Abstract

Vector vortex beams are conventionally created as the superposition of orbital angular momentum (OAM) modes with orthogonal polarizations, limiting the available degrees of freedom (DoFs) to 2, while their creation by complex optical devices such as metasurfaces, liquid crystals, and interferometers has hindered their versatility. Here we demonstrate a new class of vector vortex beam constructed from four DoFs as multiple ray-like trajectories with wave-like properties, which we create by operating a simple anisotropic microchip laser in a frequency-degenerate state. Our new structure is obtained by the superposition of two stable periodic ray trajectories, simultaneously fulfilling a completed oscillation in the cavity. By a simple external modulation, we can transform our ray trajectories into vortex beams with large OAM, multiple singularities, as well as exotic helical star-shaped patterns. Our experimental results are complemented by a complete theoretical framework for this new class of beam, revealing parallels to hybrid SU(2) coherent states. Our approach offers in principle unlimited DoFs for vectorial structured light with concomitant applications, for example, in engineering classically entangled light and in vectorial optical trapping and tweezing.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  1. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009).
    [Crossref]
  2. V. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarization-controlled optical tractor beam,” Nat. Photonics 8, 846–850 (2014).
    [Crossref]
  3. T. Omatsu, K. Miyamoto, K. Toyoda, R. Morita, Y. Arita, and K. Dholakia, “A new twist for materials science: the formation of chiral structures using the angular momentum of light,” Adv. Opt. Mater. 7, 1801672 (2019).
    [Crossref]
  4. Y. Zhao and J. Wang, “High-base vector beam encoding/decoding for visible-light communications,” Opt. Lett. 40, 4843–4846 (2015).
    [Crossref]
  5. Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light Sci. Appl. 8, 90 (2019).
    [Crossref]
  6. C. Rosales-Guzmán, B. Ndagano, and A. Forbes, “A review of complex vector light fields and their applications,” J. Opt. 20, 123001 (2018).
    [Crossref]
  7. R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
    [Crossref]
  8. A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum- like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
    [Crossref]
  9. B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
    [Crossref]
  10. T. Konrad and A. Forbes, “Quantum mechanics and classical light,” Contemp. Phys. 60, 1–22 (2019).
    [Crossref]
  11. A. Forbes, A. Aiello, and B. Ndagano, “Classically entangled light,” Prog. Opt. 64, 99–153 (2019).
    [Crossref]
  12. E. Toninelli, B. Ndagano, A. Vallés, B. Sephton, I. Nape, A. Ambrosio, F. Capasso, M. J. Padgett, and A. Forbes, “Concepts in quantum state tomography and classical implementation with intense light: a tutorial,” Adv. Opt. Photon. 11, 67–134 (2019).
    [Crossref]
  13. B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, 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, 397–402 (2017).
    [Crossref]
  14. B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Lightwave Technol. 36, 292–301 (2017).
    [Crossref]
  15. A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, B. Heim, and C. Marquardt, “High-dimensional intracity quantum cryptography with structured photons,” Optica 4, 1006–1010 (2017).
    [Crossref]
  16. A. E. Willner, “Vector-mode multiplexing brings an additional approach for capacity growth in optical fibers,” Light Sci. Appl. 7, 18002 (2018).
    [Crossref]
  17. S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
    [Crossref]
  18. V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
    [Crossref]
  19. F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
    [Crossref]
  20. X.-B. Hu, B. Zhao, Z.-H. Zhu, W. Gao, and C. Rosales-Guzmán, “In situ detection of a cooperative target’s longitudinal and angular speed using structured light,” Opt. Lett. 44, 3070–3073 (2019).
    [Crossref]
  21. G. Milione, H. Sztul, D. Nolan, and R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
    [Crossref]
  22. D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
    [Crossref]
  23. X. Yi, Y. Liu, X. Ling, X. Zhou, Y. Ke, H. Luo, S. Wen, and D. Fan, “Hybrid-order Poincaré sphere,” Phys. Rev. A 91, 023801 (2015).
    [Crossref]
  24. R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, “Arbitrary spin-to–orbital angular momentum conversion of light,” Science 358, 896–901 (2017).
    [Crossref]
  25. A. Forbes, “Structured light from lasers,” Laser Photon. Rev. 13, 1900140 (2019).
    [Crossref]
  26. 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, 18009 (2018).
    [Crossref]
  27. T.-H. Lu and C. He, “Generating orthogonally circular polarized states embedded in nonplanar geometric beams,” Opt. Express 23, 20876–20883 (2015).
    [Crossref]
  28. Y. Chen, T.-H. Lu, and K. Huang, “Hyperboloid structures formed by polarization singularities in coherent vector fields with longitudinal-transverse coupling,” Phys. Rev. Lett. 97, 233903 (2006).
    [Crossref]
  29. T.-H. Lu, Y. Chen, and K. Huang, “Generalized hyperboloid structures of polarization singularities in Laguerre–Gaussian vector fields,” Phys. Rev. A 76, 063809 (2007).
    [Crossref]
  30. Y. Chen, K. Huang, H. Lai, and Y. Lan, “Observation of vector vortex lattices in polarization states of an isotropic microcavity laser,” Phys. Rev. Lett. 90, 053904 (2003).
    [Crossref]
  31. Y. Chen, T.-H. Lu, and K. Huang, “Observation of spatially coherent polarization vector fields and visualization of vector singularities,” Phys. Rev. Lett. 96, 033901 (2006).
    [Crossref]
  32. X.-F. Zhou, J. Zhou, and C. Wu, “Vortex structures of rotating spin-orbit-coupled Bose-Einstein condensates,” Phys. Rev. A 84, 063624 (2011).
    [Crossref]
  33. A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics 9, 789–795 (2015).
    [Crossref]
  34. K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
    [Crossref]
  35. L. Du, A. Yang, A. V. Zayats, and X. Yuan, “Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum,” Nat. Phys. 15, 650–654 (2019).
    [Crossref]
  36. A. Perelomov, Generalized Coherent States and Their Applications (Springer, 2012).
  37. V. Bužek and T. Quang, “Generalized coherent state for bosonic realization of SU(2) Lie algebra,” J. Opt. Soc. Am. B 6, 2447–2449 (1989).
    [Crossref]
  38. Y. Chen, Y. Hsieh, and K. Huang, “Originating an integral formula and using the quantum fourier transform to decompose the Hermite-Laguerre-Gaussian modes into elliptical orbital modes,” OSA Continuum 1, 744–754 (2018).
    [Crossref]
  39. Y. Chen, J. Tung, P. Chiang, H. Liang, and K. Huang, “Exploring the effect of fractional degeneracy and the emergence of ray-wave duality in solid-state lasers with off-axis pumping,” Phys. Rev. A 88, 013827 (2013).
    [Crossref]
  40. N. Barré, M. Romanelli, M. Lebental, and M. Brunel, “Waves and rays in plano-concave laser cavities: I. Geometric modes in the paraxial approximation,” Eur. J. Phys. 38, 034010 (2017).
    [Crossref]
  41. Y. Shen, X. Fu, and M. Gong, “Truncated triangular diffraction lattices and orbital-angular-momentum detection of vortex SU(2) geometric modes,” Opt. Express 26, 25545–25557 (2018).
    [Crossref]
  42. Y. Shen, Z. Wan, Y. Meng, X. Fu, and M. Gong, “Polygonal vortex beams,” IEEE Photon. J. 10, 1503016 (2018).
    [Crossref]
  43. J. Tung, P. Tuan, H. Liang, K. Huang, and Y. Chen, “Fractal frequency spectrum in laser resonators and three-dimensional geometric topology of optical coherent waves,” Phys. Rev. A 94, 023811 (2016).
    [Crossref]
  44. Y. Shen, X. Yang, X. Fu, and M. Gong, “Periodic-trajectory-controlled, coherent-state-phase-switched, and wavelength-tunable SU(2) geometric modes in a frequency-degenerate resonator,” Appl. Opt. 57, 9543–9549 (2018).
    [Crossref]
  45. Y. Chen, S. Li, Y. Hsieh, J. Tung, H. Liang, and K. Huang, “Laser wave-packet representation to unify eigenmodes and geometric modes in spherical cavities,” Opt. Lett. 44, 2649–2652 (2019).
    [Crossref]
  46. Y. Shen, Y. Meng, X. Fu, and M. Gong, “Hybrid topological evolution of multi-singularity vortex beams: generalized nature for helical-Ince–Gaussian and Hermite–Laguerre–Gaussian modes,” J. Opt. Soc. Am. A 36, 578–587 (2019).
    [Crossref]
  47. Y. Chen, C. Jiang, Y. Lan, and K. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A 69, 053807 (2004).
    [Crossref]
  48. T.-H. Lu and L. Lin, “Observation of a superposition of orthogonally polarized geometric beams with a c-cut Nd: YVO4 crystal,” Appl. Phys. B 106, 863–866 (2012).
    [Crossref]
  49. M. A. Alonso and M. R. Dennis, “Ray-optical Poincaré sphere for structured Gaussian beams,” Optica 4, 476–486 (2017).
    [Crossref]
  50. M. R. Dennis and M. A. Alonso, “Swings and roundabouts: optical Poincaré spheres for polarization and Gaussian beams,” Philos. Trans. R. Soc. London, Ser. A 375, 20150441 (2017).
    [Crossref]
  51. M. R. Dennis and M. A. Alonso, “Gaussian mode families from systems of rays,” J. Phys. 1, 025003 (2019).
    [Crossref]
  52. X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
    [Crossref]

