## 2 Femtosecond Laser Filamentation and SC Generation in Water  
Femtosecond laser filamentation is an important nonlinear process in a nonlinear medium. As femtosecond laser pulses propagate in transparent media, say air, glass and water, etc., the laser intensity keeps increasing due to the Kerr self-focusing effect, and as it increases to a certain value (1013-1014 W/cm\({}^{2}\)), the medium will be ionized by the high-intensity beam, generating a large number of plasmas which play a defocusing role. A self-induced plasma channel will be formed when the Kerr self-focusing effect and the plasma defocusing effect reach a dynamical balance [40; 41], which is called filamentation. During femtosecond filamentation, nonlinear effects such as Kerr self-focusing, multiphoton/tunnelling ionization, plasma formation and recombination lead to the change of refractive index, making the laser field undergo self-phase modulation (SPM) [40]. Accord  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Vortex Beam** & **Research Content** \\ \hline
0-order BG & control of filament [27; 28; 29; 30] \\  & accompanied nonlinear phenomena [30; 31; 32] \\ \hline Higher-order BG & control of filament [37] \\  & nonlinear propagation and filamentation [38] \\ \hline Higher-order LG & multiple filaments [33; 34; 35; 36] \\  & Coherence of SC [39] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Recent works in vortex beam.  
ing to the self-phase modulation effect, the variation of the spectrum induced by SPM can be described by [39; 40]:  
\[\Delta\omega=-\frac{\partial\Delta\varphi}{\partial t}=\frac{\omega_{0}z}{c}(-n_ {2}\frac{\partial I(r,t)}{\partial t}+\frac{1}{2n_{0}\rho_{\rm c}}\frac{ \partial\rho_{e}(r,t)}{\partial t}) \tag{1}\]  
where \(\Delta\varphi\) is the variation of the phase of the laser pulse, \(\omega_{0}\), \(z\), \(n_{2}\), \(\rho_{e}(r,t)\) and \(\rho_{e}\) are the central wavelength of the laser pulse, the propagation distance in media, second-order nonlinear refractive index, plasma density and the critical plasma density (\(\rho_{c}=\varepsilon_{0}m_{e}\omega_{0}^{2}/e^{2}\)), \(\varepsilon_{0}\), \(e\) and \(m_{e}\) are permittivity of vacuum, electron charge and electron mass), respectively. As a result, the spectral broadening results from the SPM. It can be seen from Equation (1) that the spectral broadening is affected by several factors, such as the temporal evolution of laser intensity and plasma density, nonlinear refractive index of the medium and filament length, etc.  
Obviously, the degree of spectral broadening is positively related to the filament length. The span of laser spectra after spectral broadening ranges from ultraviolet to infrared wavelength covering the entire visible range [42], and therefore it is also called white light or SC [43; 44]. SC has a wide range of applications, such as time-resolved excitation and absorption spectroscopy [45], optical pulse compression [46] and the characterization of laser-induced structural transitions [47], etc. The SC generation induced by femtosecond laser in water has been extensively studied in the past few decades, both experimentally and theoretically [39; 48; 49; 50; 51; 52; 53; 54; 55; 56]. Quite a few approaches are used to study the property of the SC and to control the SC generation [39; 48; 49; 50; 51; 52; 53; 54; 55; 56]. For example, the coherence of the SC can be studied by the interference method [39; 52]. By manipulating the parameters of femtosecond laser pulses (such as spatial distribution of intensity [48; 49], polarization [50] and pulse energy [51], etc.), doping the water with other matters [53; 54], adopting a two-color scheme [56], the SC generation can be enhanced. However, these works mainly dealt with Gaussian pulses. In this paper, we mainly focus on the characteristics of the SC generated by higher-order vortex beams in water.