Abstract

We propose a method to generate specially shaped high-order singular beams of pre-designed intensity distributions. Such a method does not a priori assume a phase formula, but rather relies on the “cake-cutting and assembly” approach to achieve the azimuthal phase gradient for beam shaping, inspired by the orbital motion trajectory change of an artificial satellite. Based on our method, several typical vortex beams with desired intensity patterns are experimentally generated. As an example, we realize optical trapping and transportation of microorganisms with a triangle-shaped vortex beam, demonstrating the applicability of such unconventional vortex beams in optical trapping and manipulation.

© 2019 Chinese Laser Press

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References

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    [Crossref]
  36. Y. Lamhot, A. Barak, C. Rotschild, M. Segev, M. Saraf, E. Lifshitz, A. Marmur, R. El-Ganainy, and D. N. Christodoulides, “Optical control of thermocapillary effects in complex nanofluids,” Phys. Rev. Lett. 103, 264503 (2009).
    [Crossref]
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    [Crossref]
  38. A. Terray, J. Oakey, and D. W. M. Marr, “Microfluidic control using colloidal devices,” Science 296, 1841–1844 (2002).
    [Crossref]
  39. A. Balijepalli, J. J. Gorman, S. K. Gupta, and T. W. LeBrun, “Significantly improved trapping lifetime of nanoparticles in an optical trap using feedback control,” Nano Lett. 12, 2347–2351 (2012).
    [Crossref]

2018 (3)

Y. J. Shen, Z. S. Wan, Y. Meng, X. Fu, and M. L. Gong, “Polygonal vortex beams,” IEEE Photon. J. 10, 1503016 (2018).
[Crossref]

Y. Liang, M. Lei, S. Yan, M. Li, Y. Cai, Z. Wang, X. Yu, and B. Yao, “Rotating of low-refractive-index microparticles with a quasi-perfect optical vortex,” Appl. Opt. 57, 79–84 (2018).
[Crossref]

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7, 677–682 (2018).
[Crossref]

2017 (1)

L. Yang, D. Qian, C. Xin, Z. Hu, S. Ji, D. Wu, Y. Hu, J. Li, W. Huang, and J. Chu, “Direct laser writing of complex microtubes using femtosecond vortex beams,” Appl. Phys. Lett. 110, 221103 (2017).
[Crossref]

2016 (2)

2015 (4)

J. Zhao, I. D. Chremmos, D. Song, D. N. Christodoulides, N. K. Efremidis, and Z. Chen, “Curved singular beams for three-dimensional particle manipulation,” Sci. Rep. 5, 12086 (2015).
[Crossref]

A. Lehmuskero, P. Johansson, H. Rubinsztein-Dunlop, L. Tong, and M. Kall, “Laser trapping of colloidal metal nanoparticles,” ACS Nano 9, 3453–3469 (2015).
[Crossref]

M. Z. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Creating and probing of a perfect vortex in situ with an optically trapped particle,” Opt. Rev. 22, 162–165 (2015).
[Crossref]

M. Daly, M. Sergides, and S. N. Chormaic, “Optical trapping and manipulation of micrometer and submicrometer particles,” Laser Photon. Rev. 9, 309–329 (2015).
[Crossref]

2014 (2)

P. Li, S. Liu, T. Peng, G. Xie, X. Gan, and J. Zhao, “Spiral autofocusing Airy beams carrying power-exponent-phase vortices,” Opt. Express 22, 7598–7606 (2014).
[Crossref]

M. Gecevicius, R. Drevinskas, M. Beresna, and P. G. Kazansky, “Single beam optical vortex tweezers with tunable orbital angular momentum,” Appl. Phys. Lett. 104, 231110 (2014).
[Crossref]

2013 (2)

2012 (2)

A. Balijepalli, J. J. Gorman, S. K. Gupta, and T. W. LeBrun, “Significantly improved trapping lifetime of nanoparticles in an optical trap using feedback control,” Nano Lett. 12, 2347–2351 (2012).
[Crossref]

P. Zhang, D. Hernandez, D. Cannan, Y. Hu, S. Fardad, S. Huang, J. C. Chen, D. N. Christodoulides, and Z. Chen, “Trapping and rotating microparticles and bacteria with Moire-based optical propelling beams,” Biomed. Opt. Express 3, 1891–1897 (2012).
[Crossref]

2011 (5)

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
[Crossref]

N. Shitrit, I. Bretner, Y. Gorodetski, V. Kleiner, and E. Hasman, “Optical spin Hall effects in plasmonic chains,” Nano Lett. 11, 2038–2042 (2011).
[Crossref]

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36, 2883–2885 (2011).
[Crossref]

C. P. Lapointe, T. G. Mason, and I. I. Smalyukh, “Towards total photonic control of complex-shaped colloids by vortex beams,” Opt. Express 19, 18182–18189 (2011).
[Crossref]

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
[Crossref]

2010 (2)