2019 (11)

T. Omatsu, K. Miyamoto, K. Toyoda, R. Morita, Y. Arita, and K. Dholakia, “A new twist for materials science: the formation of chiral structures using the angular momentum of light,” Adv. Opt. Mater. 7, 1801672 (2019).
[Crossref]

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light Sci. Appl. 8, 90 (2019).
[Crossref]

T. Konrad and A. Forbes, “Quantum mechanics and classical light,” Contemp. Phys. 60, 1–22 (2019).
[Crossref]

A. Forbes, A. Aiello, and B. Ndagano, “Classically entangled light,” Prog. Opt. 64, 99–153 (2019).
[Crossref]

E. Toninelli, B. Ndagano, A. Vallés, B. Sephton, I. Nape, A. Ambrosio, F. Capasso, M. J. Padgett, and A. Forbes, “Concepts in quantum state tomography and classical implementation with intense light: a tutorial,” Adv. Opt. Photon. 11, 67–134 (2019).
[Crossref]

X.-B. Hu, B. Zhao, Z.-H. Zhu, W. Gao, and C. Rosales-Guzmán, “In situ detection of a cooperative target’s longitudinal and angular speed using structured light,” Opt. Lett. 44, 3070–3073 (2019).
[Crossref]

A. Forbes, “Structured light from lasers,” Laser Photon. Rev. 13, 1900140 (2019).
[Crossref]

L. Du, A. Yang, A. V. Zayats, and X. Yuan, “Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum,” Nat. Phys. 15, 650–654 (2019).
[Crossref]

Y. Chen, S. Li, Y. Hsieh, J. Tung, H. Liang, and K. Huang, “Laser wave-packet representation to unify eigenmodes and geometric modes in spherical cavities,” Opt. Lett. 44, 2649–2652 (2019).
[Crossref]

Y. Shen, Y. Meng, X. Fu, and M. Gong, “Hybrid topological evolution of multi-singularity vortex beams: generalized nature for helical-Ince–Gaussian and Hermite–Laguerre–Gaussian modes,” J. Opt. Soc. Am. A 36, 578–587 (2019).
[Crossref]

M. R. Dennis and M. A. Alonso, “Gaussian mode families from systems of rays,” J. Phys. 1, 025003 (2019).
[Crossref]

2018 (7)

Y. Shen, X. Yang, X. Fu, and M. Gong, “Periodic-trajectory-controlled, coherent-state-phase-switched, and wavelength-tunable SU(2) geometric modes in a frequency-degenerate resonator,” Appl. Opt. 57, 9543–9549 (2018).
[Crossref]

Y. Chen, Y. Hsieh, and K. Huang, “Originating an integral formula and using the quantum fourier transform to decompose the Hermite-Laguerre-Gaussian modes into elliptical orbital modes,” OSA Continuum 1, 744–754 (2018).
[Crossref]

Y. Shen, X. Fu, and M. Gong, “Truncated triangular diffraction lattices and orbital-angular-momentum detection of vortex SU(2) geometric modes,” Opt. Express 26, 25545–25557 (2018).
[Crossref]

Y. Shen, Z. Wan, Y. Meng, X. Fu, and M. Gong, “Polygonal vortex beams,” IEEE Photon. J. 10, 1503016 (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, 18009 (2018).
[Crossref]

A. E. Willner, “Vector-mode multiplexing brings an additional approach for capacity growth in optical fibers,” Light Sci. Appl. 7, 18002 (2018).
[Crossref]

C. Rosales-Guzmán, B. Ndagano, and A. Forbes, “A review of complex vector light fields and their applications,” J. Opt. 20, 123001 (2018).
[Crossref]

2017 (8)

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, 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, 397–402 (2017).
[Crossref]

B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Lightwave Technol. 36, 292–301 (2017).
[Crossref]

A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, B. Heim, and C. Marquardt, “High-dimensional intracity quantum cryptography with structured photons,” Optica 4, 1006–1010 (2017).
[Crossref]

R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, “Arbitrary spin-to–orbital angular momentum conversion of light,” Science 358, 896–901 (2017).
[Crossref]

N. Barré, M. Romanelli, M. Lebental, and M. Brunel, “Waves and rays in plano-concave laser cavities: I. Geometric modes in the paraxial approximation,” Eur. J. Phys. 38, 034010 (2017).
[Crossref]

X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
[Crossref]

