## 3 Experimental Characteristics of Multicolored Pulses
### Multicolored Sidebands Generated with Low Threshold  
The polarization at frequency \(\omega_{\mathrm{AS1}}\) in FWM process generating AS1 can be written as:  
\[P^{(3)}(\omega_{\mathrm{AS1}})\propto\chi_{\mathrm{eff}}^{(3)}E^{2}(\omega_{ 1})E^{*}(\omega_{2}) \tag{3.3}\]  
According to the coupled-wave equations in the FWM process, the optical field and polarization at frequency \(\omega_{\mathrm{AS1}}\) had the following relationship:  
\[\frac{\mathrm{d}E(\omega_{\mathrm{AS1}})}{\mathrm{d}z}=\frac{i\omega_{\mathrm{ AS1}}}{2\varepsilon_{0}cn_{\mathrm{AS1}}}P^{(3)}(\omega_{\mathrm{AS1}})\mathrm{e}^{- i\mathrm{d}x_{z}} \tag{3.4}\] \begin{table}
\begin{tabular}{c c c c c} \hline \(\mu\)W & \(\omega_{\mathrm{AS1}}\) & \(\omega_{\mathrm{AS1}}\) & \(\omega_{\mathrm{AS1}}\) & \(\omega_{\mathrm{AS1}}\) \\ \hline AS1 & 480 & 700 & 715 & 750 & 780 \\ AS2 & 210 & 315 & 295 & 210 & 135 \\ AS3 & 125 & 90 & 60 & 40 & 10 \\ \hline \end{tabular}
\end{table} 
Table 1: The output power of AS1-AS3 with five commonly used third-order nonlinear media. The external cross-angle of two pump beams is 1.8\({}^{\circ}\), while the input powers of Beam 1 and Beam 2 are 7 mW and 25 mW, respectively [60]. Figure 10: The power dependence of AS1 on (**a**) Beam 1 with power of Beam 2 fixed at 19 mW, and (**b**) Beam 2 with power of Beam 1 fixed at 11 mW. Power stabilities of AS1 and Beam 1 in twenty minutes are shown as the insertion of (**a**) [59]. Here, \(\Delta k\) is the wave vector mismatch in the FWM process. From Equations (3.3) and (3.4), it can be seen that the intensity of AS1 becomes higher, following the proportionality relation with the squared absolute value of the nonlinear optical susceptibility (\(\left|\chi_{\textit{eff}}^{(3)}\right|^{2}\)) of the material used. Based on this, diamond, the nonlinear optical susceptibility of which is \(\sim\)5 time larger than that of sapphire and \(\sim\)10 times larger than that of CaF\({}_{2}\)[73], was used in the experiment to obtain multicolored sidebands with a low threshold. The experiment was performed with Type-2 setup shown in Section 3.1. The spectra of two pump beams, Beam 1 and Beam 2, are depicted in Figure 11. The two spectral positions were adjusted by tuning the angle between input beams and filters. The retrieved temporal intensity profiles and phases of two pump pulses are shown in Figure 12, where the Beam 1 and Beam 2 pulse durations are 81 fs, and 47 fs, respectively. The average power of Beam 1 was set to 0.855 mW, and the average power of Beam 2 was continuously changed by a VND. Figure 13 shows the multicolored sidebands at different pump levels. The intensities of Beam 1 and Beam 2 on the diamond plate in Figure 13a were calculated to be 14.9 \(\times\) 10\({}^{9}\) W/cm\({}^{2}\) and 12.3 \(\times\) 10\({}^{9}\) W/cm\({}^{2}\) respectively. These were much lower than the threshold intensities obtained previously for multicolored sideband generation in a fused silica plate, of 60 \(\times\) 10\({}^{9}\) W/cm\({}^{2}\) and 8 \(\times\) 10\({}^{9}\) W/cm\({}^{2}\)[70]. This low pump threshold for multicolored sideband generation is important in the context of an actual experiment, because pump lasers with high repetition frequencies, _i.e._, several hundred kHz to several MHz, inevitably have a low pulse energy  
Figure 11: The spectra of Beam 1 (black), Beam 2 (red) [62]. Figure 12: The retrieved intensity profile and phase of (**a**) Beam 1; (**b**) Beam 2 [62]. when used with a conventional amplifier system. Compared to the multicolored sidebands generated with a 1 kHz amplifier, pulses with a higher repetition frequency are more useful in nonlinear microscopy. Low repetition frequencies make the image frame times unconventionally long and also lead to high noise levels. Figure 13b-d show the 2D structure obtained by increasing the power of Beam 2, a detailed discussion of which will be given in the next section. Figure 14 shows the spectra of generated multicolored sidebands obtained under pump rates of 0.855 mW and 0.856 mW for Beam 1 and Beam 2, respectively.