Abstract

A major current challenge in the field of structured light represents the development from three- (3d) to four-dimensional (4d) electric field structures, in which one exploits the transverse as well as longitudinal field components in 3d space. For this purpose, non-paraxial fields are required in order to be able to access visionary 3d topological structures as optical cones, ribbons and Möbius strips formed by 3d polarization states. We numerically demonstrate the customization of such complex topological structures by controlling generic polarization singularities in non-paraxial light fields. Our approach is based on tightly focusing tailored higher-order vector beams in combination with phase vortices. Besides demonstrating the appearance of cones and ribbons around the optical axis, we evince sculpting arrays of Möbius strips realized around off-axis generic singularities.

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2019 (3)

T. Bauer, P. Banzer, F. Bouchard, S. Orlov, L. Marrucci, E. Santamato, R. W. Boyd, E. Karimi, and G. Leuchs, “Multi-twist polarization ribbon topologies in highly-confined optical fields,” New J. Phys. 21(5), 053020 (2019).
[Crossref]

P. Huo, S. Zhang, Q. Fan, Y. Lu, and T. Xu, “Photonic spin-controlled generation and transformation of 3D optical polarization topologies enabled by all-dielectric metasurfaces,” Nanoscale 11(22), 10646–10654 (2019).
[Crossref]

C. Wan and Q. Zhan, “Generation of exotic optical polarization Möbius strips,” Opt. Express 27(8), 11516–11524 (2019).
[Crossref]

2018 (5)

E. Otte and C. Denz, “Sculpting complex polarization singularity networks,” Opt. Lett. 43(23), 5821–5824 (2018).
[Crossref]

E. Otte, C. Alpmann, and C. Denz, “Polarization singularity explosions in tailored light fields,” Laser Photonics Rev. 12(6), 1700200 (2018).
[Crossref]

E. Otte, K. Tekce, and C. Denz, “Customized focal light landscapes by complex vectorial fields for advanced optical trapping,” Proc. SPIE 10549, 105490S (2018).
[Crossref]

E. Otte, C. Rosales-Guzmán, B. Ndagano, C. Denz, and A. Forbes, “Entanglement beating in free space through spin-orbit coupling,” Light: Sci. Appl. 7(5), 18009 (2018).
[Crossref]

E. Otte, K. Tekce, and C. Denz, “Spatial multiplexing for tailored fully-structured light,” J. Opt. 20(10), 105606 (2018).
[Crossref]

2017 (8)

J. J. Nivas, F. Cardano, Z. Song, A. Rubano, R. Fittipaldi, A. Vecchione, D. Paparo, L. Marrucci, R. Bruzzese, and S. Amoruso, “Surface structuring with polarization-singular femtosecond laser beams generated by a q-plate,” Sci. Rep. 7(1), 42142 (2017).
[Crossref]

C. Alpmann, C. Schlickriede, E. Otte, and C. Denz, “Dynamic modulation of Poincaré beams,” Sci. Rep. 7(1), 8076 (2017).
[Crossref]

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
[Crossref]

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzmán, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13(4), 397–402 (2017).
[Crossref]

E. J. Galvez, I. Dutta, K. Beach, J. J. Zeosky, J. A. Jones, and B. Khajavi, “Multitwist Möbius strips and twisted ribbons in the polarization of paraxial light beams,” Sci. Rep. 7(1), 13653 (2017).
[Crossref]

A. Garcia-Etxarri, “Optical polarization Möbius strips on all-dielectric optical scatterers,” ACS Photonics 4(5), 1159–1164 (2017).
[Crossref]

E. Otte, K. Tekce, and C. Denz, “Tailored intensity landscapes by tight focusing of singular vector beams,” Opt. Express 25(17), 20194–20201 (2017).
[Crossref]

C. Rosales-Guzmán, N. Bhebhe, and A. Forbes, “Simultaneous generation of multiple vector beams on a single SLM,” Opt. Express 25(21), 25697–25706 (2017).
[Crossref]

2016 (2)