M. A. Alonso and M. R. Dennis, “Ray-optical Poincaré sphere for structured Gaussian beams,” Optica 4, 476–486 (2017).
[Crossref]

M. R. Dennis and M. A. Alonso, “Swings and roundabouts: optical Poincaré spheres for polarization and Gaussian beams,” Philos. Trans. R. Soc. London, Ser. A 375, 20150441 (2017).
[Crossref]

2016 (2)

J. Tung, P. Tuan, H. Liang, K. Huang, and Y. Chen, “Fractal frequency spectrum in laser resonators and three-dimensional geometric topology of optical coherent waves,” Phys. Rev. A 94, 023811 (2016).
[Crossref]

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

2015 (7)

X. Yi, Y. Liu, X. Ling, X. Zhou, Y. Ke, H. Luo, S. Wen, and D. Fan, “Hybrid-order Poincaré sphere,” Phys. Rev. A 91, 023801 (2015).
[Crossref]

T.-H. Lu and C. He, “Generating orthogonally circular polarized states embedded in nonplanar geometric beams,” Opt. Express 23, 20876–20883 (2015).
[Crossref]

A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics 9, 789–795 (2015).
[Crossref]

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum- like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Y. Zhao and J. Wang, “High-base vector beam encoding/decoding for visible-light communications,” Opt. Lett. 40, 4843–4846 (2015).
[Crossref]

2014 (2)

V. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarization-controlled optical tractor beam,” Nat. Photonics 8, 846–850 (2014).
[Crossref]

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

2013 (2)

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
[Crossref]

Y. Chen, J. Tung, P. Chiang, H. Liang, and K. Huang, “Exploring the effect of fractional degeneracy and the emergence of ray-wave duality in solid-state lasers with off-axis pumping,” Phys. Rev. A 88, 013827 (2013).
[Crossref]

2012 (1)

T.-H. Lu and L. Lin, “Observation of a superposition of orthogonally polarized geometric beams with a c-cut Nd: YVO4 crystal,” Appl. Phys. B 106, 863–866 (2012).
[Crossref]

2011 (2)

G. Milione, H. Sztul, D. Nolan, and R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

X.-F. Zhou, J. Zhou, and C. Wu, “Vortex structures of rotating spin-orbit-coupled Bose-Einstein condensates,” Phys. Rev. A 84, 063624 (2011).
[Crossref]

2010 (1)

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

2009 (1)

2007 (1)

T.-H. Lu, Y. Chen, and K. Huang, “Generalized hyperboloid structures of polarization singularities in Laguerre–Gaussian vector fields,” Phys. Rev. A 76, 063809 (2007).
[Crossref]

2006 (2)

Y. Chen, T.-H. Lu, and K. Huang, “Observation of spatially coherent polarization vector fields and visualization of vector singularities,” Phys. Rev. Lett. 96, 033901 (2006).
[Crossref]

Y. Chen, T.-H. Lu, and K. Huang, “Hyperboloid structures formed by polarization singularities in coherent vector fields with longitudinal-transverse coupling,” Phys. Rev. Lett. 97, 233903 (2006).
[Crossref]

2004 (1)

Y. Chen, C. Jiang, Y. Lan, and K. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A 69, 053807 (2004).
[Crossref]

2003 (1)

Y. Chen, K. Huang, H. Lai, and Y. Lan, “Observation of vector vortex lattices in polarization states of an isotropic microcavity laser,” Phys. Rev. Lett. 90, 053904 (2003).
[Crossref]

1998 (1)

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[Crossref]

1989 (1)

Aiello, A.

A. Forbes, A. Aiello, and B. Ndagano, “Classically entangled light,” Prog. Opt. 64, 99–153 (2019).
[Crossref]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics 9, 789–795 (2015).
[Crossref]

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum- like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Alfano, R.

G. Milione, H. Sztul, D. Nolan, and R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

Alonso, M. A.

M. R. Dennis and M. A. Alonso, “Gaussian mode families from systems of rays,” J. Phys. 1, 025003 (2019).
[Crossref]

M. A. Alonso and M. R. Dennis, “Ray-optical Poincaré sphere for structured Gaussian beams,” Optica 4, 476–486 (2017).
[Crossref]

M. R. Dennis and M. A. Alonso, “Swings and roundabouts: optical Poincaré spheres for polarization and Gaussian beams,” Philos. Trans. R. Soc. London, Ser. A 375, 20150441 (2017).
[Crossref]

Ambrosio, A.

Aolita, L.

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
[Crossref]

Arita, Y.

T. Omatsu, K. Miyamoto, K. Toyoda, R. Morita, Y. Arita, and K. Dholakia, “A new twist for materials science: the formation of chiral structures using the angular momentum of light,” Adv. Opt. Mater. 7, 1801672 (2019).
[Crossref]

Banzer, P.

Barré, N.

N. Barré, M. Romanelli, M. Lebental, and M. Brunel, “Waves and rays in plano-concave laser cavities: I. Geometric modes in the paraxial approximation,” Eur. J. Phys. 38, 034010 (2017).
[Crossref]

Berg-Johansen, S.

Bliokh, K. Y.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

Borghi, R.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

Bouchard, F.

Brunel, M.

N. Barré, M. Romanelli, M. Lebental, and M. Brunel, “Waves and rays in plano-concave laser cavities: I. Geometric modes in the paraxial approximation,” Eur. J. Phys. 38, 034010 (2017).
[Crossref]

Bužek, V.

Capasso, F.

Chen, Y.

Y. Chen, S. Li, Y. Hsieh, J. Tung, H. Liang, and K. Huang, “Laser wave-packet representation to unify eigenmodes and geometric modes in spherical cavities,” Opt. Lett. 44, 2649–2652 (2019).
[Crossref]

Y. Chen, Y. Hsieh, and K. Huang, “Originating an integral formula and using the quantum fourier transform to decompose the Hermite-Laguerre-Gaussian modes into elliptical orbital modes,” OSA Continuum 1, 744–754 (2018).
[Crossref]

J. Tung, P. Tuan, H. Liang, K. Huang, and Y. Chen, “Fractal frequency spectrum in laser resonators and three-dimensional geometric topology of optical coherent waves,” Phys. Rev. A 94, 023811 (2016).
[Crossref]

Y. Chen, J. Tung, P. Chiang, H. Liang, and K. Huang, “Exploring the effect of fractional degeneracy and the emergence of ray-wave duality in solid-state lasers with off-axis pumping,” Phys. Rev. A 88, 013827 (2013).
[Crossref]

T.-H. Lu, Y. Chen, and K. Huang, “Generalized hyperboloid structures of polarization singularities in Laguerre–Gaussian vector fields,” Phys. Rev. A 76, 063809 (2007).
[Crossref]

Y. Chen, T.-H. Lu, and K. Huang, “Observation of spatially coherent polarization vector fields and visualization of vector singularities,” Phys. Rev. Lett. 96, 033901 (2006).
[Crossref]

Y. Chen, T.-H. Lu, and K. Huang, “Hyperboloid structures formed by polarization singularities in coherent vector fields with longitudinal-transverse coupling,” Phys. Rev. Lett. 97, 233903 (2006).
[Crossref]

Y. Chen, C. Jiang, Y. Lan, and K. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A 69, 053807 (2004).
[Crossref]

Y. Chen, K. Huang, H. Lai, and Y. Lan, “Observation of vector vortex lattices in polarization states of an isotropic microcavity laser,” Phys. Rev. Lett. 90, 053904 (2003).
[Crossref]

Chiang, P.

