Abstract

Photons in an optical vortex usually carry orbital angular momentum, which boosts the application of the micro-rotation of absorbing particles and quantum information encoding. Such photons propagate along a straight line in free space or follow a curved trace once guided by an optical fiber. Teleportation of an optical vortex using a beam with non-diffraction and self-healing is quite challenging. We demonstrate the manipulation of the propagation trace of an optical vortex with a symmetric Airy beam (SAB) and found that the SAB experiences self-rotation with the implementation of a topological phase structure of coaxial vortex. Slight misalignment of the vortex and the SAB enables the guiding of the vortex into one of the self-accelerating channels. Multiple off-axis vortices embedded in SAB are also demonstrated to follow the trajectory of the major lobe for the SAB beam. The Poynting vector for the beams proves the direction of the energy flow corresponding to the intensity distribution. Hence, we anticipate that the proposed vortex symmetric Airy beam (VSAB) will provide new possibilities for optical manipulation and optical communication.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (5)

C.-W. Qiu and Y. Yang, “Vortex generation reaches a new plateau,” Science 357(6352), 645 (2017).
[Crossref] [PubMed]

W. Walasik, S. Z. Silahli, and N. M. Litchinitser, “Dynamics of necklace beams in nonlinear colloidal suspensions,” Sci. Rep. 7(1), 11709 (2017).
[Crossref] [PubMed]

R.-P. Chen and C.-Q. Dai, “Vortex solitons of the (3+1)-dimensional spatially modulated cubic–quintic nonlinear Schrödinger equation with the transverse modulation,” Nonlinear Dyn. 90(3), 1563–1570 (2017).
[Crossref]

R.-P. Chen and C.-Q. Dai, “Three-dimensional vector solitons and their stabilities in a Kerr medium with spatially inhomogeneous nonlinearity and transverse modulation,” Nonlinear Dyn. 88(4), 2807–2816 (2017).
[Crossref]

Y. Yang, G. Thirunavukkarasu, M. Babiker, and J. Yuan, “Orbital-angular-momentum mode selection by rotationally symmetric superposition of chiral states with application to electron vortex beams,” Phys. Rev. Lett. 119(9), 094802 (2017).
[Crossref] [PubMed]

2016 (3)

D. Garoli, P. Zilio, Y. Gorodetski, F. Tantussi, and F. De Angelis, “Beaming of helical light from plasmonic vortices via adiabatically tapered nanotip,” Nano Lett. 16(10), 6636–6643 (2016).
[Crossref] [PubMed]

L. Clark, G. Guzzinati, A. Béché, A. Lubk, and J. Verbeeck, “Symmetry-constrained electron vortex propagation,” Phys. Rev. A 93(6), 063840 (2016).
[Crossref]

X. Jiang, Y. Li, B. Liang, J.-C. Cheng, and L. Zhang, “Convert acoustic resonances to orbital angular momentum,” Phys. Rev. Lett. 117(3), 034301 (2016).
[Crossref] [PubMed]

2015 (9)

C. W. Clark, R. Barankov, M. G. Huber, M. Arif, D. G. Cory, and D. A. Pushin, “Controlling neutron orbital angular momentum,” Nature 525(7570), 504–506 (2015).
[Crossref] [PubMed]

R.-P. Chen, Z. Chen, K.-H. Chew, P.-G. Li, Z. Yu, J. Ding, and S. He, “Structured caustic vector vortex optical field: manipulating optical angular momentum flux and polarization rotation,” Sci. Rep. 5(1), 10628 (2015).
[Crossref] [PubMed]

Y.-X. Ren, R.-D. Lu, and L. Gong, “Tailoring light with a digital micromirror device,” Ann. Phys. 527(7-8), 447–470 (2015).
[Crossref]

X.-Y. Ding, Y.-X. Ren, and R.-D. Lu, “Shaping super-Gaussian beam through amplitude reflection pattern using digital micromirror device,” Sci. Chin. G: Phys. Mechanics Astronomy 58, 034202 (2015).
[Crossref]