T. Bauer, M. Neugebauer, G. Leuchs, and P. Banzer, “Optical polarization Möbius strips and points of purely transverse spin density,” Phys. Rev. Lett. 117(1), 013601 (2016).
[Crossref]

E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularitites in tailored vector beams,” J. Opt. 18(7), 074012 (2016).
[Crossref]

2015 (6)

E. Otte, C. Schlickriede, C. Alpmann, and C. Denz, “Complex light fields enter a new dimension: holographic modulation of polarization in addition to amplitude and phase,” Proc. SPIE 9379, 937908 (2015).
[Crossref]

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347(6225), 964–966 (2015).
[Crossref]

M. Mirhosseini, O. S. Maga na-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17(3), 033033 (2015).
[Crossref]

Z. Chen, T. Zeng, B. Qian, and J. Ding, “Complete shaping of optical vector beams,” Opt. Express 23(14), 17701–17710 (2015).
[Crossref]

Y.-Y. Xie, Z.-J. Cheng, X. Liu, B.-Y. Wang, Q.-Y. Yue, and C.-S. Guo, “Simple method for generation of vector beams using a small-angle birefringent beam splitter,” Opt. Lett. 40(21), 5109–5112 (2015).
[Crossref]

2014 (1)

C. J. R. Sheppard, “Jones and Stokes parameters for polarization in three dimensions,” Phys. Rev. A 90(2), 023809 (2014).
[Crossref]

2013 (4)

W. Zhang, S. Liu, P. Li, X. Jiao, and J. Zhao, “Controlling the polarization singularities of the focused azimuthally polarized beams,” Opt. Express 21(1), 974–983 (2013).
[Crossref]

W. Han, Y. Yang, W. Cheng, and Q. Zhan, “Vectorial optical field generator for the creation of arbitrarily complex fields,” Opt. Express 21(18), 20692–20706 (2013).
[Crossref]

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7(6), 839–854 (2013).
[Crossref]

C. Rosales-Guzmán, N. Hermosa, A. Belmonte, and J. P. Torres, “Experimental detection of transverse particle movement with structured light,” Sci. Rep. 3(1), 2815 (2013).
[Crossref]

2012 (1)

E. J. Galvez and S. Khadka, “Poincaré modes of light,” Proc. SPIE 8274, 82740Y (2012).
[Crossref]

2011 (3)

2010 (2)

I. Freund, “Optical Möbius strips in three-dimensional ellipse fields: I. Lines of circular polarization,” Opt. Commun. 283(1), 1–15 (2010).
[Crossref]

I. Freund, “Optical Möbius strips in three dimensional ellipse fields: II. Lines of linear polarization,” Opt. Commun. 283(1), 16–28 (2010).
[Crossref]

2009 (2)

B. Boruah and M. Neil, “Focal field computation of an arbitrarily polarized beam using fast fourier transforms,” Opt. Commun. 282(24), 4660–4667 (2009).
[Crossref]

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

2008 (1)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

2007 (2)

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86(3), 329–334 (2007).
[Crossref]

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[Crossref]

2006 (5)

2005 (1)

I. Freund, “Cones, spirals, and Möbius strips, in elliptically polarized light,” Opt. Commun. 249(1-3), 7–22 (2005).
[Crossref]

2004 (3)

M. V. Berry, M. R. Dennis, and R. L. Lee Jr, “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).
[Crossref]

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
[Crossref]

M. V. Berry, “Index formulae for singular lines of polarization,” J. Opt. A: Pure Appl. Opt. 6(7), 675–678 (2004).
[Crossref]

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref]

2002 (2)

I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201(4-6), 251–270 (2002).
[Crossref]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66(1), 016615 (2002).
[Crossref]

2001 (3)

M. Berry and M. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London, Ser. A 457(2005), 141–155 (2001).
[Crossref]

I. Freund, “Polarization flowers,” Opt. Commun. 199(1-4), 47–63 (2001).
[Crossref]

A. Mair, A. Vaziri, G. Weihs, and A. Zellinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref]

2000 (2)

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1996 (1)

1995 (1)

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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
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E. Otte, C. Schlickriede, C. Alpmann, and C. Denz, “Complex light fields enter a new dimension: holographic modulation of polarization in addition to amplitude and phase,” Proc. SPIE 9379, 937908 (2015).
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M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7(6), 839–854 (2013).
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J. J. Nivas, F. Cardano, Z. Song, A. Rubano, R. Fittipaldi, A. Vecchione, D. Paparo, L. Marrucci, R. Bruzzese, and S. Amoruso, “Surface structuring with polarization-singular femtosecond laser beams generated by a q-plate,” Sci. Rep. 7(1), 42142 (2017).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
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Banzer, P.

