Abstract

We report, for the first time to the best of our knowledge, a truncated diffraction with a triangular aperture of the SU(2) geometric modes and propose a method to detect the complicated orbital angular momentum (OAM) of an SU(2) wave-packet. As a special vortex beam, a nonplanar SU(2) mode carrying special intensity and OAM distributions brings exotic patterns in a truncated diffraction lattice. A meshy structure is unveiled therein by adjusting the illuminated aperture in the vicinity of the partial OAM regions, which can be elaborately used to evaluate the partial topological charge and OAM of an SU(2) wave-packet by counting the dark holes in the mesh. Moreover, through controlling the size and position of the aperture at the center region, the truncated triangular lattice can be close to the classical spot-array lattice for measuring the center OAM. These effects being fully validated by theoretical simulations greatly extend the versatility of topological structures detecting special beams.

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

Full Article  |  PDF Article
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References

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    [Crossref]
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    [Crossref]
  31. Y. F. Chen, T. H. Lu, and K. F. Huang, “Hyperboloid structures formed by polarization singularities in coherent vector fields with longitudinal-transverse coupling,” Phys. Rev. Lett. 97, 233903 (2006).
    [Crossref]
  32. T. H. Lu, Y. F. Chen, and K. F. Huang, “Generalized hyperboloid structures of polarization singularities in Laguerre-Gaussian vector fields,” Phys. Rev. A 76, 063809 (2007).
    [Crossref]
  33. T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Generation of multi-axis Laguerre–Gaussian beams from geometric modes of a hemiconfocal cavity,” Appl. Phys. B 103, 991–999 (2011).
    [Crossref]
  34. P. H. Tuan, Y. H. Hsieh, Y. H. Lai, K. F. Huang, and Y. F. Chen, “Characterization and generation of high-power multi-axis vortex beams by using off-axis pumped degenerate cavities with external astigmatic mode converter,” Opt. Express 26, 20481–20491(2018).
    [Crossref] [PubMed]
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    [Crossref]

2018 (5)

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light: Sci. Appl. 7, 17146 (2018).
[Crossref]

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topological charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8, 6370 (2018).
[Crossref] [PubMed]

P. H. Tuan, H. C. Liang, K. F. Huang, and Y. F. Chen, “Realizing high-pulse-energy large-angular-momentum beams by astigmatic transformation of geometric modes in an Nd:YAG/Cr4+:YAG laser,” IEEE J. Sel. Top. Quantum Electron. 24, 1600809 (2018)..
[Crossref]

Y. Shen, Z. Wan, Y. Meng, X. Fu, and M. Gong, “Polygonal vortex beams,” IEEE Photonics J. 10, 28588452018.

P. H. Tuan, Y. H. Hsieh, Y. H. Lai, K. F. Huang, and Y. F. Chen, “Characterization and generation of high-power multi-axis vortex beams by using off-axis pumped degenerate cavities with external astigmatic mode converter,” Opt. Express 26, 20481–20491(2018).
[Crossref] [PubMed]

2017 (6)

Y. F. Chen, J. C. Tung, P. H. Tuan, and K. F. Huang, “symmetry breaking induced geometric surfaces with topological curves in quantum and classical dynamics of the SU(2) coupled oscillators,” Ann. Phys. 529, 1600253 (2017).
[Crossref]

S. N. Alperin and M. E. Siemens, “Angular Momentum of Topologically Structured Darkness,” Phys. Rev. Lett. 119, 203902 (2017).
[Crossref] [PubMed]

J. C. Tung, T. Omatsu, H. C. Liang, K. F. Huang, and K. F. Huang, “Exploring the self-mode locking and vortex structures of nonplanar elliptical modes in selectively end-pumped Nd:YVO4 lasers: manifestation of large fractional orbital angular momentum,” Opt. Express. 25, 22769–22779 (2017).
[Crossref] [PubMed]

B. S. Bhargavaram, A. Sharma, and P. Senthilkumaran, “Diffraction of V-point singularities through triangular apertures,” Opt. Express 25, 10270–10275 (2017).
[Crossref]

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8, 14970 (2017).
[Crossref] [PubMed]

D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8, 14971 (2017).
[Crossref]

2016 (3)

2015 (1)

