# Generation of Complex Transverse Energy Flow Distributions with Autofocusing Optical Vortex Beams

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Analysis

## 3. Results of Modeling and Experiment

#### 3.1. Circular Airy Beams with Vortex Superposition

#### 3.2. Azimuthally Modulated Circular Vortex Airy Beams

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Grier, D.G. A revolution in optical manipulation. Nature
**2003**, 424, 810–816. [Google Scholar] [CrossRef] - Zhan, Q. Cylindrical vector beams: From mathematical concepts to applications. Adv. Opt. Photonics
**2009**, 1, 1–57. [Google Scholar] [CrossRef] - Fahrbach, F.O.; Simon, P.; Rohrbach, A. Microscopy with self-reconstructing beams. Nat. Photonics
**2010**, 4, 780–785. [Google Scholar] [CrossRef] - Padgett, M.; Bowman, R. Tweezers with a twist. Nat. Photonics
**2011**, 5, 343–348. [Google Scholar] [CrossRef] - Andrews, D.L. Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces; Academic Press: San Diego, CA, USA, 2011; p. 400. [Google Scholar]
- Wördemann, M. Structured Light Fields: Applications in Optical Trapping, Manipulation, and Organisation; Springer Science and Business Media: Berlin/Heidelberg, Germany, 2012; p. 136. [Google Scholar]
- Litchinitser, N.M. Structured light meets structured matter. Science
**2012**, 337, 1054–1055. [Google Scholar] [CrossRef] - Rubinsztein-Dunlop, H.; Forbes, A.; Berry, M.; Dennis, M.; Andrews, D.L.; Mansuripur, M.; Denz, C.; Alpmann, C.; Banzer, P.; Bauer, T.; et al. Roadmap on structured light. J. Opt.
**2017**, 19, 013001. [Google Scholar] [CrossRef] - Khonina, S.N.; Kazanskiy, N.L.; Karpeev, S.V.; Butt, M.A. Bessel beam: Significance and applications—A progressive review. Micromachines
**2020**, 11, 997. [Google Scholar] [CrossRef] - Kotlyar, V.V.; Kovalev, A.A.; Nalimov, A.G.; Porfirev, A.P. Evolution of an optical vortex with an initial fractional topological charge. Phys. Rev. A
**2020**, 102, 023516. [Google Scholar] [CrossRef] - Efremidis, N.K.; Christodoulides, D.N. Abruptly autofocusing waves. Opt. Lett.
**2010**, 35, 4045–4047. [Google Scholar] [CrossRef] - Papazoglou, D.G.; Efremidis, N.K.; Christodoulides, D.N.; Tzortzakis, S. Observation of abruptly autofocusing waves. Opt. Lett.
**2011**, 36, 1842–1844. [Google Scholar] [CrossRef] - Davis, J.A.; Cottrell, D.M.; Sand, D. Abruptly autofocusing vortex beam. Opt. Express
**2012**, 20, 13302–13310. [Google Scholar] [CrossRef] - Porfirev, A.P.; Khonina, S.N. Generation of the azimuthally modulated circular superlinear Airy beams. J. Opt. Soc. Am. B
**2017**, 34, 2544–2549. [Google Scholar] [CrossRef] - Jiang, Y.; Zhao, S.; Yu, W.; Zhu, X. Abruptly autofocusing property of circular Airy vortex beams with different initial launch angles. J. Opt. Soc. Am. A
**2018**, 35, 890–894. [Google Scholar] [CrossRef] - Ring, J.; Lindberg, J.; Mourka, A.; Mazilu, M.; Dholakia, K.; Dennis, M. Auto-focusing and self-healing of Pearcey beams. Opt. Express
**2012**, 20, 18955–18966. [Google Scholar] [CrossRef] [PubMed] - Chen, X.; Deng, D.; Zhuang, J.; Yang, X.; Liu, H.; Wang, G. Nonparaxial propagation of abruptly autofocusing circular Pearcey Gaussian beams. Appl. Opt.