T. Bauer, P. Banzer, F. Bouchard, S. Orlov, L. Marrucci, E. Santamato, R. W. Boyd, E. Karimi, and G. Leuchs, “Multi-twist polarization ribbon topologies in highly-confined optical fields,” New J. Phys. 21(5), 053020 (2019).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
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T. Bauer, M. Neugebauer, G. Leuchs, and P. Banzer, “Optical polarization Möbius strips and points of purely transverse spin density,” Phys. Rev. Lett. 117(1), 013601 (2016).
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T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347(6225), 964–966 (2015).
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Barnett, S.

Bauer, T.

T. Bauer, P. Banzer, F. Bouchard, S. Orlov, L. Marrucci, E. Santamato, R. W. Boyd, E. Karimi, and G. Leuchs, “Multi-twist polarization ribbon topologies in highly-confined optical fields,” New J. Phys. 21(5), 053020 (2019).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
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T. Bauer, M. Neugebauer, G. Leuchs, and P. Banzer, “Optical polarization Möbius strips and points of purely transverse spin density,” Phys. Rev. Lett. 117(1), 013601 (2016).
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T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347(6225), 964–966 (2015).
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Beach, K.

E. J. Galvez, I. Dutta, K. Beach, J. J. Zeosky, J. A. Jones, and B. Khajavi, “Multitwist Möbius strips and twisted ribbons in the polarization of paraxial light beams,” Sci. Rep. 7(1), 13653 (2017).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
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C. Rosales-Guzmán, N. Hermosa, A. Belmonte, and J. P. Torres, “Experimental detection of transverse particle movement with structured light,” Sci. Rep. 3(1), 2815 (2013).
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A. Belmonte and J. P. Torres, “Optical doppler shift with structured light,” Opt. Lett. 36(22), 4437–4439 (2011).
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Berry, M.

M. Berry and M. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London, Ser. A 457(2005), 141–155 (2001).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
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M. V. Berry, M. R. Dennis, and R. L. Lee Jr, “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).
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M. V. Berry, “Index formulae for singular lines of polarization,” J. Opt. A: Pure Appl. Opt. 6(7), 675–678 (2004).
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Bigelow, N. P.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
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M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
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T. Bauer, P. Banzer, F. Bouchard, S. Orlov, L. Marrucci, E. Santamato, R. W. Boyd, E. Karimi, and G. Leuchs, “Multi-twist polarization ribbon topologies in highly-confined optical fields,” New J. Phys. 21(5), 053020 (2019).
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T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347(6225), 964–966 (2015).
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M. Mirhosseini, O. S. Maga na-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17(3), 033033 (2015).
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Brown, T.

Brown, T. G.

Bruzzese, R.

J. J. Nivas, F. Cardano, Z. Song, A. Rubano, R. Fittipaldi, A. Vecchione, D. Paparo, L. Marrucci, R. Bruzzese, and S. Amoruso, “Surface structuring with polarization-singular femtosecond laser beams generated by a q-plate,” Sci. Rep. 7(1), 42142 (2017).
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Campos, J.

Cardano, F.

J. J. Nivas, F. Cardano, Z. Song, A. Rubano, R. Fittipaldi, A. Vecchione, D. Paparo, L. Marrucci, R. Bruzzese, and S. Amoruso, “Surface structuring with polarization-singular femtosecond laser beams generated by a q-plate,” Sci. Rep. 7(1), 42142 (2017).
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Chen, W.

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
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Chen, Z.

Cheng, W.

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Chong, C. T.

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Cottrell, D. M.

Courtial, J.

