# Conical Emission Induced by the Filamentation of Femtosecond Vortex Beams in Water
## 3 Results and Discussion
(**a,c,e,g,i,k**) and (**b,d,f,h,j,l**) are measured when the distance between the back wall of the cuvette and the white screen is 8.1 cm and 23.1 cm, respectively. The front of the pulse, where plasma can be neglected, induces a redshift associated with a positive \(k_{r}(t)\), i.e., converging toward the filament axis, while the contribution of the plasma in the trail of the pulse leads to a blue shift associated with a negative radial phase derivative. Hence, the conical emission is interpreted as a divergence of the anti-Stokes components of the supercontinuum induced by the plasma. No conical emission was found at Stokes shifted wavelengths. In this paper, we focus on the spectral components at the blue side of the central wavelength (i.e., visible band), \(\Delta\omega\) always takes the positive values. From Equations (2) and (4), it can be obviously seen that \(\Delta k_{r}\) is positively related to \(\Delta\omega\). As shown in Figure 4a, for a newly generated spectral component, the more its frequency deviates from the central frequency of the laser pulse \(\omega_{0}\), the larger its divergence angle with respect to the filament axis is, which can well explain the feature of the conical emission. The difference between the conical emission induced by the femtosecond LG and BG beams can be attributed to the use of a plano-convex focusing lens and axicon lens. In the experiment, after the femtosecond laser beams pass the plano-convex focusing lens and axicon lens, due to the different geometries of these lenses, the spatial distribution of the wave vectors differs much in the two cases, as shown in Figure 6b,c. Obviously, the different spatial distribution of the wave vectors results in the different spatial distribution of laser intensity affects the excitation and photoionization of the media and, thus, the accompanied light emission like fluorescence and supercontinuum. Figure 5: Spots of (**a**,**b**) 450 nm, (**c**,**d**) 500 nm, (**e**,**f**) 550 nm, (**g**,**h**) 600 nm, (**i**,**j**) 650 nm and (**k**,**l**) 700 nm spectral components after the filamentation of the LG beam with \(l=0\). (**a**,**c**,**e**,**g**,**i**,**k**) and (**b**,**d**,**f**,**h**,**j**,**l**) are measured when the distance between the back wall of the cuvette and the white screen is 8.1 cm and 23.1 cm, respectively. In our work, the focal lengths of the plano-convex focusing lens are much shorter than the focal length of the focusing lens adopted in many previous experiments about femtosecond filamentation. It leads to the tightly focusing condition, which makes the incident laser beam diverge greatly after the focus, and thus, the wave vectors point in different directions, as shown in Figure 6b. The divergence angle brought by the plano-convex focusing lens takes continuous values between \(0^{\circ}\) and \(3.8^{\circ}\). According to the focal length of the selected focusing lens, the focusing regime can be divided into linear and nonlinear ones [35], which greatly affects the femtosecond laser filamentation process and the accompanied nonlinear phenomena like fluorescence and supercontinuum radiation [34; 36]. For the tightly focusing case, linear focusing plays a dominant role in the femtosecond filamentation. Once the filament is formed, the wave vectors of the newly generated spectral components also point in different directions, and thus, the captured white light spots are circular white disks, which are completely different from the commonly observed white light spots. If we use a focusing lens with a much longer focal length, the directions of the wave vectors are almost uniform, pointing at the propagation direction. In this case, the femtosecond filamentation mainly results from self-focusing, and the normal white light spots can be observed. In contrast, after the laser beam passes the axicon lens, the direction of the wave vectors is the same (the divergence angle brought by the axicon lens is \(5^{\circ}\)), and thus the captured patterns are rings. [PERSON] et al.