Y. Chen, J. Tung, P. Chiang, H. Liang, and K. Huang, “Exploring the effect of fractional degeneracy and the emergence of ray-wave duality in solid-state lasers with off-axis pumping,” Phys. Rev. A 88, 013827 (2013).
[Crossref]

Cox, M. A.

D’ambrosio, V.

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
[Crossref]

Davoyan, A. R.

V. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarization-controlled optical tractor beam,” Nat. Photonics 8, 846–850 (2014).
[Crossref]

Del Re, L.

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
[Crossref]

Dennis, M. R.

M. R. Dennis and M. A. Alonso, “Gaussian mode families from systems of rays,” J. Phys. 1, 025003 (2019).
[Crossref]

M. R. Dennis and M. A. Alonso, “Swings and roundabouts: optical Poincaré spheres for polarization and Gaussian beams,” Philos. Trans. R. Soc. London, Ser. A 375, 20150441 (2017).
[Crossref]

M. A. Alonso and M. R. Dennis, “Ray-optical Poincaré sphere for structured Gaussian beams,” Optica 4, 476–486 (2017).
[Crossref]

Denz, C.

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, 18009 (2018).
[Crossref]

Devlin, R. C.

R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, “Arbitrary spin-to–orbital angular momentum conversion of light,” Science 358, 896–901 (2017).
[Crossref]

Dholakia, K.

T. Omatsu, K. Miyamoto, K. Toyoda, R. Morita, Y. Arita, and K. Dholakia, “A new twist for materials science: the formation of chiral structures using the angular momentum of light,” Adv. Opt. Mater. 7, 1801672 (2019).
[Crossref]

Du, L.

L. Du, A. Yang, A. V. Zayats, and X. Yuan, “Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum,” Nat. Phys. 15, 650–654 (2019).
[Crossref]

Dudley, A.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

Elser, D.

Engheta, N.

V. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarization-controlled optical tractor beam,” Nat. Photonics 8, 846–850 (2014).
[Crossref]

Fan, D.

X. Yi, Y. Liu, X. Ling, X. Zhou, Y. Ke, H. Luo, S. Wen, and D. Fan, “Hybrid-order Poincaré sphere,” Phys. Rev. A 91, 023801 (2015).
[Crossref]

Fickler, R.

Forbes, A.

T. Konrad and A. Forbes, “Quantum mechanics and classical light,” Contemp. Phys. 60, 1–22 (2019).
[Crossref]

A. Forbes, “Structured light from lasers,” Laser Photon. Rev. 13, 1900140 (2019).
[Crossref]

A. Forbes, A. Aiello, and B. Ndagano, “Classically entangled light,” Prog. Opt. 64, 99–153 (2019).
[Crossref]

E. Toninelli, B. Ndagano, A. Vallés, B. Sephton, I. Nape, A. Ambrosio, F. Capasso, M. J. Padgett, and A. Forbes, “Concepts in quantum state tomography and classical implementation with intense light: a tutorial,” Adv. Opt. Photon. 11, 67–134 (2019).
[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, 18009 (2018).
[Crossref]

C. Rosales-Guzmán, B. Ndagano, and A. Forbes, “A review of complex vector light fields and their applications,” J. Opt. 20, 123001 (2018).
[Crossref]

B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Lightwave Technol. 36, 292–301 (2017).
[Crossref]

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, 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, 397–402 (2017).
[Crossref]

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

Fu, X.

Gagnon-Bischoff, J.

Gao, W.

Giacobino, E.

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum- like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Gong, M.

Gori, F.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

Günthner, K.

He, C.

Heim, B.

Hernandez-Aranda, R. I.

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, 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, 397–402 (2017).
[Crossref]

Heshami, K.

Hnatovsky, C.

V. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarization-controlled optical tractor beam,” Nat. Photonics 8, 846–850 (2014).
[Crossref]

Hsieh, Y.

Hu, X.-B.

Huang, K.

Y. Chen, S. Li, Y. Hsieh, J. Tung, H. Liang, and K. Huang, “Laser wave-packet representation to unify eigenmodes and geometric modes in spherical cavities,” Opt. Lett. 44, 2649–2652 (2019).
[Crossref]

Y. Chen, Y. Hsieh, and K. Huang, “Originating an integral formula and using the quantum fourier transform to decompose the Hermite-Laguerre-Gaussian modes into elliptical orbital modes,” OSA Continuum 1, 744–754 (2018).
[Crossref]

X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
[Crossref]

J. Tung, P. Tuan, H. Liang, K. Huang, and Y. Chen, “Fractal frequency spectrum in laser resonators and three-dimensional geometric topology of optical coherent waves,” Phys. Rev. A 94, 023811 (2016).
[Crossref]

Y. Chen, J. Tung, P. Chiang, H. Liang, and K. Huang, “Exploring the effect of fractional degeneracy and the emergence of ray-wave duality in solid-state lasers with off-axis pumping,” Phys. Rev. A 88, 013827 (2013).
[Crossref]

T.-H. Lu, Y. Chen, and K. Huang, “Generalized hyperboloid structures of polarization singularities in Laguerre–Gaussian vector fields,” Phys. Rev. A 76, 063809 (2007).
[Crossref]

Y. Chen, T.-H. Lu, and K. Huang, “Observation of spatially coherent polarization vector fields and visualization of vector singularities,” Phys. Rev. Lett. 96, 033901 (2006).
[Crossref]

Y. Chen, T.-H. Lu, and K. Huang, “Hyperboloid structures formed by polarization singularities in coherent vector fields with longitudinal-transverse coupling,” Phys. Rev. Lett. 97, 233903 (2006).
[Crossref]

Y. Chen, C. Jiang, Y. Lan, and K. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A 69, 053807 (2004).
[Crossref]

Y. Chen, K. Huang, H. Lai, and Y. Lan, “Observation of vector vortex lattices in polarization states of an isotropic microcavity laser,” Phys. Rev. Lett. 90, 053904 (2003).
[Crossref]

Jiang, C.

Y. Chen, C. Jiang, Y. Lan, and K. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A 69, 053807 (2004).
[Crossref]

Ke, Y.

X. Yi, Y. Liu, X. Ling, X. Zhou, Y. Ke, H. Luo, S. Wen, and D. Fan, “Hybrid-order Poincaré sphere,” Phys. Rev. A 91, 023801 (2015).
[Crossref]

Konrad, T.