Davidson, N.

Davis, J. A.

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M. Berry and M. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London, Ser. A 457(2005), 141–155 (2001).
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Dennis, M. R.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
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M. R. Dennis, “Fermionic out-of-plane structure of polarization singularities,” Opt. Lett. 36(19), 3765–3767 (2011).
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M. V. Berry, M. R. Dennis, and R. L. Lee Jr, “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).
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M. R. Dennis, K. O’Holleran, and M. J. Padgett, Singular Optics: Optical Vortices and Polarization Singularities (Elsevier, 2009), vol. 53 of Progress in Optics, chap. 5, pp. 293–363.

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E. Otte, K. Tekce, and C. Denz, “Customized focal light landscapes by complex vectorial fields for advanced optical trapping,” Proc. SPIE 10549, 105490S (2018).
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E. Otte, C. Alpmann, and C. Denz, “Polarization singularity explosions in tailored light fields,” Laser Photonics Rev. 12(6), 1700200 (2018).
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E. Otte and C. Denz, “Sculpting complex polarization singularity networks,” Opt. Lett. 43(23), 5821–5824 (2018).
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E. Otte, C. Rosales-Guzmán, B. Ndagano, C. Denz, and A. Forbes, “Entanglement beating in free space through spin-orbit coupling,” Light: Sci. Appl. 7(5), 18009 (2018).
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E. Otte, K. Tekce, and C. Denz, “Spatial multiplexing for tailored fully-structured light,” J. Opt. 20(10), 105606 (2018).
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C. Alpmann, C. Schlickriede, E. Otte, and C. Denz, “Dynamic modulation of Poincaré beams,” Sci. Rep. 7(1), 8076 (2017).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
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E. Otte, K. Tekce, and C. Denz, “Tailored intensity landscapes by tight focusing of singular vector beams,” Opt. Express 25(17), 20194–20201 (2017).
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E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularitites in tailored vector beams,” J. Opt. 18(7), 074012 (2016).
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E. Otte, C. Schlickriede, C. Alpmann, and C. Denz, “Complex light fields enter a new dimension: holographic modulation of polarization in addition to amplitude and phase,” Proc. SPIE 9379, 937908 (2015).
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M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7(6), 839–854 (2013).
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E. Otte, K. Tekce, S. Lamping, B. J. Ravoo, and C. Denz, “Polarization nano-tomography of tightly focused light lansdscapes by self-assembled monolayers,” In formal peer review process Nat. Commun.NCOMMS-19-00484B, (2019).

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S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
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E. J. Galvez, I. Dutta, K. Beach, J. J. Zeosky, J. A. Jones, and B. Khajavi, “Multitwist Möbius strips and twisted ribbons in the polarization of paraxial light beams,” Sci. Rep. 7(1), 13653 (2017).
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S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
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M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7(6), 839–854 (2013).
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P. Huo, S. Zhang, Q. Fan, Y. Lu, and T. Xu, “Photonic spin-controlled generation and transformation of 3D optical polarization topologies enabled by all-dielectric metasurfaces,” Nanoscale 11(22), 10646–10654 (2019).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
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J. J. Nivas, F. Cardano, Z. Song, A. Rubano, R. Fittipaldi, A. Vecchione, D. Paparo, L. Marrucci, R. Bruzzese, and S. Amoruso, “Surface structuring with polarization-singular femtosecond laser beams generated by a q-plate,” Sci. Rep. 7(1), 42142 (2017).
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Forbes, A.

E. Otte, C. Rosales-Guzmán, B. Ndagano, C. Denz, and A. Forbes, “Entanglement beating in free space through spin-orbit coupling,” Light: Sci. Appl. 7(5), 18009 (2018).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
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Galvez, E. J.

E. J. Galvez, I. Dutta, K. Beach, J. J. Zeosky, J. A. Jones, and B. Khajavi, “Multitwist Möbius strips and twisted ribbons in the polarization of paraxial light beams,” Sci. Rep. 7(1), 13653 (2017).
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M. Mirhosseini, O. S. Maga na-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17(3), 033033 (2015).
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Gibson, G.

