## 4 Results and Discussion
For higher-order LG beams, the variation of the degree of spectral broadening with D is similar to that shown in Figure 6a (i.e., 0-order LG beam), which also exhibits plateau regions, as shown in Figure 6c-f. It can also be seen from the figure that along with the increase of TC value, the degree of spectral broadening changes little, i.e., it is not related to the TC value of LG beams, which is consistent with the work of [PERSON] et al. [39]. The SC generated by the filamentation of higher-order BG beams shows entirely different behaviors compared with that generated by higher-order LG beams. As D increases, the degree of spectral broadening firstly becomes larger, reaches its maximum at a certain position and becomes smaller gradually, as shown in Figure 7b-f. The cause for this phenomenon is actually the same as the case of the 0-order BG beam, i.e., the degree of spectral broadening is closely related to the filament length. At first, only part of the  
Figure 5: Variation of the SC induced by the filamentation of LG beams with D when the incident pulse energy is (**a,d**) 10, (**b,e**) 52 and (**c,f**) 104 \(\upmu\). (**a–c**) and (**d–f**) refers to the cases of \(l\) = 0 and \(l\) = 2. Figure 6: Variation of the SC induced by the filamentation of LG beams with D when the value of TC is (**a**) 0, (**b**) 1, (**e**) 2, (**d**) 3, (**e**) 4 and (**f**) 5. The incident pulse energy is 52 \(\upmu\)J.  
focal region of higher-order BG beams is located in water in the cuvette. As we move the cuvette (increase D), the length of the focal region of higher-order BG beams in water becomes larger, resulting in the enlargement of spectral broadening. As the focal region of higher-order BG beams is completely located in water, the degree of spectral broadening reaches its maximum, and as it gradually moves away from the cuvette, the filament length in water becomes shorter, leading to the decrease in the degree of spectral broadening. When the focal region of higher-order BG beams moves out of the cuvette (D \(>\) 60 mm), the transmitted spectrum shows little change, as shown in Figure 7. Figure 8 show the SC generated by different-order BG beams at the position where the spectral broadening reaches its maximum. According to the previous analysis, the focal region of BG beams is completely located in water at this time. It can be seen from the figure that the TC value also has little effect on the degree of the spectral broadening induced by the filament of higher-order BG beams, which is similar to the case of higher-order LG beams [39]. Both LG and BG beams belong to the vortex beam; in condensed media, the SC induced by filamentation of these beams will lose the orbital angular momentum, and therefore the spectral broadening is hardly affected by the TC value of the vortex beam. Figure 8: SC induced by the filamentation of BG beams with TC different values. The incident pulse energy is 188 \(\mu\)J.  
Figure 7: Variation of the SC induced by the filamentation of BG beams with D when the value of TC is (**a**) 0, (**b**) 1, (**c**) 2, (**d**) 3, (**e**) 4 and (**f**) 5. The incident pulse energy is 188 \(\mu\)J. It can also be seen from Figure 7 that there also exists a platform region with a constant spectral range during the filamentation of higher-order BG beams in water at different positions (spectral broadening changes little with increasing D). For example, in Figure 7b, when D is from 30 mm to 50 mm, the transmitted spectrum remains almost unchanged. It is easy to be understood since the filament formed by higher-order BG beams is shorter than the length of the cuvette. In the process of changing D, there exists a distance at which the filament formed by BG beams is completely located in water. However, the increase of TC value reduces the platform region, as shown in Figure 7a-d, which can be attributed to the fact that with the increase of TC value, the length of the Bessel region also increases, making the distance that the Bessel region is completely located in water becomes shorter.