M. Krishnan, N. Mojarad, P. Kukura, and V. Sandoghdar, “Geometry-induced electrostatic trapping of nanometric objects in a fluid,” Nature 467, 692–695 (2010).
[Crossref]

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[Crossref]

2009 (1)

Y. Lamhot, A. Barak, C. Rotschild, M. Segev, M. Saraf, E. Lifshitz, A. Marmur, R. El-Ganainy, and D. N. Christodoulides, “Optical control of thermocapillary effects in complex nanofluids,” Phys. Rev. Lett. 103, 264503 (2009).
[Crossref]

2008 (2)

E. G. Abramochkin, K. N. Afanasiev, V. G. Volostnikov, A. V. Korobtsov, S. P. Kotova, N. N. Losevsky, A. M. Mayorova, and E. V. Razueva, “Formation of vortex light fields of specified intensity for laser micromanipulation,” Biol. Bull. Russ. Acad. Sci. Phys. 72, 68–70 (2008).
[Crossref]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref]

2006 (1)

C. S. Guo, Y. Zhang, Y. J. Han, J. P. Ding, and H. T. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259, 449–454 (2006).
[Crossref]

2005 (3)

2004 (1)

E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Phys. Usp. 47, 1177–1203 (2004).
[Crossref]

2003 (2)

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

J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. 28, 872–874 (2003).
[Crossref]

2002 (1)

A. Terray, J. Oakey, and D. W. M. Marr, “Microfluidic control using colloidal devices,” Science 296, 1841–1844 (2002).
[Crossref]

2000 (1)

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[Crossref]

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, 8185–8189 (1992).
[Crossref]

Abramochkin, E. G.

E. G. Abramochkin, K. N. Afanasiev, V. G. Volostnikov, A. V. Korobtsov, S. P. Kotova, N. N. Losevsky, A. M. Mayorova, and E. V. Razueva, “Formation of vortex light fields of specified intensity for laser micromanipulation,” Biol. Bull. Russ. Acad. Sci. Phys. 72, 68–70 (2008).
[Crossref]

E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Phys. Usp. 47, 1177–1203 (2004).
[Crossref]

Afanasiev, K. N.

E. G. Abramochkin, K. N. Afanasiev, V. G. Volostnikov, A. V. Korobtsov, S. P. Kotova, N. N. Losevsky, A. M. Mayorova, and E. V. Razueva, “Formation of vortex light fields of specified intensity for laser micromanipulation,” Biol. Bull. Russ. Acad. Sci. Phys. 72, 68–70 (2008).
[Crossref]

Allen, L.

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, 8185–8189 (1992).
[Crossref]

Amato-Grill, J.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref]

Arita, Y.

M. Z. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Creating and probing of a perfect vortex in situ with an optically trapped particle,” Opt. Rev. 22, 162–165 (2015).
[Crossref]

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38, 4919–4922 (2013).
[Crossref]

Arlt, J.

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[Crossref]

Balijepalli, A.

A. Balijepalli, J. J. Gorman, S. K. Gupta, and T. W. LeBrun, “Significantly improved trapping lifetime of nanoparticles in an optical trap using feedback control,” Nano Lett. 12, 2347–2351 (2012).
[Crossref]

Barak, A.

Y. Lamhot, A. Barak, C. Rotschild, M. Segev, M. Saraf, E. Lifshitz, A. Marmur, R. El-Ganainy, and D. N. Christodoulides, “Optical control of thermocapillary effects in complex nanofluids,” Phys. Rev. Lett. 103, 264503 (2009).
[Crossref]

Bate, R. R.

R. R. Bate, D. D. Mueller, and J. E. White, Fundamentals of Astrodynamics (Dover Publications, 1971).

Battin, R. H.

R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics (American Institute of Aeronautics and Astronautics, 1999).

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, 8185–8189 (1992).
[Crossref]

Beresna, M.

M. Gecevicius, R. Drevinskas, M. Beresna, and P. G. Kazansky, “Single beam optical vortex tweezers with tunable orbital angular momentum,” Appl. Phys. Lett. 104, 231110 (2014).
[Crossref]

Bowman, R.

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
[Crossref]

Bretner, I.

N. Shitrit, I. Bretner, Y. Gorodetski, V. Kleiner, and E. Hasman, “Optical spin Hall effects in plasmonic chains,” Nano Lett. 11, 2038–2042 (2011).
[Crossref]

Cai, Y.

Cannan, D.

Chen, J. C.

Chen, M.

Chen, M. Z.

M. Z. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Creating and probing of a perfect vortex in situ with an optically trapped particle,” Opt. Rev. 22, 162–165 (2015).
[Crossref]

Chen, Z.

Cho, S.-W.

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[Crossref]

Chormaic, S. N.

M. Daly, M. Sergides, and S. N. Chormaic, “Optical trapping and manipulation of micrometer and submicrometer particles,” Laser Photon. Rev. 9, 309–329 (2015).
[Crossref]

Chremmos, I. D.