**2018**, 57, 8418–8423. [Google Scholar] [CrossRef] - Sun, C.; Deng, D.; Wang, G.; Yang, X.; Hong, W. Abruptly autofocusing properties of radially polarized circle Pearcey vortex beams. Opt. Commun.
**2020**, 457, 124690. [Google Scholar] [CrossRef] - Khonina, S.N.; Ustinov, A.V.; Porfirev, A.P. Aberration laser beams with autofocusing properties. Appl. Opt.
**2018**, 57, 1410–1416. [Google Scholar] [CrossRef] [PubMed] - Khonina, S.N. Specular and vortical Airy beams. Opt. Commun.
**2011**, 284, 4263–4271. [Google Scholar] [CrossRef] - Vaveliuk, P.; Lencina, A.; Rodrigo, J.A.; Matos, O.M. Symmetric Airy beams. Opt. Lett.
**2014**, 39, 2370–2373. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chremmos, I.; Efremidis, N.K.; Christodoulides, D.N. Pre-engineered abruptly autofocusing beams. Opt. Lett.
**2011**, 36, 1890–1892. [Google Scholar] [CrossRef] - Li, P.; Liu, S.; Peng, T.; Xie, G.; Gan, X.; Zhao, J. Spiral autofocusing Airy beams carrying power-exponent phase vortices. Opt. Express
**2014**, 22, 7598–7606. [Google Scholar] [CrossRef] [PubMed] - Khonina, S.N.; Ustinov, A.V. Fractional Airy beams. J. Opt. Soc. Am. A
**2017**, 34, 1991–1999. [Google Scholar] [CrossRef] [PubMed] - Khonina, S.N.; Porfirev, A.P.; Ustinov, A.V. Sudden autofocusing of superlinear chirp beams. J. Opt.
**2018**, 20, 025605. [Google Scholar] [CrossRef] - Ddholakia, K.; Čižmár, T. Shaping the future of manipulation. Nat. Photonics
**2011**, 5, 335–342. [Google Scholar] [CrossRef] - Jiang, Y.F.; Huang, K.K.; Lu, X.H. Radiation force of abruptly autofocusing Airy beams on a Rayleigh particle. Opt. Express
**2013**, 21, 24413–24421. [Google Scholar] [CrossRef] - Lu, W.; Sun, X.; Chen, H.; Liu, S.; Lin, Z. Abruptly autofocusing property and optical manipulation of circular Airy beams. Phys. Rev. A
**2019**, 99, 013817. [Google Scholar] [CrossRef] - Panagiotopoulos, P.; Papazoglou, D.G.; Couairon, A.; Tzortzakis, S. Abruptly autofocusing beams enable advanced multiscale photo-polymerization. Nat. Commun.
**2013**, 4, 2622. [Google Scholar] [CrossRef] [Green Version] - Panagiotopoulos, P.; Couairon, A.; Kolesik, M.; Papazoglou, D.G.; Moloney, J.V.; Tzortzakis, S. Nonlinear plasma-assisted collapse of ring-Airy wave packets. Phys. Rev. A
**2016**, 93, 033808. [Google Scholar] [CrossRef] [Green Version] - Liu, D.; Wang, Y.; Zhai, Z.; Fang, Z.; Tao, Q.; Perrie, W.; Edwarson, S.P.; Dearden, G. Dynamic laser beam shaping for material processing using hybrid holograms. Opt. Laser Technol.
**2018**, 102, 68–73. [Google Scholar] [CrossRef] - Ustinov, A.V.; Khonina, S.N.; Khorin, P.A.; Porfirev, A.P. Control of the intensity distribution along the light spiral generated by a generalized spiral phase plate. J. Opt. Soc. Am. B
**2021**, 38, 420–427. [Google Scholar] [CrossRef] - Zhang, Y.; Li, P.; Liu, S.; Han, L.; Cheng, H.; Zhao, J. Manipulating spin-dependent splitting of vector abruptly autofocusing beam by encoding cosine-azimuthal variant phases. Opt. Express
**2016**, 24, 28409–28418. [Google Scholar] [CrossRef] - Yan, X.; Guo, L.; Cheng, M.; Chai, S. Free-space propagation of autofocusing Airy vortex beams with controllable intensity gradients. Chin. Opt. Lett.
**2019**, 17, 040101. [Google Scholar] [CrossRef] - Brimis, A.; Makris, K.G.; Papazoglou, D.G. Tornado waves. Opt. Lett.