T. Konrad and A. Forbes, “Quantum mechanics and classical light,” Contemp. Phys. 60, 1–22 (2019).
[Crossref]

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, 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, 397–402 (2017).
[Crossref]

Krolikowski, W.

V. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarization-controlled optical tractor beam,” Nat. Photonics 8, 846–850 (2014).
[Crossref]

Kwek, L. C.

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
[Crossref]

Lai, H.

Y. Chen, K. Huang, H. Lai, and Y. Lan, “Observation of vector vortex lattices in polarization states of an isotropic microcavity laser,” Phys. Rev. Lett. 90, 053904 (2003).
[Crossref]

Lan, Y.

Y. Chen, C. Jiang, Y. Lan, and K. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A 69, 053807 (2004).
[Crossref]

Y. Chen, K. Huang, H. Lai, and Y. Lan, “Observation of vector vortex lattices in polarization states of an isotropic microcavity laser,” Phys. Rev. Lett. 90, 053904 (2003).
[Crossref]

Larocque, H.

Lebental, M.

N. Barré, M. Romanelli, M. Lebental, and M. Brunel, “Waves and rays in plano-concave laser cavities: I. Geometric modes in the paraxial approximation,” Eur. J. Phys. 38, 034010 (2017).
[Crossref]

Leuchs, G.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum- like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics 9, 789–795 (2015).
[Crossref]

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Li, S.

Li, Y.

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
[Crossref]

Liang, H.

Y. Chen, S. Li, Y. Hsieh, J. Tung, H. Liang, and K. Huang, “Laser wave-packet representation to unify eigenmodes and geometric modes in spherical cavities,” Opt. Lett. 44, 2649–2652 (2019).
[Crossref]

J. Tung, P. Tuan, H. Liang, K. Huang, and Y. Chen, “Fractal frequency spectrum in laser resonators and three-dimensional geometric topology of optical coherent waves,” Phys. Rev. A 94, 023811 (2016).
[Crossref]

Y. Chen, J. Tung, P. Chiang, H. Liang, and K. Huang, “Exploring the effect of fractional degeneracy and the emergence of ray-wave duality in solid-state lasers with off-axis pumping,” Phys. Rev. A 88, 013827 (2013).
[Crossref]

Lin, L.

T.-H. Lu and L. Lin, “Observation of a superposition of orthogonally polarized geometric beams with a c-cut Nd: YVO4 crystal,” Appl. Phys. B 106, 863–866 (2012).
[Crossref]

Ling, X.

X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
[Crossref]

X. Yi, Y. Liu, X. Ling, X. Zhou, Y. Ke, H. Luo, S. Wen, and D. Fan, “Hybrid-order Poincaré sphere,” Phys. Rev. A 91, 023801 (2015).
[Crossref]

Litvin, I.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

Liu, Q.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light Sci. Appl. 8, 90 (2019).
[Crossref]

Liu, Y.

X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
[Crossref]

X. Yi, Y. Liu, X. Ling, X. Zhou, Y. Ke, H. Luo, S. Wen, and D. Fan, “Hybrid-order Poincaré sphere,” Phys. Rev. A 91, 023801 (2015).
[Crossref]

Lu, T.-H.

T.-H. Lu and C. He, “Generating orthogonally circular polarized states embedded in nonplanar geometric beams,” Opt. Express 23, 20876–20883 (2015).
[Crossref]

T.-H. Lu and L. Lin, “Observation of a superposition of orthogonally polarized geometric beams with a c-cut Nd: YVO4 crystal,” Appl. Phys. B 106, 863–866 (2012).
[Crossref]

T.-H. Lu, Y. Chen, and K. Huang, “Generalized hyperboloid structures of polarization singularities in Laguerre–Gaussian vector fields,” Phys. Rev. A 76, 063809 (2007).
[Crossref]

Y. Chen, T.-H. Lu, and K. Huang, “Observation of spatially coherent polarization vector fields and visualization of vector singularities,” Phys. Rev. Lett. 96, 033901 (2006).
[Crossref]

Y. Chen, T.-H. Lu, and K. Huang, “Hyperboloid structures formed by polarization singularities in coherent vector fields with longitudinal-transverse coupling,” Phys. Rev. Lett. 97, 233903 (2006).
[Crossref]

Luo, H.

X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
[Crossref]

X. Yi, Y. Liu, X. Ling, X. Zhou, Y. Ke, H. Luo, S. Wen, and D. Fan, “Hybrid-order Poincaré sphere,” Phys. Rev. A 91, 023801 (2015).
[Crossref]

Marquardt, C.

Marrucci, L.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
[Crossref]

McLaren, M.

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, 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, 397–402 (2017).
[Crossref]

Meng, Y.

Milione, G.

G. Milione, H. Sztul, D. Nolan, and R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

Min, C.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light Sci. Appl. 8, 90 (2019).
[Crossref]

Miyamoto, K.

T. Omatsu, K. Miyamoto, K. Toyoda, R. Morita, Y. Arita, and K. Dholakia, “A new twist for materials science: the formation of chiral structures using the angular momentum of light,” Adv. Opt. Mater. 7, 1801672 (2019).
[Crossref]

Morita, R.

T. Omatsu, K. Miyamoto, K. Toyoda, R. Morita, Y. Arita, and K. Dholakia, “A new twist for materials science: the formation of chiral structures using the angular momentum of light,” Adv. Opt. Mater. 7, 1801672 (2019).
[Crossref]

Mouane, O.

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, 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, 397–402 (2017).
[Crossref]

Mueller, J. B.

R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, “Arbitrary spin-to–orbital angular momentum conversion of light,” Science 358, 896–901 (2017).
[Crossref]

Mukunda, N.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

Naidoo, D.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

Nape, I.

Ndagano, B.

E. Toninelli, B. Ndagano, A. Vallés, B. Sephton, I. Nape, A. Ambrosio, F. Capasso, M. J. Padgett, and A. Forbes, “Concepts in quantum state tomography and classical implementation with intense light: a tutorial,” Adv. Opt. Photon. 11, 67–134 (2019).
[Crossref]

A. Forbes, A. Aiello, and B. Ndagano, “Classically entangled light,” Prog. Opt. 64, 99–153 (2019).
[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, 18009 (2018).
[Crossref]

C. Rosales-Guzmán, B. Ndagano, and A. Forbes, “A review of complex vector light fields and their applications,” J. Opt. 20, 123001 (2018).
[Crossref]

B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Lightwave Technol. 36, 292–301 (2017).
[Crossref]

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, 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, 397–402 (2017).
[Crossref]

Neugebauer, M.

A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics 9, 789–795 (2015).
[Crossref]

Nolan, D.

G. Milione, H. Sztul, D. Nolan, and R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

Nori, F.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

Omatsu, T.

T. Omatsu, K. Miyamoto, K. Toyoda, R. Morita, Y. Arita, and K. Dholakia, “A new twist for materials science: the formation of chiral structures using the angular momentum of light,” Adv. Opt. Mater. 7, 1801672 (2019).
[Crossref]

Ornigotti, M.