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

2013 (2)

Y. F. Chen, J. C. Tung, P. Y. Chiang, H. C. Liang, and K. F. 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]

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

2012 (4)

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

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science. 388, 640–643 (2012).
[Crossref]

T. H. Lu and L. H. 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]

Y. C. Lin, T. H. Lu, K. F. Huang, and Y. F. Chen, “Model of commensurate harmonic oscillators with SU(2) coupling interactions: Analogous observation in laser transverse modes,” Phys. Rev. E 85, 046217 (2012).
[Crossref]

2011 (4)

T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Generation of multi-axis Laguerre–Gaussian beams from geometric modes of a hemiconfocal cavity,” Appl. Phys. B 103, 991–999 (2011).
[Crossref]

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

L. E. E. de Araujo and M. E. Anderson, “Measuring vortex charge with a triangular aperture,” Opt. Lett. 36, 787–789 (2011).
[Crossref] [PubMed]

A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express 19, 5760–5771 (2011).
[Crossref] [PubMed]

2010 (2)

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[Crossref]

Y. F. Chen, Y. C. Lin, K. F. Huang, and T. H. Lu, “Spatial transformation of coherent optical waves with orbital morphologies,” Phys. Rev. A 82, 043801 (2010).
[Crossref]

2008 (2)

T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Three-dimensional coherent optical waves localized on trochoidal parametric surfaces,” Phys. Rev. Lett. 101, 233901 (2008).
[Crossref] [PubMed]

M. S. Kumar and B. Dutta-Roy, “Commensurate anisotropic oscillator, SU(2) coherent states and the classical limit,” J. Phys. A: Math. Theor. 41, 075306 (2008).
[Crossref]

2007 (1)

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

2006 (2)

Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, “Devil’s Staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces,” Phys. Rev. Lett. 96, 213902 (2006).
[Crossref]

Y. F. Chen, T. H. Lu, and K. F. 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. F. Chen, C. H. Jiang, Y. P. Lan, and K. F. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A 69, 053870 (2004).
[Crossref]

2001 (1)

J. Dingjan, M. P. van Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188, 345–351 (2001).
[Crossref]

1993 (1)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Adhikary, G.

D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8, 14971 (2017).
[Crossref]

Ahmed, N.

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

Allen, L.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Alperin, S. N.

S. N. Alperin and M. E. Siemens, “Angular Momentum of Topologically Structured Darkness,” Phys. Rev. Lett. 119, 203902 (2017).
[Crossref] [PubMed]

Alpmann, C.

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

Anderson, M. E.

Arissian, L.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8, 14970 (2017).
[Crossref] [PubMed]

Baumgartl, J.

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Bhargavaram, B. S.

Bouchard, F.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8, 14970 (2017).
[Crossref] [PubMed]

Bowman, R.

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

Boyd, R. W.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8, 14970 (2017).
[Crossref] [PubMed]

Brown, G. G.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8, 14970 (2017).
[Crossref] [PubMed]

Camper, A.

D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8, 14971 (2017).
[Crossref]

Chappuis, C.

D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8, 14971 (2017).
[Crossref]

Chávez-Cerda, S.

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topological charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8, 6370 (2018).
[Crossref] [PubMed]

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[Crossref]

Chen, Y. F.

P. H. Tuan, H. C. Liang, K. F. Huang, and Y. F. Chen, “Realizing high-pulse-energy large-angular-momentum beams by astigmatic transformation of geometric modes in an Nd:YAG/Cr4+:YAG laser,” IEEE J. Sel. Top. Quantum Electron. 24, 1600809 (2018)..
[Crossref]

P. H. Tuan, Y. H. Hsieh, Y. H. Lai, K. F. Huang, and Y. F. Chen, “Characterization and generation of high-power multi-axis vortex beams by using off-axis pumped degenerate cavities with external astigmatic mode converter,” Opt. Express 26, 20481–20491(2018).
[Crossref] [PubMed]

Y. F. Chen, J. C. Tung, P. H. Tuan, and K. F. Huang, “symmetry breaking induced geometric surfaces with topological curves in quantum and classical dynamics of the SU(2) coupled oscillators,” Ann. Phys. 529, 1600253 (2017).
[Crossref]