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
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Figures (5)

Fig. 1.
Fig. 1. Higher-order vector beams with singularity index (a) $\sigma _{12}=8$ and (b) $\sigma _{12}=-8$ forming vectorial flowers and spider webs, respectively. Illustrated are the linear polarization distributions including red flow lines and the corresponding Stokes field phase $\phi _{12}$.
Fig. 2.
Fig. 2. Polarization topologies around the optical axis: We show the intensity distribution of (a) each component $|\mathcal {E}_{x,y,z}|^{2}\in [0,1]$ and their respective phase $\varphi _{x,y,z}\in [0,2\pi ]$ as well as (b) the total intensity structure $|\boldsymbol {\mathcal {{E}}}|^{2}\in [0,1]$ of a tightly focused vector beam with the singularity index $\sigma _{12}=8$ and a topological charge of $l=1$ ($z=0$). All intensity structures are normalized with respect to their maximum, whereby the ratio of the maximum of $|\mathcal {E}_{x,y,z}|^{2}$ to the one of $|\boldsymbol {\mathcal {{E}}}|^{2}$ is shown within the intensity images in (a). Furthermore, (c)-(e) 3d polarization topologies inside the focal 2d plane are shown, whereby each topology is observed by tracing the major axis (blue lines with blue and green endpoints) on three different circles (white dashed line in (b) $|\boldsymbol {\mathcal {{E}}}|^{2}$ and its components (a) $|\mathcal {E}_{x,y,z}|^{2}$; $z=0$). The black circles in (c), (d), and (e) correspond to the small, medium, and large white circle in (b) at $z=0$, respectively.
Fig. 3.
Fig. 3. Generic singularities in focal structures: We show the zero-lines inside the focal plane of a tightly focused higher-order vector beam with singularity index $\sigma _{12}=8$ and additional topological charge $l \in [0,5]$ in (a)-(f) with increasing $l$. The background shows the normalized intensity structure $|\boldsymbol {\mathcal {{E}}}|^{2}\in [0,1]$. By examining the intersection points of the zero-lines, 1) C-points (blue, orange line: $\Re (\Psi _C)=0$, $\Im (\Psi _C)=0$) as well as 2) L-points (red, yellow, green line: $\gamma _{x,y,z}=0$) can be detected. (color version online)
Fig. 4.
Fig. 4. Projection of the major axis $\boldsymbol {\alpha }$ of a polarization ellipse onto the (a) radial projection plane $\tau _0$ and the (b) tangential projection plane $\pi _0$.
Fig. 5.
Fig. 5. Topologies around off-axis singularities: We explore a tightly focused vectorial flower with $\sigma _{12}=8$ and phase vortex with $l=1$, whose intensity structure $|\boldsymbol {\mathcal {{E}}}|^{2}$ is shown in (a). Presented are the (b) 3d polarization topology around a C-point (white dashed line in (a)), which is formed by tracing the major axis, and the corresponding projections onto the planes (c) $\tau _0$ and (d) $\pi _0$. Moreover, the overall polarization topology around all C-points is illustrated in (e), where the background corresponds to the total normalized intensity $|\boldsymbol {\mathcal {{E}}}(x,y,0)|^{2}$, revealing an optical polarization Möbius strip array.

Equations (6)

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Σ 12 = S 1 + i S 2 = A 12 exp ( i ϕ 12 )
E i n = [ cos ( σ 12 2 φ ) , sin ( σ 12 2 φ ) ] T exp ( i l φ )
α = 1 | E E | ( E E E ) , β = 1 | E E | ( E E E ) , γ = ( E × E ) ,
Ψ C = E E = ( α 2 β 2 ) e i 2 θ
I ) l = 0 : no polarization singularity, II ) 0 < l < | σ 12 | / 2 1 : | σ 12 2 | C- and L-points, III ) l = | σ 12 / 2 1 | : one L-point in the center, IV ) l = | σ 12 | / 2 : one C-point in the center.
τ α = δ ( 2 π ) δ ( 0 ) 2 π , π α = δ ( 2 π ) δ ( 0 ) 2 π ,

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