M. Mohammad, S. M.-L. Omar, N. O. S. Malcolm, R. Brandon, M. Mehul, P. J. L. Martin, J. P. Miles, J. G. Daniel, and W. B. Robert, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17(3), 033033 (2015).
[Crossref]

Y.-X. Ren, Z.-X. Fang, L. Gong, K. Huang, Y. Chen, and R.-D. Lu, “Dynamic generation of Ince-Gaussian modes with a digital micromirror device,” J. Appl. Phys. 117(13), 133106 (2015).
[Crossref]

Y.-X. Ren, Z.-X. Fang, L. Gong, K. Huang, Y. Chen, and R.-D. Lu, “Digital generation and control of Hermite–Gaussian modes with an amplitude digital micromirror device,” J. Opt. 17(12), 125604 (2015).
[Crossref]

Z.-X. Fang, Y.-X. Ren, L. Gong, P. Vaveliuk, Y. Chen, and R.-D. Lu, “Shaping symmetric Airy beam through binary amplitude modulation for ultralong needle focus,” J. Appl. Phys. 118(20), 203102 (2015).
[Crossref]

Y. Chen, Z.-X. Fang, Y.-X. Ren, L. Gong, and R.-D. Lu, “Generation and characterization of a perfect vortex beam with a large topological charge through a digital micromirror device,” Appl. Opt. 54(27), 8030–8035 (2015).
[Crossref] [PubMed]

2014 (5)

P. Vaveliuk, A. Lencina, J. A. Rodrigo, and O. Martinez Matos, “Symmetric Airy beams,” Opt. Lett. 39(8), 2370–2373 (2014).
[Crossref] [PubMed]

X.-Y. Ding, Y.-X. Ren, L. Gong, Z.-X. Fang, and R.-D. Lu, “Microscopic lithography with pixelate diffraction of a digital micro-mirror device for micro-lens fabrication,” Appl. Opt. 53(24), 5307–5311 (2014).
[Crossref] [PubMed]

R. Jáuregui and P. A. Quinto-Su, “On the general properties of symmetric incomplete Airy beams,” J. Opt. Soc. Am. A 31(11), 2484–2488 (2014).
[Crossref] [PubMed]

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. K. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref] [PubMed]

L. Gong, Y.-X. Ren, W.-W. Liu, M. Wang, M.-C. Zhong, Z.-Q. Wang, and Y.-M. Li, “Generation of cylindrically polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116(18), 183105 (2014).
[Crossref]

2013 (5)

L. Clark, A. Béché, G. Guzzinati, A. Lubk, M. Mazilu, R. Van Boxem, and J. Verbeeck, “Exploiting lens aberrations to create electron-vortex beams,” Phys. Rev. Lett. 111(6), 064801 (2013).
[Crossref] [PubMed]

C. Rosales-Guzmán, M. Mazilu, J. Baumgartl, V. Rodríguez-Fajardo, R. Ramos-García, and K. Dholakia, “Collision of propagating vortices embedded within Airy beams,” J. Opt. 15(4), 044001 (2013).
[Crossref]

P. Rose, F. Diebel, M. Boguslawski, and C. Denz, “Airy beam induced optical routing,” Appl. Phys. Lett. 102(10), 101101 (2013).
[Crossref]

R.-P. Chen, K.-H. Chew, and S. He, “Dynamic control of collapse in a vortex Airy beam,” Sci. Rep. 3(1), 1406 (2013).
[Crossref] [PubMed]

J. Zhao, P. Zhang, D. Deng, J. Liu, Y. Gao, I. D. Chremmos, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Observation of self-accelerating Bessel-like optical beams along arbitrary trajectories,” Opt. Lett. 38(4), 498–500 (2013).
[Crossref] [PubMed]

2012 (3)

