# Conical Emission Induced by the Filamentation of Femtosecond Vortex Beams in Water
## 3 Results and Discussion
numerically simulated the propagation of a femtosecond BG beam in glass and found that the BG beam can generate a ring-shaped filamentary ionized channel, forming multiple helical filaments [37]. They also found that the ring-shaped filamentary ionized channel is merely affected by the TC value. As a consequence, the white light generated by the multiple helical filaments exhibits the rainbow-like conical emission. To conclude, the conical emission induced by the two kinds of vortex beams differs from each other, showing different profiles: the conical emission induced by the BG beams shows a conical surface with a hollow interior for all the monochromatic components, while that induced by the LG beams show a cone with a solid interior for all the monochromatic components. The different spatial intensity distributions for the two kinds of vortex beams can account for the difference. In a tightly focusing condition, there exists spin-to-orbital angular momentum conversion [38]. Though the spin angular momentum carried by the vortex beams prepared by the SLM is zero, the axicon and plano-convex focusing lens cause strong focusing. In this case,  
Figure 6: (**a**) Illustration of the general wave vector. Spatial distribution of the wave vector of the laser beam after it passes the (**b**) plano-convex focusing lens and (**c**) axicon lens. the polarization state of the vortex beam may be changed due to the spin-to-orbital angular momentum coupling. As is known, the polarization state of a laser affects the ionization efficiency of the media: the linearly polarized pulse is more efficient in ionization than the circularly polarized one when they have the same power [39]. To study the role that the optical angular momentum plays in the conical emission induced by femtosecond vortex beams in water, we try to obtain the variation of the divergence angle of the supercontinuum with the wavelength and the TC. Through the radii of the light spots measured at the two selected positions, we can obtain the maximum divergence angle of the supercontinuum by using Equation (1), as shown in Figures 7 and 8. Here, we use the divergence angle rather than the conical angle because the use of the lens contributes partly to the divergence angle. As has been stated in the previous part, the divergence angle \(\theta\) brought by the plano-convex focusing lens and axicon lens are 0\({}^{\circ}\)-3.8\({}^{\circ}\) and 5\({}^{\circ}\), respectively. Therefore, the real conical angle is \(\Theta=\theta_{d}-\theta\). It can be seen from Figures 7a and 8a that the divergence angle decreases with increasing wavelength, which agrees with the quantitative analysis performed with the SPM. Interestingly, for all the TC values adopted in our work, the change of the divergence angles of the supercontinuum induced by the femtosecond vortex beams with the wavelength is coincident. We also draw the curves showing the variation of the divergence angle with the TC value, as shown in Figures 7b and 8b. It can be seen from the figures that within the error, the divergence angle does not change with the TC value for all the selected wavelengths. The above results suggest that the OAM does not affect the supercontinuum generation and conical emission induced by the filamentation of femtosecond vortex beams. Though the OAM affects the filamentation threshold [23; 24; 26], and the OAM can be transferred from the input pulse to the atoms/molecules in surrounding media during the interaction between laser and matter [11; 12; 13], in condensed state matters, the TC value does not affect the spectral broadening, provided the filament is formed [26; 27]. According to the interpretation of the SPM, see Equations (2) and (4), the helical phase carried by the light waves does not evolve in time and radial direction in space. As a result, the OAM does not bring any additional information to the supercontinuum and conical emission.