# Conical Emission Induced by the Filamentation of Femtosecond Vortex Beams in Water
## 2 Experimental Setup  
To explore the characteristics of the colored conical emission induced by the filamentation of femtosecond vortex beams in water, we design an experimental setup schematically shown in Figure 1. The experiment is carried out using an amplified Ti: sapphire laser (coherent Libra). The duration, central wavelength, energy and repetition rate of the emitted pulses are 50 fs, 800 nm, 3.0 mJ and 1 kHz, respectively. An energy attenuator, which consists of a half-wave plate, H and a Glan prism, G, adjusts the pulse energy to a desired value. Since the pulse energy emitted from the laser is too high for the femtosecond filamentation in water, it is attenuated to 188 uJ and fixed at this value throughout this paper. The laser beam spot size is adjusted by an iris, which is set as 10 mm in the experiment, and therefore, the light beam can perfectly enter the liquid crystal screen of a spatial light modulator SLM (Hamamatsu, LCOS-SLM, X13138-02, Shizuoka, Japan). The femtosecond laser pulses emitted from the laser system are linearly polarized, and their polarization direction is horizontal, which is identical to the alignment direction of liquid crystal molecules in the SLM. Vortex beams with different TCs can be generated by loading the corresponding phase diagrams (e.g., figures 2(a\({}^{\prime}\))-2(c\({}^{\prime}\)) in Ref. [26]; figures 2(a)-2(c) in Ref. [28]) to the liquid crystal screen of the SLM. Since the SLM only modulates the phase of the laser beam and does not change the polarization state of the laser beam, the laser beams reflected by the SLM (i.e., the generated vortex beams) are still linearly polarized. In other words, the spin angular momentum of the generated vortex beam is zero. In the experiment, to study the characteristics of the colored conical emission induced by the femtosecond LG and BG beams, the vortex beams are focused by two kinds of optical lenses (which are placed at the dashed green frame): one is a plano-convex focusing lens (fused silica) with a focal length of f = 75 mm (generation LG beams), and the other one is an axicon lens with a top angle of 170 degrees (generation BG beams). The vortex beams are focused into a quartz cuvette C (50 mm \(\times\) 10 mm \(\times\) 40 mm) filled with water and form filaments, generating the supercontinuum. To avoid contribution from the cuvette walls, the beams are focused close to the center of the cuvette. The femtosecond filament has a certain length along the propagation direction of femtosecond laser pulses, and the filament length is affected by many factors, such as the parameters of the laser beam, the type and nature of the transparent medium and the focusing condition. It is not easy to determine the specific location of the filament in the water. More importantly, each part of the filament emits the supercontinuum. These factors make it difficult to determine the tip of the conical emission. As a result, we designed an indirect method to measure the divergence angle of supercontinuum induced by femtosecond vortex beams. The schematic diagram of the method is shown in Figure 2. A white screen S is placed on a moving stage, and its surface is perpendicular to the propagation direction of the femtosecond laser beam. By moving the white screen along the propagation direction, the light spots are captured at two selected positions, at which the distances between the back wall of the cuvette and the white screen are 8.1 cm and 23.1 cm, respectively. The spacing between the two positions is \(D\) = 15 cm. Once the radius of the light spot at the two positions is measured, we can obtain the maximum divergence angle of the supercontinuum \(\theta_{\rm d}\):  
\[\theta_{\rm d}=\arctan(R_{1}-R_{2})/D \tag{1}\]  
Figure 1: Schematic diagram of the experimental setup to study the colored conical emission induced by the filamentation of femtosecond vortex beams in water.