X. Chu, “Propagation of an Airy beam with a spiral phase,” Opt. Lett. 37(24), 5202–5204 (2012).
[Crossref] [PubMed]

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012).
[Crossref]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

2011 (4)

2010 (6)

2008 (2)

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008).
[Crossref] [PubMed]

2007 (3)

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

T. Higashiuchi and H. Sakaguchi, “Vortex interaction and the two-dimensional dark soliton,” Laser Phys. 17(2), 221–225 (2007).
[Crossref]

2006 (1)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref] [PubMed]

1999 (2)

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Ahmed, N.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Alfano, R. R.

Allen, L.

L. Allen, M. J. Padgett, and M. Babiker, “IV The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Arecchi, F. T.

F. Lenzini, S. Residori, F. T. Arecchi, and U. Bortolozzo, “Optical vortex interaction and generation via nonlinear wave mixing,” Phys. Rev. A 84(6), 061801 (2011).
[Crossref]

Arif, M.

C. W. Clark, R. Barankov, M. G. Huber, M. Arif, D. G. Cory, and D. A. Pushin, “Controlling neutron orbital angular momentum,” Nature 525(7570), 504–506 (2015).
[Crossref] [PubMed]

Babiker, M.

Y. Yang, G. Thirunavukkarasu, M. Babiker, and J. Yuan, “Orbital-angular-momentum mode selection by rotationally symmetric superposition of chiral states with application to electron vortex beams,” Phys. Rev. Lett. 119(9), 094802 (2017).
[Crossref] [PubMed]

L. Allen, M. J. Padgett, and M. Babiker, “IV The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[Crossref]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Barankov, R.

C. W. Clark, R. Barankov, M. G. Huber, M. Arif, D. G. Cory, and D. A. Pushin, “Controlling neutron orbital angular momentum,” Nature 525(7570), 504–506 (2015).
[Crossref] [PubMed]

Baumgartl, J.

C. Rosales-Guzmán, M. Mazilu, J. Baumgartl, V. Rodríguez-Fajardo, R. Ramos-García, and K. Dholakia, “Collision of propagating vortices embedded within Airy beams,” J. Opt. 15(4), 044001 (2013).
[Crossref]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Béché, A.

L. Clark, G. Guzzinati, A. Béché, A. Lubk, and J. Verbeeck, “Symmetry-constrained electron vortex propagation,” Phys. Rev. A 93(6), 063840 (2016).
[Crossref]

L. Clark, A. Béché, G. Guzzinati, A. Lubk, M. Mazilu, R. Van Boxem, and J. Verbeeck, “Exploiting lens aberrations to create electron-vortex beams,” Phys. Rev. Lett. 111(6), 064801 (2013).
[Crossref] [PubMed]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Boguslawski, M.

