}{}^{(-1)}\). As the lower-order signals will participate in the generation of adjacent higher-order signals as pump pulses, this process is called CFWM.  
Based on the phase-matching condition we have expressed above, the output parameters, such as wavelength and output angle, of the generated sidebands can be calculated to explain and inform experimental work. In our experiments, the wavelength range of the two pump beams were 660-740 nm (Beam 1) and 800 nm (Beam 2). The nonlinear medium was assumed to be fused silica plate with a thickness of 1 mm. The simulations were performed under these conditions. The wavelength dependence of Beam 1 for optimal phase-matching on the order number at different cross-angles is shown in Figure 2a. To fulfill the phase-matching condition, the wavelength of Beam 1 should redshift for higher order anti-Stokes sidebands. The wavelengths of generated sidebands for different cross-angles are shown in Figure 2b, which clearly shows that the wavelengths of same order sidebands can be tunable by changing the cross-angle of the two pump beams, and the tuning range covered the wavelength gap between adjacent sidebands. The exit angles of the generated sidebands are plotted against the order number at different cross-angles in Figure 2c. The difference in exit angle between the multicolored sidebands was large enough for easy separation, even for adjacent sidebands. The dependence of the exit angle on the center wavelength of the generated sidebands at different cross-angles in different materials is shown in Figure 2d. The phase mismatching for the first four anti-Stokes sidebands at two different angles, 1.87\({}^{\circ}\) and 2.34\({}^{\circ}\), are shown in Figure 2e. The increase of the slope of the curves with the order numbers means the reduction of the gain bandwidth for the sidebands, which was confirmed by our experiment. The calculations based on the phase-matching condition of CFWM agree with the experimental results, which will be given in next section.  
Figure 1: Schematic of multicolored sidebands generation using CFWM process. The phase-matching geometries for (**a**) AS1–AS3 and S1; (**b**) AS1; (**c**) AS2; (**d**) AS3, and (**e**) S1 [60].