**2020**, 45, 280–283. [Google Scholar] [CrossRef] - Wu, Y.; Xu, C.; Lin, Z.; Qiu, H.; Fu, X.; Chen, K.; Deng, D. Abruptly autofocusing polycyclic tornado ring Airy beam. New J. Phys.
**2020**, 22, 093045. [Google Scholar] [CrossRef] - Zhu, J.; Chen, Y.; Zhang, Y.; Cai, X.; Yu, S. Spin and orbital angular momentum and their conversion in cylindrical vector vortices. Opt. Lett.
**2014**, 39, 4435–4438. [Google Scholar] [CrossRef] [PubMed] - Bliokh, K.Y.; Rodriguez-Fortuno, F.; Nori, F.; Zayats, A.V. Spin-orbit interactions of light. Nat. Photonics
**2015**, 9, 796–808. [Google Scholar] [CrossRef] - Porfirev, A.P.; Ustinov, A.V.; Khonina, S.N. Polarization conversion when focusing cylindrically polarized vortex beams. Sci. Rep.
**2016**, 6, 1–9. [Google Scholar] [CrossRef] [Green Version] - Shi, P.; Du, L.; Yuan, X. Structured spin angular momentum in highly focused cylindrical vector vortex beams for optical manipulation. Opt. Express
**2018**, 26, 23449–23459. [Google Scholar] [CrossRef] - Bekshaev, A.Y.; Soskin, M.S. Transverse energy flows in vectorial fields of paraxial beams with singularities. Opt. Commun.
**2007**, 271, 332–348. [Google Scholar] [CrossRef] - Bekshaev, A.Y. Internal energy flows and instantaneous field of a monochromatic paraxial light beam. Appl. Opt.
**2012**, 51, C13–C16. [Google Scholar] [CrossRef] [PubMed] - Khonina, S.N.; Ustinov, A.V.; Logachev, V.I.; Porfirev, A.P. Properties of vortex light fields generated by generalized spiral phase plates. Phys. Rev. A
**2020**, 101, 043829. [Google Scholar] [CrossRef] - Kotlyar, V.V.; Soifer, V.A.; Khonina, S.N. An algorithm for the generation of laser beams with longitudinal periodicity: Rotating images. J. Mod. Opt.
**1997**, 44, 1409–1416. [Google Scholar] [CrossRef] - Khonina, S.N.; Kotlyar, V.V.; Soifer, V.A.; Lautanen, J.; Honkanen, M.; Turunen, J. Generating a couple of rotating nondiffracting beams using a binary-phase DOE. Optik
**1999**, 110, 137–144. [Google Scholar] - Zhang, P.; Prakash, J.; Zhang, Z.; Mills, M.S.; Efremidis, N.K.; Christodoulides, D.N.; Chen, Z.G. Trapping and guiding microparticles with morphing autofocusing Airy beams. Opt. Lett.
**2011**, 36, 2883–2885. [Google Scholar] [CrossRef] [PubMed] - Chen, M.; Huang, S.; Liu, X.; Chen, Y.; Shao, W. Optical trapping and rotating of micro-particles using the circular Airy vortex beams. Appl. Phys. B
**2019**, 125, 184. [Google Scholar] [CrossRef] - Vallee, O. Airy Functions and Applications in Physics; Vallee, O., Soares., M., Eds.; Imperial College Press: London, UK, 2004; p. 194. ISBN 978-1-86094-478-9. [Google Scholar]
- Degtyarev, S.A.; Volotovsky, S.G.; Khonina, S.N. Sublinearly chirped metalenses for forming abruptly autofocusing cylindrically polarized beams. J. Opt. Soc. Am. B
**2018**, 35, 1963–1969. [Google Scholar] [CrossRef] - Almazov, A.A.; Khonina, S.N.; Kotlyar, V.V. Using phase diffraction optical elements to shape and select laser beams consisting of a superposition of an arbitrary number of angular harmonics. J. Opt. Technol.
**2005**, 72, 391–399. [Google Scholar] [CrossRef] - Khonina, S.N.; Kotlyar, V.V.; Soifer, V.A.; Jefimovs, K.; Turunen, J. Generation and selection of laser beams represented by a superposition of two angular harmonics. J. Mod. Opt.
**2004**, 51, 761–773. [Google Scholar] [CrossRef]

