# Conical Emission Induced by the Filamentation of Femtosecond Vortex Beams in Water
## 3 Results and Discussion
In the case of LG beams, the spot of each monochromatic component from the white light induced by the filamentation of the femtosecond LG beam is a circular disk, as shown in Figure 5. The different spectral components are mixed together, making the white light spots circular white disks (see Figure 3d-f), which are different from the commonly observed white light spots. Before conducting further analysis and discussion, we shall review the physical mechanism that accounts for the conical emission. Since the self-phase modulation (SPM) is taken as the predominant mechanism for the supercontinuum generation [16, 17] and the conical emission is a characteristic phenomenon of supercontinuum generation, we prefer the SPM scheme to qualitatively interpret the conical emission in the following part (in effect, it has been carried out by numerous work). The variation of the frequency induced by the SPM can be described by [16, 17, 26, 27]:  
\[\Delta\omega=\omega-\omega_{0}=-\frac{\partial\Delta\varphi}{\partial t}\sim \frac{\omega_{0}z}{c}\left(-n_{2}\frac{\partial I(r,t)}{\partial t}+\frac{1} {2n_{0}\rho_{\rm c}}\frac{\partial\rho_{\rm e}(r,t)}{\partial t}\right) \tag{2}\]  
where \(\Delta\varphi\) is the variation of the phase of the laser pulse, \(\omega_{0}\), \(z\), \(n_{2}\), \(\rho_{\rm e}(r,t)\) and \(\rho_{\rm c}\) are the central frequency of the laser pulse, the propagation distance in media, second order nonlinear refractive index of the transparent medium, plasma density and the critical plasma density (\(\rho_{\rm c}=\epsilon_{0}m_{\rm e}\omega_{0}^{2}/e^{2}\), \(\epsilon_{0}\), \(e\) and \(m_{\rm e}\) are permittivity of vacuum, electron charge and electron mass), respectively. For the 800 nm pulses, the value of \(\rho_{\rm c}\) is about \(2\times 10^{21}\) cm\({}^{-3}\), though it is a little lower than the number density of water molecule \(\rho_{\rm water}=3.3\times 10^{22}\) cm\({}^{-3}\). Limited by the ionization rate during femtosecond filamentation (about 1%), the electron number density \(\rho_{\rm e}\) is still much lower than the critical electron number density \(\rho_{\rm c}\). Besides, due to the high number density of water molecules, the distance between water molecules  
Figure 3: Patterns of the supercontinuum induced by the filamentation of the femtosecond (**a–c**) BG and (**d–f**) LG beams in water. The TC values of the vortex beams are (**a,d**) \(l=0\), (**b,e**) \(l=1\), and (**c,f**) \(l=2\). is about \(1/10\) of that between air molecules. As a result, the plasma will recombine shortly after they are generated. During the femtosecond laser filamentation, the temporal evolution of plasma and laser intensity also brings variation to the wave vector, and the general wave vector \(\overset{\rightarrow}{k}\) is given by:  
\[\overset{\rightarrow}{k}=(k_{z}+\Delta k_{z})\overset{\rightarrow}{\mathbf{e} }_{z}+(k_{r}+\Delta k_{r})\overset{\rightarrow}{\mathbf{e}}_{r} \tag{3}\]  
where \(k_{z}\) and \(k_{r}\) are the initial longitudinal and radial wave vectors of the incident femtosecond pulses, \(\Delta k_{z}\) and \(\Delta k_{r}\) are the variation in the longitudinal and radial wave vectors, \(\overset{\rightarrow}{\mathbf{e}}_{z}\) and \(\overset{\rightarrow}{\mathbf{e}}_{r}\) are the unit vectors of the longitudinal and radial components. Both \(\Delta k_{z}\) and \(\Delta k_{r}\) are related to the plasma generation and temporal evolution of laser intensity [16; 17]. Since the conical emission arises from the variation in radial wave vector, here, we mainly consider the variation in radial wave vector, which is given by [16]:  
\[\Delta k_{r}=-\frac{\partial\Delta\varphi}{\partial r}\sim\frac{\omega_{0}z}{c }\bigg{(}-n_{2}\frac{\partial I(r,t)}{\partial r}+\frac{1}{2n_{0}\rho_{\rm c} }\frac{\partial\rho_{\rm e}(r,t)}{\partial r}\bigg{)} \tag{4}\]  
Figure 4: Spots of the (**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 BG beam with \(l=0\).