# Conical Emission Induced by the Filamentation of Femtosecond Vortex Beams in Water
## 1 Introduction  
Apart from spin angular momentum, photons can also carry another form of angular momentum: orbital angular momentum (OAM) [1]. It is related to the helical phase front and doughnut transversal profile of light. The helical phase front is expressed in \(\exp(-il\phi)\), where \(l\) is the topological charge (TC) and equals the orbital angular quantum number, and \(\phi\) is the azimuth angle. Accordingly, a light beam carrying OAM is often called the vortex beam (i.e., the TC is larger than 0). Each photon in the vortex beams carries an OAM of \(l\hbar\). The doughnut transversal profile of the spatial distribution in intensity comes from the phase singularity at the center of the beam. The specific nature and characteristics of the vortex beam make it widely applied in many important and promising aspects, such as the optical manipulation of particles [2; 3; 4], information transmission and cryptography [5; 6], super-resolution microscopy [7; 8], and micro-processing of materials [9; 10]. These applications rely on the interaction between the vortex laser beams and matter. It has been observed that the OAM carried by the vortex beams can be transferred to matter [11; 12; 13]. When the vortex laser beam is intense enough to induce some nonlinear parametric optical processes, such as frequency doubling, optical parametric down-conversion and four-wave mixing, spectral components with new frequency are generated, and the OAM transfer from the fundamental laser to the newly generated spectral components has been demonstrated in these nonlinear optical processes [14; 15]. Since these newly generated components possess OAM, different phenomena can be observed, which in turn provides an approach to studying the nonlinear optical processes. However, when the vortex beam is too intense, it will result in photoionization of the media, generating numerous plasmas, which makes the system complicated. As a result, fluorescence, terahertz wave, supercontinuum and other electromagnetic radiation are emitted during the interaction between high-intensity laser and matter, generating many new spectral components. Among the interactions between intense laser and matter, femtosecond laser filamentation is a typical one. Femtosecond filament is formed when the Kerr self-focusing effect and plasma-defocusing effect reach a dynamic balance [16; 17]. Femtosecond filamentation has been observed in various transparent nonlinear media, including gas (e.g., air), liquid (e.g., water) and solidity (e.g., sapphire, glass). Compared with the gaseous media, it is much easier to generate filament in the condensed media due to their higher nonlinear refractivity. Theoretical and experimental research on the property, control and application of femtosecond filamentation have been conducted in the past few decades. Recently, the introduction of OAM has made the filamentation of femtosecond vortex beams a hot research topic [18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28], for the OAM can provide more freedom to study strong-field physics. Besides, the OAM can cause changes in the electron transition probability and selection rule within the atoms and molecules, affecting the excitation and photoionization of atoms and molecules [29; 30; 31]. In our recent work, we have studied the supercontinuum and fluorescence induced by the filamentation of femtosecond vortex beams in water and air [26; 27; 28]. By studying the influence of TC on the supercontinuum and fluorescence emission, we discussed the formation mechanisms of supercontinuum and fluorescence emission. The white light generated during the femtosecond filamentation in gases or condensed media generally consists of a white central part surrounded by a rainbow-like conical emission. The radial order of the spectral components is inverse of diffraction, with bluer frequencies appearing on the outside rings. Conical emission is a signature of femtosecond filamentation. As far as we know, few works focus on the conical emission induced by femtosecond vortex beams.