# Dynamics of Fractional Vortex Beams at Fraunhofer Diffraction Zone

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## Abstract

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## 1. Introduction

## 2. Theoretical Insights and Numerical Results

## 3. Experimental Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

FVBs | Fractional vortex beams |

TC | Topological charge |

SLM | Spatial light modulator |

FWHM | Full width at half maximum |

CMOS | Complementary Metal-oxide semiconductor |

CCD | Charge-coupled device |

NF | Neutral filter |

## References

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**Figure 1.**Numerical simulations. (

**a**) Fractional vortex beam with $\alpha =2.15$: (

**a.1**) initial phase profile, (

**a.2**) far-field phase profile, (

**a.3**) far-field intensity profile, and (

**a.4**) far-field intensity profile in logarithmic scale. (

**b**) Fractional vortex beam with $\alpha =3.63$: (

**b.1**) initial phase profile, (

**b.2**) far-field phase profile, (

**b.3**) far-field intensity profile, and (

**b.4**) far-field intensity profile in logarithmic scale. (

**c**) Normalized distance measured from the position of the unit vortices to the center. (

**c.1**) Distance for the resident vortices ${\mathrm{v}}_{r}$ in transitions from $1\to 2$ to $7\to 8$. (

**c.2**) Distance for the tourist vortices ${\mathrm{v}}_{t}$, present only in transitions starting with odd n, from $1\to 2$ to $7\to 8$.

**Figure 2.**(

**a**) Experimental setup: a collimated beam (HeNe, $\lambda =632.8$ nm) hits the SLM programmed with one of the phase profiles. The focused field is imaged by means of the CMOS camera through a 2f system (f $=400$ mm). Notation: NF, neutral filter; BS, beam splitter; AP, aperture. (

**b**,

**c**) Sample intensity profiles (logarithmic scale) for transition [$\alpha ,1\to 2$], (

**b.1**) $\alpha =1.38$, and (

**b.2**) $\alpha =1.82$. Transition [$\alpha ,2\to 3$], (

**c.1**) $\alpha =2.34$, and (

**c.2**) $\alpha =2.88$.

**Figure 3.**Distances of resident vortices ${\mathrm{v}}_{r}$ with fractional steps of $0.5$. (

**a**) Numerical simulations for different waists ${\omega}_{0}$: $0.3$ (red square), $0.6$ (blue circle), and $0.9$ mm (black asterisk). Experimental results (green triangle) for the measured waist ${\omega}_{0}=0.77$ mm. Top right: focal plane intensity profile sample (logarithmic scale) for $\alpha =1.50$. (

**b**) Experimental Gaussian beam profile (HeNe, THORLABS HNL100L, $\lambda =632.8$ nm).

**Figure 4.**(

**a**) Distances of resident vortices with fractional steps of $0.5$; numerical simulation (red square), and experimental measurements (green triangle) comparison. (

**b**) Normalized focal-plane intensity profiles for $\alpha =3.50$, ${\omega}_{0}=0.77$ mm, $\lambda =632.8$ nm. (

**b.1**) Numerical simulation and (

**b.2**) experimental results. (

**c**) Normalized focal plane intensity profiles (logarithmic scale) for $\alpha =3.50$, ${\omega}_{0}=0.77$ mm, $\lambda =632.8$ nm. (

**c.1**) Numerical simulation and (

**c.2**) experimental results.

**Figure 5.**(

**a**) Distances of resident vortices ${\mathrm{v}}_{r}$ (${\omega}_{0}=0.77$ mm, $\lambda =632.8$ nm), for phase transitions $2\to 3$ and $4\to 5$. Numerical simulations (red circles for $2\to 3$ and green square for $4\to 5$) and experimental results (black triangles for $2\to 3$ and blue asterisks for $4\to 5$) comparison. (

**b**) Experimental results: distances of resident vortices for transitions $1\to 2$, $2\to 3$, $3\to 4$, $4\to 5$, and $5\to 6$ with $\alpha $ steps of $0.02$.

**Figure 6.**Experimental results: far-field intensity profiles (logarithmic scale). (

**a**) Phase transition $\alpha ,2\to 3$. (

**a.1**) $\alpha =2.18$. (

**a.2**) $\alpha =2.86$. (

**b**) Phase transition $\alpha ,4\to 5$. (

**b.1**) $\alpha =4.38$. (

**b.2**) $\alpha =4.74$. Top right: zoomed local minimum for resident vortex position.

**Table 1.**Estimated topological charge $\left|\alpha \right|$ via interpolation from resident vortex experimental distance curves, d${}_{exp}$(µm). Data correspond to Figure 6 samples: $\alpha =2.18$, $\alpha =2.86$, $\alpha =4.38$, and $\alpha =4.74$.

Input $\mathit{\alpha}$ | d (µm) | d${}_{\mathit{exp}}$ (µm) | ${\mathit{\alpha}}_{\mathit{exp}}$ | Err $\mathbf{\Delta}\mathit{\alpha}$ |
---|---|---|---|---|

2.18 | 462.3 | 462.0 ± 35.2 | 2.18 ± 0.02 | <1% |

2.86 | 88.0 | 83.6 ± 4.4 | 2.87 ± 0.01 | <0.7% |

4.38 | 352.3 | 358.6 ± 13.2 | 4.40 ± 0.04 | <1% |

4.74 | 176.1 | 176.0 ± 2.2 | 4.74 ± 0.02 | <0.4% |

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**MDPI and ACS Style**

Peters, E.; Funes, G.; Martínez-León, L.; Tajahuerce, E.
Dynamics of Fractional Vortex Beams at Fraunhofer Diffraction Zone. *Photonics* **2022**, *9*, 479.
https://doi.org/10.3390/photonics9070479

**AMA Style**

Peters E, Funes G, Martínez-León L, Tajahuerce E.
Dynamics of Fractional Vortex Beams at Fraunhofer Diffraction Zone. *Photonics*. 2022; 9(7):479.
https://doi.org/10.3390/photonics9070479

**Chicago/Turabian Style**

Peters, Eduardo, Gustavo Funes, L. Martínez-León, and Enrique Tajahuerce.
2022. "Dynamics of Fractional Vortex Beams at Fraunhofer Diffraction Zone" *Photonics* 9, no. 7: 479.
https://doi.org/10.3390/photonics9070479