P. Rose, F. Diebel, M. Boguslawski, and C. Denz, “Airy beam induced optical routing,” Appl. Phys. Lett. 102(10), 101101 (2013).
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L. Clark, A. Béché, G. Guzzinati, A. Lubk, M. Mazilu, R. Van Boxem, and J. Verbeeck, “Exploiting lens aberrations to create electron-vortex beams,” Phys. Rev. Lett. 111(6), 064801 (2013).
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C. Rosales-Guzmán, M. Mazilu, J. Baumgartl, V. Rodríguez-Fajardo, R. Ramos-García, and K. Dholakia, “Collision of propagating vortices embedded within Airy beams,” J. Opt. 15(4), 044001 (2013).
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J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
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M. Mohammad, S. M.-L. Omar, N. O. S. Malcolm, R. Brandon, M. Mehul, P. J. L. Martin, J. P. Miles, J. G. Daniel, and W. B. Robert, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17(3), 033033 (2015).
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M. Mohammad, S. M.-L. Omar, N. O. S. Malcolm, R. Brandon, M. Mehul, P. J. L. Martin, J. P. Miles, J. G. Daniel, and W. B. Robert, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17(3), 033033 (2015).
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M. Mohammad, S. M.-L. Omar, N. O. S. Malcolm, R. Brandon, M. Mehul, P. J. L. Martin, J. P. Miles, J. G. Daniel, and W. B. Robert, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17(3), 033033 (2015).
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T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. K. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
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Y.-X. Ren, Z.-X. Fang, L. Gong, K. Huang, Y. Chen, and R.-D. Lu, “Dynamic generation of Ince-Gaussian modes with a digital micromirror device,” J. Appl. Phys. 117(13), 133106 (2015).
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Y.-X. Ren, Z.-X. Fang, L. Gong, K. Huang, Y. Chen, and R.-D. Lu, “Digital generation and control of Hermite–Gaussian modes with an amplitude digital micromirror device,” J. Opt. 17(12), 125604 (2015).
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A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
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F. Lenzini, S. Residori, F. T. Arecchi, and U. Bortolozzo, “Optical vortex interaction and generation via nonlinear wave mixing,” Phys. Rev. A 84(6), 061801 (2011).
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M. Mohammad, S. M.-L. Omar, N. O. S. Malcolm, R. Brandon, M. Mehul, P. J. L. Martin, J. P. Miles, J. G. Daniel, and W. B. Robert, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17(3), 033033 (2015).
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Rodríguez-Fajardo, V.

C. Rosales-Guzmán, M. Mazilu, J. Baumgartl, V. Rodríguez-Fajardo, R. Ramos-García, and K. Dholakia, “Collision of propagating vortices embedded within Airy beams,” J. Opt. 15(4), 044001 (2013).
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C. Rosales-Guzmán, M. Mazilu, J. Baumgartl, V. Rodríguez-Fajardo, R. Ramos-García, and K. Dholakia, “Collision of propagating vortices embedded within Airy beams,” J. Opt. 15(4), 044001 (2013).
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G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
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Y. Yang, G. Thirunavukkarasu, M. Babiker, and J. Yuan, “Orbital-angular-momentum mode selection by rotationally symmetric superposition of chiral states with application to electron vortex beams,” Phys. Rev. Lett. 119(9), 094802 (2017).
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J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
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Van Boxem, R.

L. Clark, A. Béché, G. Guzzinati, A. Lubk, M. Mazilu, R. Van Boxem, and J. Verbeeck, “Exploiting lens aberrations to create electron-vortex beams,” Phys. Rev. Lett. 111(6), 064801 (2013).
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Z.-X. Fang, Y.-X. Ren, L. Gong, P. Vaveliuk, Y. Chen, and R.-D. Lu, “Shaping symmetric Airy beam through binary amplitude modulation for ultralong needle focus,” J. Appl. Phys. 118(20), 203102 (2015).
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P. Vaveliuk, A. Lencina, J. A. Rodrigo, and O. Martinez Matos, “Symmetric Airy beams,” Opt. Lett. 39(8), 2370–2373 (2014).
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L. Clark, G. Guzzinati, A. Béché, A. Lubk, and J. Verbeeck, “Symmetry-constrained electron vortex propagation,” Phys. Rev. A 93(6), 063840 (2016).
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L. Clark, A. Béché, G. Guzzinati, A. Lubk, M. Mazilu, R. Van Boxem, and J. Verbeeck, “Exploiting lens aberrations to create electron-vortex beams,” Phys. Rev. Lett. 111(6), 064801 (2013).
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W. Walasik, S. Z. Silahli, and N. M. Litchinitser, “Dynamics of necklace beams in nonlinear colloidal suspensions,” Sci. Rep. 7(1), 11709 (2017).
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A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
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L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
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Phys. Rev. A (3)

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G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
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Sci. Chin. G: Phys. Mechanics Astronomy (1)

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Figures (6)