J. C. Tung, H. C. Liang, T. H. Lu, K. F. Huang, and Y. F. Chen, “Exploring vortex structures in orbital-angular-momentum beams generated from planar geometric modes with a mode converter,” Opt. Express 24, 22796–22805 (2016).
[Crossref] [PubMed]

Y. F. Chen, J. C. Tung, P. Y. Chiang, H. C. Liang, and K. F. 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]

Y. C. Lin, T. H. Lu, K. F. Huang, and Y. F. Chen, “Model of commensurate harmonic oscillators with SU(2) coupling interactions: Analogous observation in laser transverse modes,” Phys. Rev. E 85, 046217 (2012).
[Crossref]

T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Generation of multi-axis Laguerre–Gaussian beams from geometric modes of a hemiconfocal cavity,” Appl. Phys. B 103, 991–999 (2011).
[Crossref]

Y. F. Chen, Y. C. Lin, K. F. Huang, and T. H. Lu, “Spatial transformation of coherent optical waves with orbital morphologies,” Phys. Rev. A 82, 043801 (2010).
[Crossref]

T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Three-dimensional coherent optical waves localized on trochoidal parametric surfaces,” Phys. Rev. Lett. 101, 233901 (2008).
[Crossref] [PubMed]

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

Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, “Devil’s Staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces,” Phys. Rev. Lett. 96, 213902 (2006).
[Crossref]

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

Y. F. Chen, C. H. Jiang, Y. P. Lan, and K. F. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A 69, 053870 (2004).
[Crossref]

Chiang, P. Y.

Y. F. Chen, J. C. Tung, P. Y. Chiang, H. C. Liang, and K. F. 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]

Corkum, P. B.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8, 14970 (2017).
[Crossref] [PubMed]

Cucini, R.

D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8, 14971 (2017).
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de Araujo, L. E. E.

De Ninno, G.

D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8, 14971 (2017).
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M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7, 839–854 (2013).
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DiMauro, L. F.

D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8, 14971 (2017).
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J. Dingjan, M. P. van Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188, 345–351 (2001).
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J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Dovillaire, G.

D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8, 14971 (2017).
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Dutta-Roy, B.

M. S. Kumar and B. Dutta-Roy, “Commensurate anisotropic oscillator, SU(2) coherent states and the classical limit,” J. Phys. A: Math. Theor. 41, 075306 (2008).
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M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light: Sci. Appl. 7, 17146 (2018).
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Esseling, M.

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

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

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light: Sci. Appl. 7, 17146 (2018).
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R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science. 388, 640–643 (2012).
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Fonseca, E. J. S.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
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Frassetto, F.

D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8, 14971 (2017).
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Fu, X.

Y. Shen, Z. Wan, Y. Meng, X. Fu, and M. Gong, “Polygonal vortex beams,” IEEE Photonics J. 10, 28588452018.

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D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8, 14971 (2017).
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Gbur, G.

Géneaux, R.

D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8, 14971 (2017).
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Gong, M.

Y. Shen, Z. Wan, Y. Meng, X. Fu, and M. Gong, “Polygonal vortex beams,” IEEE Photonics J. 10, 28588452018.

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F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8, 14970 (2017).
[Crossref] [PubMed]

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

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[Crossref]

Hsieh, Y. H.

Huang, H.

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

Huang, K. F.

P. H. Tuan, H. C. Liang, K. F. Huang, and Y. F. Chen, “Realizing high-pulse-energy large-angular-momentum beams by astigmatic transformation of geometric modes in an Nd:YAG/Cr4+:YAG laser,” IEEE J. Sel. Top. Quantum Electron. 24, 1600809 (2018)..
[Crossref]

P. H. Tuan, Y. H. Hsieh, Y. H. Lai, K. F. Huang, and Y. F. Chen, “Characterization and generation of high-power multi-axis vortex beams by using off-axis pumped degenerate cavities with external astigmatic mode converter,” Opt. Express 26, 20481–20491(2018).
[Crossref] [PubMed]

J. C. Tung, T. Omatsu, H. C. Liang, K. F. Huang, and K. F. Huang, “Exploring the self-mode locking and vortex structures of nonplanar elliptical modes in selectively end-pumped Nd:YVO4 lasers: manifestation of large fractional orbital angular momentum,” Opt. Express. 25, 22769–22779 (2017).
[Crossref] [PubMed]