J. Zhao, I. D. Chremmos, D. Song, D. N. Christodoulides, N. K. Efremidis, and Z. Chen, “Curved singular beams for three-dimensional particle manipulation,” Sci. Rep. 5, 12086 (2015).
[Crossref]

Christodoulides, D. N.

J. Zhao, I. D. Chremmos, D. Song, D. N. Christodoulides, N. K. Efremidis, and Z. Chen, “Curved singular beams for three-dimensional particle manipulation,” Sci. Rep. 5, 12086 (2015).
[Crossref]

P. Zhang, D. Hernandez, D. Cannan, Y. Hu, S. Fardad, S. Huang, J. C. Chen, D. N. Christodoulides, and Z. Chen, “Trapping and rotating microparticles and bacteria with Moire-based optical propelling beams,” Biomed. Opt. Express 3, 1891–1897 (2012).
[Crossref]

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36, 2883–2885 (2011).
[Crossref]

Y. Lamhot, A. Barak, C. Rotschild, M. Segev, M. Saraf, E. Lifshitz, A. Marmur, R. El-Ganainy, and D. N. Christodoulides, “Optical control of thermocapillary effects in complex nanofluids,” Phys. Rev. Lett. 103, 264503 (2009).
[Crossref]

Chu, J.

L. Yang, D. Qian, C. Xin, Z. Hu, S. Ji, D. Wu, Y. Hu, J. Li, W. Huang, and J. Chu, “Direct laser writing of complex microtubes using femtosecond vortex beams,” Appl. Phys. Lett. 110, 221103 (2017).
[Crossref]

Cizmar, T.

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
[Crossref]

Curtis, J. E.

Daly, M.

M. Daly, M. Sergides, and S. N. Chormaic, “Optical trapping and manipulation of micrometer and submicrometer particles,” Laser Photon. Rev. 9, 309–329 (2015).
[Crossref]

Dholakia, K.

M. Z. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Creating and probing of a perfect vortex in situ with an optically trapped particle,” Opt. Rev. 22, 162–165 (2015).
[Crossref]

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38, 4919–4922 (2013).
[Crossref]

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
[Crossref]

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[Crossref]

Ding, J. P.

C. S. Guo, Y. Zhang, Y. J. Han, J. P. Ding, and H. T. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259, 449–454 (2006).
[Crossref]

Dong, Y.

Dowling, J. P.

K. T. Kapale and J. P. Dowling, “Vortex phase qubit: generating arbitrary, counterrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95, 173601 (2005).
[Crossref]

Drevinskas, R.

M. Gecevicius, R. Drevinskas, M. Beresna, and P. G. Kazansky, “Single beam optical vortex tweezers with tunable orbital angular momentum,” Appl. Phys. Lett. 104, 231110 (2014).
[Crossref]

Efremidis, N. K.

J. Zhao, I. D. Chremmos, D. Song, D. N. Christodoulides, N. K. Efremidis, and Z. Chen, “Curved singular beams for three-dimensional particle manipulation,” Sci. Rep. 5, 12086 (2015).
[Crossref]

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36, 2883–2885 (2011).
[Crossref]

El-Ganainy, R.

Y. Lamhot, A. Barak, C. Rotschild, M. Segev, M. Saraf, E. Lifshitz, A. Marmur, R. El-Ganainy, and D. N. Christodoulides, “Optical control of thermocapillary effects in complex nanofluids,” Phys. Rev. Lett. 103, 264503 (2009).
[Crossref]

Fardad, S.

Fu, X.

Y. J. Shen, Z. S. Wan, Y. Meng, X. Fu, and M. L. Gong, “Polygonal vortex beams,” IEEE Photon. J. 10, 1503016 (2018).
[Crossref]

Gan, X.

Gecevicius, M.

M. Gecevicius, R. Drevinskas, M. Beresna, and P. G. Kazansky, “Single beam optical vortex tweezers with tunable orbital angular momentum,” Appl. Phys. Lett. 104, 231110 (2014).
[Crossref]

Gong, M. L.

Y. J. Shen, Z. S. Wan, Y. Meng, X. Fu, and M. L. Gong, “Polygonal vortex beams,” IEEE Photon. J. 10, 1503016 (2018).
[Crossref]

Gorman, J. J.

A. Balijepalli, J. J. Gorman, S. K. Gupta, and T. W. LeBrun, “Significantly improved trapping lifetime of nanoparticles in an optical trap using feedback control,” Nano Lett. 12, 2347–2351 (2012).
[Crossref]

Gorodetski, Y.

N. Shitrit, I. Bretner, Y. Gorodetski, V. Kleiner, and E. Hasman, “Optical spin Hall effects in plasmonic chains,” Nano Lett. 11, 2038–2042 (2011).
[Crossref]

Grier, D. G.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref]

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

J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. 28, 872–874 (2003).
[Crossref]

Guo, C. S.

C. S. Guo, Y. Zhang, Y. J. Han, J. P. Ding, and H. T. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259, 449–454 (2006).
[Crossref]

Gupta, S. K.

