# Conical Emission Induced by the Filamentation of Femtosecond Vortex Beams in Water
## 3 Results and Discussion  
Figure 3 shows the patterns of the supercontinuum induced by the femtosecond BG and LG beams in water. These patterns are the results of the affine transformation of the light spots captured by the camera. Water is chosen as the medium in this paper because it can generate stable, ultrabroad, high-intensity supercontinuum at a low cost, standing out among the condensed state media to generate supercontinuum. The critical power for self-focusing \(P_{\rm cr}=3.77\lambda_{0}^{2}/8\pi n_{0}n_{2}\) (the linear and nonlinear refractive index of water are \(n_{0}=1.33\) and \(n_{2}=4.1\times 10^{-16}\) cm\({}^{2}\)/W) in water for 800 nm pulse is \(P_{\rm cr}=1.761\) MW [33], which is much lower than the filamentation threshold due to the strong dispersion effect [34]. In our previous work, the dependence of the filamentation threshold on the TC in the case of the LG beam was studied by interferometry [26]. It is found that the filamentation threshold of the femtosecond vortex pulse increases linearly with the TC value, and for the LG beam with \(l=10\), its filamentation threshold is 140 MW [26]. For the input femtosecond pulse, its energy is attenuated to 188 \(\upmu\)J, and the corresponding pulse power is 3.76 GW, which is much higher than the filamentation threshold of the femtosecond vortex beams in water. In this case, multiple filaments are generated. The white light produced by the BG beams is distributed in a set of concentric rings like a rainbow with a white center, as shown in Figure 3a-c. In contrast, the spots of the white light induced by the LG beams are completely different, which are circular white disks, as shown in Figure 3d-f. Obviously, they are different from the white light spots that have been reported in many previous literature, which generally consist of a white central part surrounded by a set of rainbow-like rings. It can also be seen from Figure 3 that by increasing the TC value of the vortex beams, the characteristics of the white light patterns are almost the same as the zero-order 'vortex' beams (\(l=0\)) for both the LG and BG beams. To make out the origin of the difference between the spots of white light induced by the two kinds of vortex beams, especially the abnormal white light spots in the case of the LG beams, in the following part, using a set of bandpass filters (450 nm, 500 nm, 550 nm, 600 nm, 650 nm and 700 nm), we study the light spots for different monochromatic components. Figure 2: Schematic diagram of the method to measure the divergence angle of supercontinuum induced by femtosecond vortex beams. \(\theta_{\rm d}\) is the divergence angle. By moving the white screen, the light spot is recorded at two selected positions along the propagation direction of the femtosecond beam. \(R_{1}\) and \(R_{2}\) are the radii of the light spots captured at the two positions. We take the white light induced by the filament generated by the zero-order BG and LG beams as examples and display the light spot patterns for different spectral components of the white light, as shown in Figures 4 and 5. It can be seen from the figures that the radius of the light spot decreases with the increase in the wavelength both in the case of LG and BG beams, which is the feature of conical emission. However, there exists a significant difference between the profiles of light spots of the monochromatic components from the white light induced by the two kinds of vortex beams. In the case of BG beams, the white light spots are composed of a central bright spot and several concentric rings, as shown in Figure 4b,d,f,h,j,l). It results in a rainbow of concentric rings of white light when all spectral components are mixed together. The central part of the rings contains all the spectral components, making it a white dot, as shown in Figure 3a-c, which is similar to the commonly observed white light spots.