J. C. Tung, T. Omatsu, H. C. Liang, K. F. Huang, and K. F. Huang, “Exploring the self-mode locking and vortex structures of nonplanar elliptical modes in selectively end-pumped Nd:YVO4 lasers: manifestation of large fractional orbital angular momentum,” Opt. Express. 25, 22769–22779 (2017).
[Crossref] [PubMed]

Y. F. Chen, J. C. Tung, P. H. Tuan, and K. F. Huang, “symmetry breaking induced geometric surfaces with topological curves in quantum and classical dynamics of the SU(2) coupled oscillators,” Ann. Phys. 529, 1600253 (2017).
[Crossref]

J. C. Tung, H. C. Liang, T. H. Lu, K. F. Huang, and Y. F. Chen, “Exploring vortex structures in orbital-angular-momentum beams generated from planar geometric modes with a mode converter,” Opt. Express 24, 22796–22805 (2016).
[Crossref] [PubMed]

Y. F. Chen, J. C. Tung, P. Y. Chiang, H. C. Liang, and K. F. 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]

Y. C. Lin, T. H. Lu, K. F. Huang, and Y. F. Chen, “Model of commensurate harmonic oscillators with SU(2) coupling interactions: Analogous observation in laser transverse modes,” Phys. Rev. E 85, 046217 (2012).
[Crossref]

T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Generation of multi-axis Laguerre–Gaussian beams from geometric modes of a hemiconfocal cavity,” Appl. Phys. B 103, 991–999 (2011).
[Crossref]

Y. F. Chen, Y. C. Lin, K. F. Huang, and T. H. Lu, “Spatial transformation of coherent optical waves with orbital morphologies,” Phys. Rev. A 82, 043801 (2010).
[Crossref]

T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Three-dimensional coherent optical waves localized on trochoidal parametric surfaces,” Phys. Rev. Lett. 101, 233901 (2008).
[Crossref] [PubMed]

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

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

Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, “Devil’s Staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces,” Phys. Rev. Lett. 96, 213902 (2006).
[Crossref]

Y. F. Chen, C. H. Jiang, Y. P. Lan, and K. F. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A 69, 053870 (2004).
[Crossref]

Jesus-Silva, A. J.

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topological charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8, 6370 (2018).
[Crossref] [PubMed]

Jiang, C. H.

Y. F. Chen, C. H. Jiang, Y. P. Lan, and K. F. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A 69, 053870 (2004).
[Crossref]

Karimi, E.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8, 14970 (2017).
[Crossref] [PubMed]

Ko, D. H.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8, 14970 (2017).
[Crossref] [PubMed]

Kong, F.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8, 14970 (2017).
[Crossref] [PubMed]

Krenn, M.

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light: Sci. Appl. 7, 17146 (2018).
[Crossref]

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science. 388, 640–643 (2012).
[Crossref]

Kumar, M. S.

M. S. Kumar and B. Dutta-Roy, “Commensurate anisotropic oscillator, SU(2) coherent states and the classical limit,” J. Phys. A: Math. Theor. 41, 075306 (2008).
[Crossref]

Lai, Y. H.

Lan, Y. P.

Y. F. Chen, C. H. Jiang, Y. P. Lan, and K. F. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A 69, 053870 (2004).
[Crossref]

Lapkiewicz, R.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science. 388, 640–643 (2012).
[Crossref]

Li, Z.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8, 14970 (2017).
[Crossref] [PubMed]

Liang, H. C.

P. H. Tuan, H. C. Liang, K. F. Huang, and Y. F. Chen, “Realizing high-pulse-energy large-angular-momentum beams by astigmatic transformation of geometric modes in an Nd:YAG/Cr4+:YAG laser,” IEEE J. Sel. Top. Quantum Electron. 24, 1600809 (2018)..
[Crossref]

J. C. Tung, T. Omatsu, H. C. Liang, K. F. Huang, and K. F. Huang, “Exploring the self-mode locking and vortex structures of nonplanar elliptical modes in selectively end-pumped Nd:YVO4 lasers: manifestation of large fractional orbital angular momentum,” Opt. Express. 25, 22769–22779 (2017).
[Crossref] [PubMed]

J. C. Tung, H. C. Liang, T. H. Lu, K. F. Huang, and Y. F. Chen, “Exploring vortex structures in orbital-angular-momentum beams generated from planar geometric modes with a mode converter,” Opt. Express 24, 22796–22805 (2016).
[Crossref] [PubMed]

Y. F. Chen, J. C. Tung, P. Y. Chiang, H. C. Liang, and K. F. 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. H.

T. H. Lu and L. H. 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]

Lin, Y. C.