A. Balijepalli, J. J. Gorman, S. K. Gupta, and T. W. LeBrun, “Significantly improved trapping lifetime of nanoparticles in an optical trap using feedback control,” Nano Lett. 12, 2347–2351 (2012).
[Crossref]

Han, Y. J.

C. S. Guo, Y. Zhang, Y. J. Han, J. P. Ding, and H. T. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259, 449–454 (2006).
[Crossref]

Hasman, E.

N. Shitrit, I. Bretner, Y. Gorodetski, V. Kleiner, and E. Hasman, “Optical spin Hall effects in plasmonic chains,” Nano Lett. 11, 2038–2042 (2011).
[Crossref]

Hernandez, D.

Hitomi, T.

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[Crossref]

Hu, Y.

L. Yang, D. Qian, C. Xin, Z. Hu, S. Ji, D. Wu, Y. Hu, J. Li, W. Huang, and J. Chu, “Direct laser writing of complex microtubes using femtosecond vortex beams,” Appl. Phys. Lett. 110, 221103 (2017).
[Crossref]

P. Zhang, D. Hernandez, D. Cannan, Y. Hu, S. Fardad, S. Huang, J. C. Chen, D. N. Christodoulides, and Z. Chen, “Trapping and rotating microparticles and bacteria with Moire-based optical propelling beams,” Biomed. Opt. Express 3, 1891–1897 (2012).
[Crossref]

Hu, Z.

L. Yang, D. Qian, C. Xin, Z. Hu, S. Ji, D. Wu, Y. Hu, J. Li, W. Huang, and J. Chu, “Direct laser writing of complex microtubes using femtosecond vortex beams,” Appl. Phys. Lett. 110, 221103 (2017).
[Crossref]

Huang, S.

Huang, W.

L. Yang, D. Qian, C. Xin, Z. Hu, S. Ji, D. Wu, Y. Hu, J. Li, W. Huang, and J. Chu, “Direct laser writing of complex microtubes using femtosecond vortex beams,” Appl. Phys. Lett. 110, 221103 (2017).
[Crossref]

Ji, S.

L. Yang, D. Qian, C. Xin, Z. Hu, S. Ji, D. Wu, Y. Hu, J. Li, W. Huang, and J. Chu, “Direct laser writing of complex microtubes using femtosecond vortex beams,” Appl. Phys. Lett. 110, 221103 (2017).
[Crossref]

Johansson, P.

A. Lehmuskero, P. Johansson, H. Rubinsztein-Dunlop, L. Tong, and M. Kall, “Laser trapping of colloidal metal nanoparticles,” ACS Nano 9, 3453–3469 (2015).
[Crossref]

Kall, M.

A. Lehmuskero, P. Johansson, H. Rubinsztein-Dunlop, L. Tong, and M. Kall, “Laser trapping of colloidal metal nanoparticles,” ACS Nano 9, 3453–3469 (2015).
[Crossref]

Kang, M.

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[Crossref]

Kapale, K. T.

K. T. Kapale and J. P. Dowling, “Vortex phase qubit: generating arbitrary, counterrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95, 173601 (2005).
[Crossref]

Kazansky, P. G.

M. Gecevicius, R. Drevinskas, M. Beresna, and P. G. Kazansky, “Single beam optical vortex tweezers with tunable orbital angular momentum,” Appl. Phys. Lett. 104, 231110 (2014).
[Crossref]

Kim, H.

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[Crossref]

Kleiner, V.

N. Shitrit, I. Bretner, Y. Gorodetski, V. Kleiner, and E. Hasman, “Optical spin Hall effects in plasmonic chains,” Nano Lett. 11, 2038–2042 (2011).
[Crossref]

Korobtsov, A. V.

E. G. Abramochkin, K. N. Afanasiev, V. G. Volostnikov, A. V. Korobtsov, S. P. Kotova, N. N. Losevsky, A. M. Mayorova, and E. V. Razueva, “Formation of vortex light fields of specified intensity for laser micromanipulation,” Biol. Bull. Russ. Acad. Sci. Phys. 72, 68–70 (2008).
[Crossref]

Kotova, S. P.

E. G. Abramochkin, K. N. Afanasiev, V. G. Volostnikov, A. V. Korobtsov, S. P. Kotova, N. N. Losevsky, A. M. Mayorova, and E. V. Razueva, “Formation of vortex light fields of specified intensity for laser micromanipulation,” Biol. Bull. Russ. Acad. Sci. Phys. 72, 68–70 (2008).
[Crossref]

Krishnan, M.

M. Krishnan, N. Mojarad, P. Kukura, and V. Sandoghdar, “Geometry-induced electrostatic trapping of nanometric objects in a fluid,” Nature 467, 692–695 (2010).
[Crossref]

Kukura, P.

M. Krishnan, N. Mojarad, P. Kukura, and V. Sandoghdar, “Geometry-induced electrostatic trapping of nanometric objects in a fluid,” Nature 467, 692–695 (2010).
[Crossref]

Lamhot, Y.