Fig. 1
Fig. 1 Numerical simulation for on-axis VSAB (l = 1) propagating in free space. (a)The sliced maps that are surrounded by two side-view profiles, the blue dashed arrows and blue lines indicate the beam propagation dynamics; The side-view profile of an on-axis VSAB (l = 1), its corresponding intensity profiles (b1)~(b4) and topological phase (b5)~(b8) at the planes marked by dashed lines in (b) with ξ = 0.05, 1.08, 2, and 2.7, respectively; Panels b1-b4 correspond to slices a1, a2, a4 and a5, respectively.
Fig. 2
Fig. 2 Schematic of the experimental setup. Insets near the DMD and camera show the displayed binary amplitude hologram and the experimental beam profile of an off-axis VSAB (l = 1).
Fig. 3
Fig. 3 Experimental demonstration of on-axis VSABs. (a) a side view of an on-axis VSAB (l = 1); (a1)-(a4) transverse beam profiles at the planes z=0.05 z 0 , 1.1 z 0 , 2 z 0 , and 2.7 z 0 ( z 0 =1.8cm), marked by the dashed lines in (a), respectively; (b) the side view of an on-axis VSAB (l = 2) and the corresponding snapshots (b1-b4) at the planes z=0.05 z 0 , 1.25 z 0 , 2.1 z 0 and 2.8 z 0 , respectively. The scale bars for all figures are the same as shown in (b4). The rotation angles as a function of propagation distance are shown in (c) for l = 1 and (d) l = 2.
Fig. 4
Fig. 4 Propagation dynamics of an off-axis VSAB in free space. (a) Simulation and (b) experimental side-view profiles, panels (a1-a4, b1-b4) show the transverse beam profiles at the positions ξ = 0.05, 1, 2.1 and 2.8 marked by the dashed lines, and panels (a5-a8) map the transverse phase structure at the corresponding positions. The fork-like phase structures are marked by a red circle on each phase panel.
Fig. 5
Fig. 5 Numerical demonstrations of the energy flow (white arrows) superimposed on the transversal intensity maps for VSABs at different distances. The Poynting vectors for on-axis VSAB with topological charge (a) l = 1, and (b) l = 2 are pointing to the azimuthal direction near the self-focusing plane, and away from the beam axis near the major lobe at a position far away from the beam focus. In contrast, the Poynting vector for an off-axis VSAB points to the beam axis before and near focus with a small amount of component circulating the beam axis. At far field, the major lobe without vortex shows outward energy flow, while the lobe imbedded with vortex exhibits circulation on the Poynting vector map. The propagation coordinates for (a, b) are the same as described in Fig. 2, while positions in (c) are the same as those in Fig. 4.
Fig. 6
Fig. 6 The simulation (a) intensity and (b) phase, (c) experimental intensity profiles for multiple vortices imbedded in the symmetric Airy beam. Each column shows the results for VSAB with vortex pair on the side, vortex pair along the diagonal, triple vortices, and four vortices. Red circles in (b) mark the positions of phase singularities.

Equations (7)

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u 0 (s,ξ=0)=Ai(s)exp(as)
U 1 (k)=exp(a k 2 )exp( i 3 ( | k | 3 3 a 2 | k |i a 3 ))
u 1 (s)= 1 2π U 1 (k) exp(iks)dk = exp(as) 2π 0 exp( i k 3 3 ks) dk+ exp(as) 2π 0 exp( i k 3 3 +ks) dk = exp(as) 2 [ Ai(s)+iGi(s) ]+ exp(as) 2 [ Ai(s)+iGi(s) ]
Gi(η)= 1 π 0 sin( 1 3 t 3 +ηt) dt
u 2 (x,y,z=0)= u 1 ( s x ) u 1 ( s y )
u(x,y)= u 2 (x,y) j=1 N [ (x x j )+i(y y j ) ] l [ (x x j ) 2 + (y y j ) 2 ] l/2
S = c 4π E × B = c 8π [ iω(u u u u)+2ωk | u | 2 e z ]

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