Otte, E.

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, 18009 (2018).
[Crossref]

Padgett, M. J.

Perelomov, A.

A. Perelomov, Generalized Coherent States and Their Applications (Springer, 2012).

Perez-Garcia, B.

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, 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, 397–402 (2017).
[Crossref]

Peuntinger, C.

Piccirillo, B.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

Qiu, C.-W.

X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
[Crossref]

Quang, T.

Rodríguez-Fortuño, F. J.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

Romanelli, M.

N. Barré, M. Romanelli, M. Lebental, and M. Brunel, “Waves and rays in plano-concave laser cavities: I. Geometric modes in the paraxial approximation,” Eur. J. Phys. 38, 034010 (2017).
[Crossref]

Rosales-Guzman, C.

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, 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, 397–402 (2017).
[Crossref]

B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Lightwave Technol. 36, 292–301 (2017).
[Crossref]

Rosales-Guzmán, C.

X.-B. Hu, B. Zhao, Z.-H. Zhu, W. Gao, and C. Rosales-Guzmán, “In situ detection of a cooperative target’s longitudinal and angular speed using structured light,” Opt. Lett. 44, 3070–3073 (2019).
[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, 18009 (2018).
[Crossref]

C. Rosales-Guzmán, B. Ndagano, and A. Forbes, “A review of complex vector light fields and their applications,” J. Opt. 20, 123001 (2018).
[Crossref]

Roux, F. S.

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, 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, 397–402 (2017).
[Crossref]

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

Rubin, N. A.

R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, “Arbitrary spin-to–orbital angular momentum conversion of light,” Science 358, 896–901 (2017).
[Crossref]

Santarsiero, M.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

Sciarrino, F.

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
[Crossref]

Sephton, B.

Shen, Y.

Shvedov, V.

V. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarization-controlled optical tractor beam,” Nat. Photonics 8, 846–850 (2014).
[Crossref]

Simon, B. N.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

Simon, R.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

Simon, S.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

Sit, A.

Slussarenko, S.

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
[Crossref]

Spagnolo, N.

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
[Crossref]

Spreeuw, R. J. C.

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[Crossref]

Stiller, B.

Sztul, H.

G. Milione, H. Sztul, D. Nolan, and R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

Toninelli, E.

Töppel, F.

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum- like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Toyoda, K.

T. Omatsu, K. Miyamoto, K. Toyoda, R. Morita, Y. Arita, and K. Dholakia, “A new twist for materials science: the formation of chiral structures using the angular momentum of light,” Adv. Opt. Mater. 7, 1801672 (2019).
[Crossref]

Tuan, P.

J. Tung, P. Tuan, H. Liang, K. Huang, and Y. Chen, “Fractal frequency spectrum in laser resonators and three-dimensional geometric topology of optical coherent waves,” Phys. Rev. A 94, 023811 (2016).
[Crossref]

Tung, J.

Y. Chen, S. Li, Y. Hsieh, J. Tung, H. Liang, and K. Huang, “Laser wave-packet representation to unify eigenmodes and geometric modes in spherical cavities,” Opt. Lett. 44, 2649–2652 (2019).
[Crossref]

J. Tung, P. Tuan, H. Liang, K. Huang, and Y. Chen, “Fractal frequency spectrum in laser resonators and three-dimensional geometric topology of optical coherent waves,” Phys. Rev. A 94, 023811 (2016).
[Crossref]

Y. Chen, J. Tung, P. Chiang, H. Liang, and K. Huang, “Exploring the effect of fractional degeneracy and the emergence of ray-wave duality in solid-state lasers with off-axis pumping,” Phys. Rev. A 88, 013827 (2013).
[Crossref]

Vallés, A.

Walborn, S. P.

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
[Crossref]

Wan, Z.

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

Wang, J.

Wang, X.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light Sci. Appl. 8, 90 (2019).
[Crossref]

Wen, S.

X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
[Crossref]

X. Yi, Y. Liu, X. Ling, X. Zhou, Y. Ke, H. Luo, S. Wen, and D. Fan, “Hybrid-order Poincaré sphere,” Phys. Rev. A 91, 023801 (2015).
[Crossref]

Willner, A. E.

A. E. Willner, “Vector-mode multiplexing brings an additional approach for capacity growth in optical fibers,” Light Sci. Appl. 7, 18002 (2018).
[Crossref]

Wu, C.

X.-F. Zhou, J. Zhou, and C. Wu, “Vortex structures of rotating spin-orbit-coupled Bose-Einstein condensates,” Phys. Rev. A 84, 063624 (2011).
[Crossref]

Xie, Z.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light Sci. Appl. 8, 90 (2019).
[Crossref]

Yang, A.

L. Du, A. Yang, A. V. Zayats, and X. Yuan, “Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum,” Nat. Phys. 15, 650–654 (2019).
[Crossref]

Yang, X.

Yi, X.

X. Yi, Y. Liu, X. Ling, X. Zhou, Y. Ke, H. Luo, S. Wen, and D. Fan, “Hybrid-order Poincaré sphere,” Phys. Rev. A 91, 023801 (2015).
[Crossref]

Yuan, X.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light Sci. Appl. 8, 90 (2019).
[Crossref]

L. Du, A. Yang, A. V. Zayats, and X. Yuan, “Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum,” Nat. Phys. 15, 650–654 (2019).
[Crossref]

Zayats, A. V.

L. Du, A. Yang, A. V. Zayats, and X. Yuan, “Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum,” Nat. Phys. 15, 650–654 (2019).
[Crossref]

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

Zhan, Q.

Zhang, Y.

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, 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, 397–402 (2017).
[Crossref]

Zhao, B.

Zhao, Y.

Zhou, J.

X.-F. Zhou, J. Zhou, and C. Wu, “Vortex structures of rotating spin-orbit-coupled Bose-Einstein condensates,” Phys. Rev. A 84, 063624 (2011).
[Crossref]

Zhou, X.

X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
[Crossref]

X. Yi, Y. Liu, X. Ling, X. Zhou, Y. Ke, H. Luo, S. Wen, and D. Fan, “Hybrid-order Poincaré sphere,” Phys. Rev. A 91, 023801 (2015).
[Crossref]

Zhou, X.-F.

X.-F. Zhou, J. Zhou, and C. Wu, “Vortex structures of rotating spin-orbit-coupled Bose-Einstein condensates,” Phys. Rev. A 84, 063624 (2011).
[Crossref]

Zhu, Z.-H.