Y. Lamhot, A. Barak, C. Rotschild, M. Segev, M. Saraf, E. Lifshitz, A. Marmur, R. El-Ganainy, and D. N. Christodoulides, “Optical control of thermocapillary effects in complex nanofluids,” Phys. Rev. Lett. 103, 264503 (2009).
[Crossref]

Lao, G.

Lapointe, C. P.

LeBrun, T. W.

A. Balijepalli, J. J. Gorman, S. K. Gupta, and T. W. LeBrun, “Significantly improved trapping lifetime of nanoparticles in an optical trap using feedback control,” Nano Lett. 12, 2347–2351 (2012).
[Crossref]

Lee, B.

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[Crossref]

Lee, S.-Y.

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[Crossref]

Lehmuskero, A.

A. Lehmuskero, P. Johansson, H. Rubinsztein-Dunlop, L. Tong, and M. Kall, “Laser trapping of colloidal metal nanoparticles,” ACS Nano 9, 3453–3469 (2015).
[Crossref]

Lei, M.

Li, J.

L. Yang, D. Qian, C. Xin, Z. Hu, S. Ji, D. Wu, Y. Hu, J. Li, W. Huang, and J. Chu, “Direct laser writing of complex microtubes using femtosecond vortex beams,” Appl. Phys. Lett. 110, 221103 (2017).
[Crossref]

Li, M.

Li, P.

Liang, Y.

Lifshitz, E.

Y. Lamhot, A. Barak, C. Rotschild, M. Segev, M. Saraf, E. Lifshitz, A. Marmur, R. El-Ganainy, and D. N. Christodoulides, “Optical control of thermocapillary effects in complex nanofluids,” Phys. Rev. Lett. 103, 264503 (2009).
[Crossref]

Lin, J.

Liu, S.

Losevsky, N. N.

E. G. Abramochkin, K. N. Afanasiev, V. G. Volostnikov, A. V. Korobtsov, S. P. Kotova, N. N. Losevsky, A. M. Mayorova, and E. V. Razueva, “Formation of vortex light fields of specified intensity for laser micromanipulation,” Biol. Bull. Russ. Acad. Sci. Phys. 72, 68–70 (2008).
[Crossref]

Lu, X.

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7, 677–682 (2018).
[Crossref]

Marion, J. B.

S. T. Thornton and J. B. Marion, Classical Dynamics of Particles and Systems, 5th ed. (Brooks Cole, 2003).

Marmur, A.

Y. Lamhot, A. Barak, C. Rotschild, M. Segev, M. Saraf, E. Lifshitz, A. Marmur, R. El-Ganainy, and D. N. Christodoulides, “Optical control of thermocapillary effects in complex nanofluids,” Phys. Rev. Lett. 103, 264503 (2009).
[Crossref]

Marr, D. W. M.

A. Terray, J. Oakey, and D. W. M. Marr, “Microfluidic control using colloidal devices,” Science 296, 1841–1844 (2002).
[Crossref]

Mason, T. G.

Mayorova, A. M.

E. G. Abramochkin, K. N. Afanasiev, V. G. Volostnikov, A. V. Korobtsov, S. P. Kotova, N. N. Losevsky, A. M. Mayorova, and E. V. Razueva, “Formation of vortex light fields of specified intensity for laser micromanipulation,” Biol. Bull. Russ. Acad. Sci. Phys. 72, 68–70 (2008).
[Crossref]

Mazilu, M.

M. Z. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Creating and probing of a perfect vortex in situ with an optically trapped particle,” Opt. Rev. 22, 162–165 (2015).
[Crossref]

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38, 4919–4922 (2013).
[Crossref]

Meng, Y.

Y. J. Shen, Z. S. Wan, Y. Meng, X. Fu, and M. L. Gong, “Polygonal vortex beams,” IEEE Photon. J. 10, 1503016 (2018).
[Crossref]

Mills, M. S.

Mojarad, N.

M. Krishnan, N. Mojarad, P. Kukura, and V. Sandoghdar, “Geometry-induced electrostatic trapping of nanometric objects in a fluid,” Nature 467, 692–695 (2010).
[Crossref]

Mueller, D. D.

R. R. Bate, D. D. Mueller, and J. E. White, Fundamentals of Astrodynamics (Dover Publications, 1971).

Niu, H. B.

Oakey, J.

A. Terray, J. Oakey, and D. W. M. Marr, “Microfluidic control using colloidal devices,” Science 296, 1841–1844 (2002).
[Crossref]

Padgett, M.

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
[Crossref]

Park, J.

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[Crossref]

Peng, T.

Peng, X.

Prakash, J.

Qian, D.

L. Yang, D. Qian, C. Xin, Z. Hu, S. Ji, D. Wu, Y. Hu, J. Li, W. Huang, and J. Chu, “Direct laser writing of complex microtubes using femtosecond vortex beams,” Appl. Phys. Lett. 110, 221103 (2017).
[Crossref]

Razueva, E. V.

