## 4 2-D Multicolored Sidebands Arrays
Figure 16: **(a)** Schematics of the generation of 2-D multicolored arrays; **(b)** A photograph of the 2-D multicolored arrays generated in sapphire plate; **(c)** 2-D multicolored arrays are defined as B\({}_{\text{m},\text{n}}\), in which B\({}_{\text{0},0}\) and B\({}^{-1,0}\) refer to Beam 1 and Beam 2 respectively [56]. o.a., optical axis. Figure 17: The spectra of sidebands on **(a)** the center row B\({}_{\text{m},0}\); and **(b)** the second column B\({}_{\text{2,n}}\)[56]. During the experiment, the power of Beam 1 was amplified from 0.1 to 0.17 mW, which means that the FWM process could also be used for parametric amplification. The power stability of several sidebands in different arrays, measured with a photodiode, is shown in Figure 18b. The stabilities are all in the range 0.5%-2% in RMS within 200 s.  
In 2013, 2-dimensional multicolored arrays were observed with a low pump rate in a diamond plate, as shown in Figure 13. The experimental setup was the same as the Type-2 setup shown in Section 3.1. Increasing the pulse energy of Beam 2 to 0.856 \(\upmu\)J created 2-D multicolored arrays, as shown in Figure 13b. The threshold energy was much lower than that for a sapphire plate, \(\text{<2}\)\(\upmu\)J for a diamond plate and 25 \(\upmu\)J for a sapphire plate [56,62]. More sidebands, lines and brighter arrays were observed when the energy of Beam 2 was increased further, as shown in Figure 13c,d. The mechanism for the generation of 2D structure is not yet fully clear, although we have given some explanations based on the CFWM and beam breakup [62]. The detailed simulations of spontaneous breakup of elliptical laser beams were performed by [PERSON] and his coworkers [76]. They attributed this breakup to multistep four-wave and parametric amplification of certain components occurring in the spatial spectrum of the self-focusing laser beam. Interestingly, beam breakup was observed even with a near circular (the ellipticity \(e=1.09\)) input beam when the input power was \(\sim\)20 times larger than \(P_{\text{cr}}\), which is defined as follows [77]:  
\[P_{\text{cr}}=3.77\uplambda^{2}/(8\pi nn_{2}) \tag{1}\]  
Here, \(\uplambda\) is the laser wavelength in vacuum, \(n\) is the refractive index, and \(n_{2}\) is the nonlinear refractive index. The pump beams have an ellipticity of \(\sim\)1.2 in the experiment, due to asymmetric focusing with several concave mirrors. The summation of the peak powers of the two input beams is about \(\sim\)60 \(P_{\text{cr}}\) (\(P_{\text{cr}}=0.4\) MW for diamond plate), and would be large enough for beam breakup [76]. The combination of this beam breakup and CFWM is the main cause of this 2-D multicolor structure. Figure 18: **(a)** The output power of sibebands on the B\({}_{\text{m},0}\), B\({}_{\text{m},1}\), B\({}_{\text{m},-1}\), B\({}_{\text{m},-2}\), B\({}_{\text{m},2}\), with pump power of 0.1 mW and 25 mW for Beam 1 and Beam 2, respectively. The inset shows the power dependence of three different sidebands, B\({}_{1,0}\), B\({}_{2,0}\), and B\({}_{2,1}\), on power of Beam 2; **(b)** The power stabilities of three sidebands B\({}_{3,1}\), B\({}_{4,0}\) and B\({}_{1,0}\)[56]. Beam breakup can also occur when the pump beams are circular. [PERSON] has investigated the interaction of two noncollinear femtosecond laser filaments in sapphire both numerically and experimentally [78]. The simulation was based on the nonlinear Schrodinger equation. The experiment was performed with a Ti:sapphire amplifier with a pulse width of 120 fs. The two incident beams, which are all from the same laser source, have a cross-angle of 4.64\({}^{\circ}\). Because of the asymmetric distributions of refractive index changes in the nonlinear material due to the Kerr effect of the pump beams, additional \"hot points\" or plasma channels arise in the plane oriented perpendicular to the pulse propagation plane with input pulse power above 10 \(P_{\mathrm{cr}}\) (corresponding to a pulse energy of 3\(\sim\)4 \(\upmu\)J). Clear beam breakup was observed.