Adv. Opt. Mater. (1)

T. Omatsu, K. Miyamoto, K. Toyoda, R. Morita, Y. Arita, and K. Dholakia, “A new twist for materials science: the formation of chiral structures using the angular momentum of light,” Adv. Opt. Mater. 7, 1801672 (2019).
[Crossref]

Adv. Opt. Photon. (2)

Appl. Opt. (1)

Appl. Phys. B (1)

T.-H. Lu and L. Lin, “Observation of a superposition of orthogonally polarized geometric beams with a c-cut Nd: YVO4 crystal,” Appl. Phys. B 106, 863–866 (2012).
[Crossref]

Contemp. Phys. (1)

T. Konrad and A. Forbes, “Quantum mechanics and classical light,” Contemp. Phys. 60, 1–22 (2019).
[Crossref]

Eur. J. Phys. (1)

N. Barré, M. Romanelli, M. Lebental, and M. Brunel, “Waves and rays in plano-concave laser cavities: I. Geometric modes in the paraxial approximation,” Eur. J. Phys. 38, 034010 (2017).
[Crossref]

Found. Phys. (1)

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[Crossref]

IEEE Photon. J. (1)

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

J. Lightwave Technol. (1)

J. Opt. (1)

C. Rosales-Guzmán, B. Ndagano, and A. Forbes, “A review of complex vector light fields and their applications,” J. Opt. 20, 123001 (2018).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

J. Phys. (1)

M. R. Dennis and M. A. Alonso, “Gaussian mode families from systems of rays,” J. Phys. 1, 025003 (2019).
[Crossref]

Laser Photon. Rev. (1)

A. Forbes, “Structured light from lasers,” Laser Photon. Rev. 13, 1900140 (2019).
[Crossref]

Light Sci. Appl. (3)

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, 18009 (2018).
[Crossref]

A. E. Willner, “Vector-mode multiplexing brings an additional approach for capacity growth in optical fibers,” Light Sci. Appl. 7, 18002 (2018).
[Crossref]

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light Sci. Appl. 8, 90 (2019).
[Crossref]

Nat. Commun. (1)

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
[Crossref]

Nat. Photonics (4)

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics 9, 789–795 (2015).
[Crossref]

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

V. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarization-controlled optical tractor beam,” Nat. Photonics 8, 846–850 (2014).
[Crossref]

Nat. Phys. (2)

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, 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, 397–402 (2017).
[Crossref]

L. Du, A. Yang, A. V. Zayats, and X. Yuan, “Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum,” Nat. Phys. 15, 650–654 (2019).
[Crossref]

New J. Phys. (2)

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum- like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Optica (3)

OSA Continuum (1)

Philos. Trans. R. Soc. London, Ser. A (1)

M. R. Dennis and M. A. Alonso, “Swings and roundabouts: optical Poincaré spheres for polarization and Gaussian beams,” Philos. Trans. R. Soc. London, Ser. A 375, 20150441 (2017).
[Crossref]

Phys. Rev. A (6)

Y. Chen, J. Tung, P. Chiang, H. Liang, and K. Huang, “Exploring the effect of fractional degeneracy and the emergence of ray-wave duality in solid-state lasers with off-axis pumping,” Phys. Rev. A 88, 013827 (2013).
[Crossref]

J. Tung, P. Tuan, H. Liang, K. Huang, and Y. Chen, “Fractal frequency spectrum in laser resonators and three-dimensional geometric topology of optical coherent waves,” Phys. Rev. A 94, 023811 (2016).
[Crossref]

Y. Chen, C. Jiang, Y. Lan, and K. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A 69, 053807 (2004).
[Crossref]

X. Yi, Y. Liu, X. Ling, X. Zhou, Y. Ke, H. Luo, S. Wen, and D. Fan, “Hybrid-order Poincaré sphere,” Phys. Rev. A 91, 023801 (2015).
[Crossref]

X.-F. Zhou, J. Zhou, and C. Wu, “Vortex structures of rotating spin-orbit-coupled Bose-Einstein condensates,” Phys. Rev. A 84, 063624 (2011).
[Crossref]

T.-H. Lu, Y. Chen, and K. Huang, “Generalized hyperboloid structures of polarization singularities in Laguerre–Gaussian vector fields,” Phys. Rev. A 76, 063809 (2007).
[Crossref]

Phys. Rev. Lett. (5)

Y. Chen, K. Huang, H. Lai, and Y. Lan, “Observation of vector vortex lattices in polarization states of an isotropic microcavity laser,” Phys. Rev. Lett. 90, 053904 (2003).
[Crossref]

Y. Chen, T.-H. Lu, and K. Huang, “Observation of spatially coherent polarization vector fields and visualization of vector singularities,” Phys. Rev. Lett. 96, 033901 (2006).
[Crossref]

Y. Chen, T.-H. Lu, and K. Huang, “Hyperboloid structures formed by polarization singularities in coherent vector fields with longitudinal-transverse coupling,” Phys. Rev. Lett. 97, 233903 (2006).
[Crossref]

G. Milione, H. Sztul, D. Nolan, and R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

Prog. Opt. (1)

A. Forbes, A. Aiello, and B. Ndagano, “Classically entangled light,” Prog. Opt. 64, 99–153 (2019).
[Crossref]

Rep. Prog. Phys. (1)

X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
[Crossref]

Science (1)

R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, “Arbitrary spin-to–orbital angular momentum conversion of light,” Science 358, 896–901 (2017).
[Crossref]

Other (1)

A. Perelomov, Generalized Coherent States and Their Applications (Springer, 2012).

Supplementary Material (4)

NameDescription
» Supplement 1       Supplemental document
» Visualization 1       Experimental result of the dynamic process of generating hybrid SU(2) mode at Q=5.
» Visualization 2       Experimental result of the dynamic process of generating hybrid SU(2) mode at Q=6.
» Visualization 3       Concept video

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Figures (6)