E. G. Abramochkin, K. N. Afanasiev, V. G. Volostnikov, A. V. Korobtsov, S. P. Kotova, N. N. Losevsky, A. M. Mayorova, and E. V. Razueva, “Formation of vortex light fields of specified intensity for laser micromanipulation,” Biol. Bull. Russ. Acad. Sci. Phys. 72, 68–70 (2008).
[Crossref]

Roichman, Y.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref]

Rotschild, C.

Y. Lamhot, A. Barak, C. Rotschild, M. Segev, M. Saraf, E. Lifshitz, A. Marmur, R. El-Ganainy, and D. N. Christodoulides, “Optical control of thermocapillary effects in complex nanofluids,” Phys. Rev. Lett. 103, 264503 (2009).
[Crossref]

Rubinsztein-Dunlop, H.

A. Lehmuskero, P. Johansson, H. Rubinsztein-Dunlop, L. Tong, and M. Kall, “Laser trapping of colloidal metal nanoparticles,” ACS Nano 9, 3453–3469 (2015).
[Crossref]

Sandoghdar, V.

M. Krishnan, N. Mojarad, P. Kukura, and V. Sandoghdar, “Geometry-induced electrostatic trapping of nanometric objects in a fluid,” Nature 467, 692–695 (2010).
[Crossref]

Saraf, M.

Y. Lamhot, A. Barak, C. Rotschild, M. Segev, M. Saraf, E. Lifshitz, A. Marmur, R. El-Ganainy, and D. N. Christodoulides, “Optical control of thermocapillary effects in complex nanofluids,” Phys. Rev. Lett. 103, 264503 (2009).
[Crossref]

Segev, M.

Y. Lamhot, A. Barak, C. Rotschild, M. Segev, M. Saraf, E. Lifshitz, A. Marmur, R. El-Ganainy, and D. N. Christodoulides, “Optical control of thermocapillary effects in complex nanofluids,” Phys. Rev. Lett. 103, 264503 (2009).
[Crossref]

Sergides, M.

M. Daly, M. Sergides, and S. N. Chormaic, “Optical trapping and manipulation of micrometer and submicrometer particles,” Laser Photon. Rev. 9, 309–329 (2015).
[Crossref]

Shen, Y. J.

Y. J. Shen, Z. S. Wan, Y. Meng, X. Fu, and M. L. Gong, “Polygonal vortex beams,” IEEE Photon. J. 10, 1503016 (2018).
[Crossref]

Shitrit, N.

N. Shitrit, I. Bretner, Y. Gorodetski, V. Kleiner, and E. Hasman, “Optical spin Hall effects in plasmonic chains,” Nano Lett. 11, 2038–2042 (2011).
[Crossref]

Smalyukh, I. I.

Song, D.

J. Zhao, I. D. Chremmos, D. Song, D. N. Christodoulides, N. K. Efremidis, and Z. Chen, “Curved singular beams for three-dimensional particle manipulation,” Sci. Rep. 5, 12086 (2015).
[Crossref]

Spreeuw, R. J. C.

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, 8185–8189 (1992).
[Crossref]

Sun, B.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref]

Tao, S. H.

Terray, A.

A. Terray, J. Oakey, and D. W. M. Marr, “Microfluidic control using colloidal devices,” Science 296, 1841–1844 (2002).
[Crossref]

Thornton, S. T.

S. T. Thornton and J. B. Marion, Classical Dynamics of Particles and Systems, 5th ed. (Brooks Cole, 2003).

Tong, L.

A. Lehmuskero, P. Johansson, H. Rubinsztein-Dunlop, L. Tong, and M. Kall, “Laser trapping of colloidal metal nanoparticles,” ACS Nano 9, 3453–3469 (2015).
[Crossref]

Volostnikov, V. G.

E. G. Abramochkin, K. N. Afanasiev, V. G. Volostnikov, A. V. Korobtsov, S. P. Kotova, N. N. Losevsky, A. M. Mayorova, and E. V. Razueva, “Formation of vortex light fields of specified intensity for laser micromanipulation,” Biol. Bull. Russ. Acad. Sci. Phys. 72, 68–70 (2008).
[Crossref]

E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Phys. Usp. 47, 1177–1203 (2004).
[Crossref]

Wan, Z. S.

Y. J. Shen, Z. S. Wan, Y. Meng, X. Fu, and M. L. Gong, “Polygonal vortex beams,” IEEE Photon. J. 10, 1503016 (2018).
[Crossref]

Wang, H. T.

C. S. Guo, Y. Zhang, Y. J. Han, J. P. Ding, and H. T. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259, 449–454 (2006).
[Crossref]

Wang, J.

Wang, Z.

White, J. E.

R. R. Bate, D. D. Mueller, and J. E. White, Fundamentals of Astrodynamics (Dover Publications, 1971).

Woerdman, J. P.

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, 8185–8189 (1992).
[Crossref]

Wright, E. M.

M. Z. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Creating and probing of a perfect vortex in situ with an optically trapped particle,” Opt. Rev. 22, 162–165 (2015).
[Crossref]

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38, 4919–4922 (2013).
[Crossref]

Wu, D.