**Figure 1.**Experimental setup for the investigation of the generation and propagation of designed autofocusing optical vortex (OV) beams: PH, pinhole (aperture size of 40 µm); L1, L2, and L3 are lenses (f

_{1}= 350, f

_{2}= 350, and f

_{3}= 150 mm, respectively); SLM, spatial light modulator (HOLOEYE, PLUTO VIS with a 1920 × 1080 pixel resolution); D, circular aperture; and CAM, video camera.

**Figure 2.**Simulation results for the field defined by Equation (1), which does not have any angular dependence: (

**a**) amplitude and (

**b**) phase of the input field ($x,y\in [-1\mathrm{mm},1\mathrm{mm}]$); (

**c**) longitudinal distribution of the amplitude ($z\in [20\mathrm{mm},500\mathrm{mm}]$, $y\in [-1\mathrm{mm},1\mathrm{mm}]$), and transverse patterns ($x,y\in [-0.25\mathrm{mm},0.25\mathrm{mm}]$) of the field amplitude at different distances (top line): (

**d**) z = 150 mm, (

**e**) ${z}_{foc}=200\mathrm{mm}$, and (

**f**) z = 250 mm, as well as the corresponding distributions of the transverse energy flow density (TEFD) amplitude in these planes (bottom line). The red color corresponds to the radial component $\left|{F}_{\mathsf{\rho}}\left(\mathsf{\rho},\mathsf{\theta}\right)\right|$, the blue color corresponds to the angular component $\left|{F}_{\mathsf{\theta}}\left(\mathsf{\rho},\mathsf{\theta}\right)\right|$, and the arrows show the direction of flow.

**Figure 3.**Simulation results for the field defined by Equation (1) in the presence of a single vortex phase of order ${m}_{1}=1$ (the rest of the description is as in Figure 2).

**Figure 4.**Simulation results for the field defined by Equation (17) with azimuthal modulation of the order of q = 2 (the rest of the description is as in Figure 2).

**Table 1.**Results of simulations and experiments for the field defined by Equation (14) with different orders of vortex terms m

_{1}, m

_{2}.

Input Amplitude and Phase (2 mm × 2 mm) | Longitudinal Intensity Distribution (2 mm × 300 mm) | Transverse Distribution | ||
---|---|---|---|---|

z = 150 mm | z = 200 mm | z = 250 mm | ||

m_{1} = 2, m_{2} = 1 | Simulation Experiment | Simulation: intensity (1.6 mm × 1.6 mm) Simulation: TEFD (1 mm × 1 mm) Experiment: intensity (1.6 mm × 1.6 mm) | ||

m_{1} = 2, m_{2} = −1 | Simulation Experiment | Simulation: intensity (1.6 mm × 1.6 mm) Simulation: TEFD (1 mm × 1 mm) Experiment: intensity (1.6 mm × 1.6 mm) |

**Table 2.**Results of modeling and experiment for the field defined by Equation (18) with azimuthal modulation of order q = 2 with different orders of additional vortex phase singularity m.

Input Amplitude and Phase (2 mm × 2 mm) |
Longitudinal Intensity Distribution (1.6 mm × 300 mm) |
Transverse Intensity Distribution (1.6 mm × 1.6 mm) | ||
---|---|---|---|---|

z = 100 mm | z = 200 mm | z = 300 mm | ||

q = 2, m = 1 | Simulation | Simulation | ||

Experiment | Experiment | |||

q = 2, m = 2 | Simulation | Simulation | ||

Experiment | Experiment | |||

**Table 3.**Results of TEFD simulation for the field defined by Equation (18) with azimuthal modulation of order q = 2 and with different orders of additional vortex phase singularity m.

Input Amplitude and Phase (2 mm × 2 mm) | Transverse Distribution (1 mm × 1 mm) | |
---|---|---|

Value | z = 150 mm z = 175 mm z = 200 mm z = 225 mm z = 250 mm | |

q = 2, m = 1 | $\left|E\left(\mathsf{\rho},\mathsf{\theta}\right)\right|$ | |

$\left|F\left(\mathsf{\rho},\mathsf{\theta}\right)\right|$ | ||

q = 2, m = 2 | $\left|E\left(\mathsf{\rho},\mathsf{\theta}\right)\right|$ | |

$\left|F\left(\mathsf{\rho},\mathsf{\theta}\right)\right|$ |

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## Share and Cite

**MDPI and ACS Style**

Khonina, S.N.; Porfirev, A.P.; Ustinov, A.V.; Butt, M.A.
Generation of Complex Transverse Energy Flow Distributions with Autofocusing Optical Vortex Beams. *Micromachines* **2021**, *12*, 297.
https://doi.org/10.3390/mi12030297

**AMA Style**

Khonina SN, Porfirev AP, Ustinov AV, Butt MA.
Generation of Complex Transverse Energy Flow Distributions with Autofocusing Optical Vortex Beams. *Micromachines*. 2021; 12(3):297.
https://doi.org/10.3390/mi12030297

**Chicago/Turabian Style**

Khonina, Svetlana N., Alexey P. Porfirev, Andrey V. Ustinov, and Muhammad Ali Butt.
2021. "Generation of Complex Transverse Energy Flow Distributions with Autofocusing Optical Vortex Beams" *Micromachines* 12, no. 3: 297.
https://doi.org/10.3390/mi12030297