Fig. 1.
Fig. 1. Frequency degeneracy and SU(2) oscillating orbits. (a) For a concave mirror with a curvature of radius $R$ and a cavity length of $L$ that is precisely controlled, different periodic oscillating trajectories in a cavity with different round-trip periods $Q$ would emerge at some special cavity lengths corresponding to various frequency-degenerate states $|\Omega = P/Q\rangle$. (b) The distribution of frequency-degenerate states can be revealed by the fractal frequency spectrum, i.e., frequency difference ratio $({f_{n,m,l}} - {f_{{n_0},{m_0},{l_0}}})/\Delta\! {f_L}$ as a function of $L/R$ for the range of $|n - {n_0}| \le 12$, $|m - {m_0}| \le 12$, and $|l - {l_0}| \le 12$, where some degeneracy states $|\Omega = P/Q\rangle$ are marked at corresponding positions. (c) A certain periodic trajectory is determined by not only the degenerate state $|\Omega = P/Q\rangle$ but also the phase state $|\phi \rangle$. For the state $|\Omega = P/Q\rangle$, an auxiliary circle with the vertices of an equilateral $Q$-gon located on it is helpful to determine the starting points of oscillating orbits, where the starting points at the $z = 0$ plane corresponds to the projection of the vertices on the auxiliary circle. For the two eigenphase states $|\phi = 0\rangle$ and $|\phi = \pi \rangle$, there are always coincident projected vertices. Rotating the auxiliary circle for an angle $\pi /Q$ of one eigen-phase-state will produce another eigen-phase-state. If the two trajectories of the eigen-phase-states are superposed together and sharing at least a pair of coincident projected points, the hybrid SU(2) oscillating orbits can be obtained, where the yellow auxiliary lines connecting the $Q$-gon vertices on the auxiliary circle contribute possible coincident projections and result in some equilateral star shape. The yellow dots marked in the hybrid trajectories show the shared coincident projections corresponding to the actual positions of a pump spot in the experimental generation.
Fig. 2.
Fig. 2. Controlling planar and vortex SU(2) geometric orbits. (a) The experimental setup of generating a SU(2) geometric mode in a degenerate cavity. This is a case of the $|\Omega = 1/4\rangle$ state, and the period-4 oscillating orbits are schematically depicted in the cavity. (b) A schematic of the birefringent effects for the geometric beams in c-cut ${\rm Nd}{:}{{\rm YVO}_4}$. I. The index ellipsoid is depicted for determining the effective refractive index of an input beam. II. For an obliquely incident beam, the vertical and horizontal polarized components yield ordinary and effective refractive indices, respectively, and this difference can introduce a polarization modulation effect in the light beam. III. For normal incidence, there would not be a polarization modulation because there is always an ordinary refractive index. (c) The planar geometric modes can be converted into corresponding vortex geometric modes via focusing it into an astigmatic mode converter constituted by two cylindrical lenses. The illustrations of the formation of hybrid SU(2) trajectories for both cases of planar and vortex geometric modes are theoretically depicted at degenerate states (d) $|\Omega = 1/5\rangle$ and (e) $|\Omega = 1/6\rangle$, where the plot range of the $z$ axis is from 0 to $L$.
Fig. 3.
Fig. 3. Generation of hybrid SU(2) geometric modes. (a) The simulated and experimental results of the evolution from a pure SU(2) geometric mode to a hybrid SU(2) geometric mode are shown for the $|\Omega = 1/5\rangle$ and $|\Omega = 1/6\rangle$ states. For the $|\Omega = 1/5\rangle$ state, the hybrid SU(2) orbits appear at the transition from smaller-scale to larger-scale $|\phi = 0\rangle$ orbits, where the $|\phi = 0\rangle$ orbit is continuously varying and a larger-scale $|\phi = \pi \rangle$ orbit is added, forming the hybrid orbit. For the $|\Omega = 1/6\rangle$ state, the hybrid SU(2) orbits appear at the transition from the smaller-scale $|\phi = 0\rangle$ to the larger-scale $|\phi = \pi \rangle$ orbits, where the $|\phi = 0\rangle$ orbit is switched into another $|\phi = \pi \rangle$ orbit and they exist simultaneously at the transition state. (b) The simulated and experimental results of the evolution of hybrid SU(2) geometric modes for the $|\Omega = 1/5\rangle$ state (see the experimentally recorded dynamic process in Visualization 1) and (c) that for the $|\Omega = 1/6\rangle$ state (see the experimentally recorded dynamic process in Visualization 2), where the experimental patterns are marked with the corresponding actual pump off-axis displacement $\Delta x$, and the experimental results in figures (b) and (c) are captured near the beam waist with the white bars marked at the bottom left showing the scale for reference of the beam size [unit: micrometers (µm)].
Fig. 4.
Fig. 4. Vectorial properties of hybrid SU(2) vortex beams. (a),(b) The theoretical 3D wave packet of hybrid SU(2) vortex beams is plotted with clear hyperbolic twisted structures for the (a) $|\Omega = 1/5\rangle$ and (b) $|\Omega = 1/6\rangle$ states, where the plot range of the $z$ axis is from ${-}2L$ to $2L$. (c)–(f) The experimental and theoretical results of the transverse patterns (first column) with a measure after a polarizer at four different orientations (2nd to 5th columns) for the beams generated at (c) the $|\Omega = 1/5\rangle$ state with $\Delta x = 600 \; {\unicode{x00B5}{\rm m}}$ and (d) $\Delta x = 500 \; {\unicode{x00B5}{\rm m}}$, and at the (e) $|\Omega = 1/6\rangle$ state with $\Delta x = 700 \; {\unicode{x00B5}{\rm m}}$ and (f) $\Delta x = 620 \, {\unicode{x00B5}{\rm m}}$, where the white arrows indicate the linear polarizer orientations (vertical, diagonal, horizontal, and antidiagonal directions).
Fig. 5.
Fig. 5. Polarization singularities and topological phase of hybrid SU(2) vortex beams. (a)–(d) The distributions of phase difference angle defined by ${\Theta _1} = {\rm arctan} (|{E_x}|/|{E_y}|)$ for beams A–D, which show that the beams A–C are VBs and beam D is a scalar beam. (e)–(h) The distributions of phase difference angle defined by ${\Theta _2} = {\rm arg} ({E_x}/{E_y})$ for beams A–D, which demonstrate the distribution of polarization singularities (for VBs A–C) and phase singularities (for scalar beam D), where the insets show zoomed-in details.
Fig. 6.
Fig. 6. Complicated hybrid SU(2) modes with multiple trajectory-combination numbers and multitrajectory superposition. (a)–(d) The auxiliary-circle representations of the possible hybrid trajectories for degenerate states (a) $|\Omega = 1/7\rangle$, (b) $|\Omega = 1/8\rangle$, (c) $|\Omega = 1/9\rangle$, and (d) $|\Omega = 1/10\rangle$. (e) The auxiliary circle with intracavity oscillation trajectory representation for a complicated hybrid SU(2) mode superposed by three decomposed SU(2) trajectories at $|\Omega = 1/8\rangle$ state. The black dashed lines represent the coincident inflection points (shared pumping spots) of the three decomposed SU(2) trajectories in a hybrid SU(2) mode.

Equations (7)

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| ϕ = 1 2 N / 2 K = 0 N ( N K ) 1 / 2 e i K ϕ | K , N ,
| Ψ n , m ( ϕ ) = 1 2 N / 2 K = 0 N ( N K ) 1 / 2 e i K ϕ | ψ n + QK,m,l PK ( α , β ) ,
| ψ n , m , l ( α , β ) = ( b x ) n n ! ( b y ) m m ! | ψ 0 , 0 , l ,
[ b x b y ] = [ e i α / 2 cos ( β / 2 ) e i α / 2 sin ( β / 2 ) e i α / 2 sin ( β / 2 ) e i α / 2 cos ( β / 2 ) ] [ a x a y ] ,
| ψ = | ψ 1 + | ψ 2 = | Ω | N 1 | ϕ + | Ω | N 2 | ϕ + π .
| ψ = | ψ 1 + | ψ 2 = | Ω | N 1 | ϕ | H + | Ω | N 2 | ϕ + π | V ,
| ψ = | ψ 1 + | ψ 2 = | Ω | 1 | ϕ | H + | Ω | 2 | ϕ + π | V ,