L. Yang, D. Qian, C. Xin, Z. Hu, S. Ji, D. Wu, Y. Hu, J. Li, W. Huang, and J. Chu, “Direct laser writing of complex microtubes using femtosecond vortex beams,” Appl. Phys. Lett. 110, 221103 (2017).
[Crossref]

Xie, G.

Xin, C.

L. Yang, D. Qian, C. Xin, Z. Hu, S. Ji, D. Wu, Y. Hu, J. Li, W. Huang, and J. Chu, “Direct laser writing of complex microtubes using femtosecond vortex beams,” Appl. Phys. Lett. 110, 221103 (2017).
[Crossref]

Yan, S.

Yang, L.

L. Yang, D. Qian, C. Xin, Z. Hu, S. Ji, D. Wu, Y. Hu, J. Li, W. Huang, and J. Chu, “Direct laser writing of complex microtubes using femtosecond vortex beams,” Appl. Phys. Lett. 110, 221103 (2017).
[Crossref]

Yang, Y.

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7, 677–682 (2018).
[Crossref]

Y. Yang, Y. Dong, C. Zhao, and Y. Cai, “Generation and propagation of an anomalous vortex beam,” Opt. Lett. 38, 5418–5421 (2013).
[Crossref]

Yao, B.

Yu, X.

Yuan, X. C.

Zeng, J.

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7, 677–682 (2018).
[Crossref]

Zhang, P.

Zhang, Y.

C. S. Guo, Y. Zhang, Y. J. Han, J. P. Ding, and H. T. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259, 449–454 (2006).
[Crossref]

Zhang, Z.

Zhao, C.

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7, 677–682 (2018).
[Crossref]

Y. Yang, Y. Dong, C. Zhao, and Y. Cai, “Generation and propagation of an anomalous vortex beam,” Opt. Lett. 38, 5418–5421 (2013).
[Crossref]

Zhao, D.

Zhao, J.

J. Zhao, I. D. Chremmos, D. Song, D. N. Christodoulides, N. K. Efremidis, and Z. Chen, “Curved singular beams for three-dimensional particle manipulation,” Sci. Rep. 5, 12086 (2015).
[Crossref]

P. Li, S. Liu, T. Peng, G. Xie, X. Gan, and J. Zhao, “Spiral autofocusing Airy beams carrying power-exponent-phase vortices,” Opt. Express 22, 7598–7606 (2014).
[Crossref]

Zhu, X.

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7, 677–682 (2018).
[Crossref]

ACS Nano (1)

A. Lehmuskero, P. Johansson, H. Rubinsztein-Dunlop, L. Tong, and M. Kall, “Laser trapping of colloidal metal nanoparticles,” ACS Nano 9, 3453–3469 (2015).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (1)

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[Crossref]

Appl. Phys. Lett. (2)

M. Gecevicius, R. Drevinskas, M. Beresna, and P. G. Kazansky, “Single beam optical vortex tweezers with tunable orbital angular momentum,” Appl. Phys. Lett. 104, 231110 (2014).
[Crossref]

L. Yang, D. Qian, C. Xin, Z. Hu, S. Ji, D. Wu, Y. Hu, J. Li, W. Huang, and J. Chu, “Direct laser writing of complex microtubes using femtosecond vortex beams,” Appl. Phys. Lett. 110, 221103 (2017).
[Crossref]

Biol. Bull. Russ. Acad. Sci. Phys. (1)

E. G. Abramochkin, K. N. Afanasiev, V. G. Volostnikov, A. V. Korobtsov, S. P. Kotova, N. N. Losevsky, A. M. Mayorova, and E. V. Razueva, “Formation of vortex light fields of specified intensity for laser micromanipulation,” Biol. Bull. Russ. Acad. Sci. Phys. 72, 68–70 (2008).
[Crossref]

Biomed. Opt. Express (1)

IEEE Photon. J. (1)

Y. J. Shen, Z. S. Wan, Y. Meng, X. Fu, and M. L. Gong, “Polygonal vortex beams,” IEEE Photon. J. 10, 1503016 (2018).
[Crossref]

Laser Photon. Rev. (1)

M. Daly, M. Sergides, and S. N. Chormaic, “Optical trapping and manipulation of micrometer and submicrometer particles,” Laser Photon. Rev. 9, 309–329 (2015).
[Crossref]

Nano Lett. (3)

N. Shitrit, I. Bretner, Y. Gorodetski, V. Kleiner, and E. Hasman, “Optical spin Hall effects in plasmonic chains,” Nano Lett. 11, 2038–2042 (2011).
[Crossref]

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[Crossref]

A. Balijepalli, J. J. Gorman, S. K. Gupta, and T. W. LeBrun, “Significantly improved trapping lifetime of nanoparticles in an optical trap using feedback control,” Nano Lett. 12, 2347–2351 (2012).
[Crossref]

Nanophotonics (1)

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7, 677–682 (2018).
[Crossref]

