# Conical Emission Induced by the Filamentation of Femtosecond Vortex Beams in Water

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup

_{d}:

## 3. Results and Discussion

_{0}= 1.33 and n

_{2}= 4.1 × 10

^{−16}cm

^{2}/W) in water for 800 nm pulse is P

_{cr}= 1.761 MW [33], which is much lower than the filamentation threshold due to the strong dispersion effect [34]. In our previous work, the dependence of the filamentation threshold on the TC in the case of the LG beam was studied by interferometry [26]. It is found that the filamentation threshold of the femtosecond vortex pulse increases linearly with the TC value, and for the LG beam with l = 10, its filamentation threshold is 140 MW [26]. For the input femtosecond pulse, its energy is attenuated to 188 μJ, and the corresponding pulse power is 3.76 GW, which is much higher than the filamentation threshold of the femtosecond vortex beams in water. In this case, multiple filaments are generated.

_{2}, ${\rho}_{\mathrm{e}}(r,t)$ and ${\rho}_{\mathrm{c}}$ 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 (${\rho}_{\mathrm{c}}={\epsilon}_{0}{m}_{\mathrm{e}}{\omega}_{0}^{2}/{e}^{2}$, ${\epsilon}_{0}$, e and m

_{e}are permittivity of vacuum, electron charge and electron mass), respectively. For the 800 nm pulses, the value of ${\rho}_{\mathrm{c}}$ is about 2 × 10

^{21}cm

^{−3}, though it is a little lower than the number density of water molecule ${\rho}_{\mathrm{water}}=3.3\times {10}^{22}{\mathrm{cm}}^{-3}$. Limited by the ionization rate during femtosecond filamentation (about 1‰), the electron number density ${\rho}_{\mathrm{e}}$ is still much lower than the critical electron number density ${\rho}_{\mathrm{c}}$. Besides, due to the high number density of water molecules, the distance between water molecules is about 1/10 of that between air molecules. As a result, the plasma will recombine shortly after they are generated.

_{r}(t), i.e., converging toward the filament axis, while the contribution of the plasma in the trail of the pulse leads to a blue shift associated with a negative radial phase derivative. Hence, the conical emission is interpreted as a divergence of the anti-Stokes components of the supercontinuum induced by the plasma. No conical emission was found at Stokes shifted wavelengths. In this paper, we focus on the spectral components at the blue side of the central wavelength (i.e., visible band), $\Delta \omega $ always takes the positive values. From Equations (2) and (4), it can be obviously seen that $\Delta {k}_{r}$ is positively related to $\Delta \omega $. As shown in Figure 4a, for a newly generated spectral component, the more its frequency deviates from the central frequency of the laser pulse ${\omega}_{0}$, the larger its divergence angle with respect to the filament axis is, which can well explain the feature of the conical emission.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of the experimental setup to study the colored conical emission induced by the filamentation of femtosecond vortex beams in water. M

_{1}, M

_{2}: plane mirrors, H: half-wave plate, G: Glan prism, SLM: spatial light modulator, A: axicon lens (the dashed green rectangle indicates it can be replaced by a plano-convex focusing lens); C: quartz cuvette; F: band-pass filter; S: white screen.

**Figure 2.**Schematic diagram of the method to measure the divergence angle of supercontinuum induced by femtosecond vortex beams. θ

_{d}is the divergence angle. By moving the white screen, the light spot is recorded at two selected positions along the propagation direction of the femtosecond beam. R

_{1}and R

_{2}are the radii of the light spots captured at the two positions.

**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.

**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. (

**a**,

**c**,

**e**,

**g**,

**i**,

**k**) and (

**b**,

**d**,

**f**,

**h**,

**j**,

**l**) are measured when the distance between the back wall of the cuvette and the white screen is 8.1 cm and 23.1 cm, respectively.

**Figure 5.**Spots of (

**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 LG beam with l = 0. (

**a**,

**c**,

**e**,

**g**,

**i**,

**k**) and (

**b**,

**d**,

**f**,

**h**,

**j**,

**l**) are measured when the distance between the back wall of the cuvette and the white screen is 8.1 cm and 23.1 cm, respectively.

**Figure 6.**(

**a**) Illustration of the general wave vector. Spatial distribution of the wave vector of the laser beam after it passes the (

**b**) plano-convex focusing lens and (

**c**) axicon lens.

**Figure 7.**(

**a**) Variation of the maximum divergence angle of the supercontinuum induced by the BG beams with the wavelength. (

**b**) Variation of the maximum divergence angle of different spectral components of the supercontinuum induced by the BG beams with the TC value.

**Figure 8.**(

**a**) Variation of the maximum divergence angle of the supercontinuum induced by the LG beams with the wavelength. (

**b**) Variation of the maximum divergence angle of different spectral components of the supercontinuum induced by the LG beams with the TC value.

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**MDPI and ACS Style**

Liu, Y.; Huo, Y.; Zhu, L.; Jin, M.; Zhang, H.; Li, S.; Hua, W.
Conical Emission Induced by the Filamentation of Femtosecond Vortex Beams in Water. *Appl. Sci.* **2023**, *13*, 12435.
https://doi.org/10.3390/app132212435

**AMA Style**

Liu Y, Huo Y, Zhu L, Jin M, Zhang H, Li S, Hua W.
Conical Emission Induced by the Filamentation of Femtosecond Vortex Beams in Water. *Applied Sciences*. 2023; 13(22):12435.
https://doi.org/10.3390/app132212435

**Chicago/Turabian Style**

Liu, Yang, Yuchi Huo, Lin Zhu, Mingxing Jin, He Zhang, Suyu Li, and Wei Hua.
2023. "Conical Emission Induced by the Filamentation of Femtosecond Vortex Beams in Water" *Applied Sciences* 13, no. 22: 12435.
https://doi.org/10.3390/app132212435