Nat. Photonics (2)

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
[Crossref]

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
[Crossref]

Nature (2)

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

M. Krishnan, N. Mojarad, P. Kukura, and V. Sandoghdar, “Geometry-induced electrostatic trapping of nanometric objects in a fluid,” Nature 467, 692–695 (2010).
[Crossref]

Opt. Commun. (1)

C. S. Guo, Y. Zhang, Y. J. Han, J. P. Ding, and H. T. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259, 449–454 (2006).
[Crossref]

Opt. Express (5)

Opt. Lett. (4)

Opt. Rev. (1)

M. Z. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Creating and probing of a perfect vortex in situ with an optically trapped particle,” Opt. Rev. 22, 162–165 (2015).
[Crossref]

Photon. Res. (1)

Phys. Rev. A (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, 8185–8189 (1992).
[Crossref]

Phys. Rev. Lett. (3)

K. T. Kapale and J. P. Dowling, “Vortex phase qubit: generating arbitrary, counterrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95, 173601 (2005).
[Crossref]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref]

Y. Lamhot, A. Barak, C. Rotschild, M. Segev, M. Saraf, E. Lifshitz, A. Marmur, R. El-Ganainy, and D. N. Christodoulides, “Optical control of thermocapillary effects in complex nanofluids,” Phys. Rev. Lett. 103, 264503 (2009).
[Crossref]

Phys. Usp. (1)

E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Phys. Usp. 47, 1177–1203 (2004).
[Crossref]

Sci. Rep. (1)

J. Zhao, I. D. Chremmos, D. Song, D. N. Christodoulides, N. K. Efremidis, and Z. Chen, “Curved singular beams for three-dimensional particle manipulation,” Sci. Rep. 5, 12086 (2015).
[Crossref]

Science (1)

A. Terray, J. Oakey, and D. W. M. Marr, “Microfluidic control using colloidal devices,” Science 296, 1841–1844 (2002).
[Crossref]

Other (3)

S. T. Thornton and J. B. Marion, Classical Dynamics of Particles and Systems, 5th ed. (Brooks Cole, 2003).

R. R. Bate, D. D. Mueller, and J. E. White, Fundamentals of Astrodynamics (Dover Publications, 1971).

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Supplementary Material (1)

NameDescription
» Visualization 1       The visualization shows the motion of the trapped yeast cell along a triangular path as driven by the ACV beam.

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Figures (6)

Fig. 1.
Fig. 1. (a) Schematic of trajectory change of an artificial satellite undergoing uniform circular motion. If the OAM of the artificial satellite is increased sufficiently, the artificial satellite will fly away from its original trajectory as shown by the blue curve. (b)–(d) Sketched trajectories for a “predesigned” artificial satellite motion, forming triangle, square, and hexagon patterns. (e)–(g) Desired phase distributions for generation of vortex beams with different intensity patterns corresponding to those trajectories in (b)–(d). The corner stages are marked with white dashed lines in (e)–(g).
Fig. 2.
Fig. 2. Illustration of the “cake-cutting” method of phase engineering for generation of SOV beams. (a)–(c) Phase diagrams of vortex beams of topological charge l=6,7,8, respectively. (d) Assembled phase diagram from slices in (a)–(c) for generation of a triangle-shaped beam as shown in Figs. 1(b) and 1(e).
Fig. 3.
Fig. 3. Experimental setup for the generation of SOV beams and their application in cell transportation. SLM, spatial light modulator; Lens L1 and L2, 4f system; F, spatial filter; CCD, charge-coupled device; BS, beam splitter; O, oil immersion objective lens; Lens L3, condenser lens; WLS, white light source.
Fig. 4.
Fig. 4. Experimental demonstrations of the SOV beams. (a)–(c) show the holograms used to generate the SOV beams. (d)–(f) show corresponding intensity patterns in triangle, square, and hexagon shapes without fine-shaping. We take the topological charges l=6, 7, 8 in (a) and (d), and l=10, 11, 12 in (b), (c) and (e), (f).
Fig. 5.
Fig. 5. Experimental demonstrations of the SOV beams. (a1)–(a5) show the phase distributions and (b1)–(b5) show the corresponding holograms used to generate the SOV beams; (c1)–(c5) are the corresponding intensity patterns in triangle, square, hexagon, irregular quadrilateral, and spiral-like shapes after fine-shaping with phase engineering. In (a4), the topological charge of the vortex is indicated, and in (b4), the difference in fringe numbers shows also the topological charge l=28. The corner stages are marked with white dashed lines in (a1)–(a3) and (a5).
Fig. 6.
Fig. 6. Demonstration of optical trapping and transporting of a yeast cell by the SOV beam. (a)–(d) are snapshots showing the motion of the trapped yeast cell along a triangular path as driven by the SOV beam (Visualization 1). The dashed black arrow is added to show the direction of the motion.

Equations (3)

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rH=(r1+r2)/2,
ΔV=μr1(2r2r1+r21),
ΔV=μr2(12r1r1+r2),