Y. C. Lin, T. H. Lu, K. F. Huang, and Y. F. Chen, “Model of commensurate harmonic oscillators with SU(2) coupling interactions: Analogous observation in laser transverse modes,” Phys. Rev. E 85, 046217 (2012).
[Crossref]

T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Generation of multi-axis Laguerre–Gaussian beams from geometric modes of a hemiconfocal cavity,” Appl. Phys. B 103, 991–999 (2011).
[Crossref]

Y. F. Chen, Y. C. Lin, K. F. Huang, and T. H. Lu, “Spatial transformation of coherent optical waves with orbital morphologies,” Phys. Rev. A 82, 043801 (2010).
[Crossref]

T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Three-dimensional coherent optical waves localized on trochoidal parametric surfaces,” Phys. Rev. Lett. 101, 233901 (2008).
[Crossref] [PubMed]

Lu, T. H.

J. C. Tung, H. C. Liang, T. H. Lu, K. F. Huang, and Y. F. Chen, “Exploring vortex structures in orbital-angular-momentum beams generated from planar geometric modes with a mode converter,” Opt. Express 24, 22796–22805 (2016).
[Crossref] [PubMed]

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

T. H. Lu and L. H. 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]

Y. C. Lin, T. H. Lu, K. F. Huang, and Y. F. Chen, “Model of commensurate harmonic oscillators with SU(2) coupling interactions: Analogous observation in laser transverse modes,” Phys. Rev. E 85, 046217 (2012).
[Crossref]

T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Generation of multi-axis Laguerre–Gaussian beams from geometric modes of a hemiconfocal cavity,” Appl. Phys. B 103, 991–999 (2011).
[Crossref]

Y. F. Chen, Y. C. Lin, K. F. Huang, and T. H. Lu, “Spatial transformation of coherent optical waves with orbital morphologies,” Phys. Rev. A 82, 043801 (2010).
[Crossref]

T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Three-dimensional coherent optical waves localized on trochoidal parametric surfaces,” Phys. Rev. Lett. 101, 233901 (2008).
[Crossref] [PubMed]

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

Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, “Devil’s Staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces,” Phys. Rev. Lett. 96, 213902 (2006).
[Crossref]

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

Melo, L. A.

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topological charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8, 6370 (2018).
[Crossref] [PubMed]

Meng, Y.

Y. Shen, Z. Wan, Y. Meng, X. Fu, and M. Gong, “Polygonal vortex beams,” IEEE Photonics J. 10, 28588452018.

Miotti, P.

D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8, 14971 (2017).
[Crossref]

Mourka, A.

Omatsu, T.

J. C. Tung, T. Omatsu, H. C. Liang, K. F. Huang, and K. F. Huang, “Exploring the self-mode locking and vortex structures of nonplanar elliptical modes in selectively end-pumped Nd:YVO4 lasers: manifestation of large fractional orbital angular momentum,” Opt. Express. 25, 22769–22779 (2017).
[Crossref] [PubMed]

Padgett, M.

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

Plick, W. N.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science. 388, 640–643 (2012).
[Crossref]

Poletto, L.

D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8, 14971 (2017).
[Crossref]

Ramelow, S.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science. 388, 640–643 (2012).
[Crossref]

Ren, Y.

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

Ressel, B.

D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8, 14971 (2017).
[Crossref]

Ribeiro, P. H. S.

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topological charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8, 6370 (2018).
[Crossref] [PubMed]

Ribic, P. R.

D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8, 14971 (2017).
[Crossref]

Ruchon, T.

D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8, 14971 (2017).
[Crossref]

Schaeff, C.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science. 388, 640–643 (2012).
[Crossref]

Senthilkumaran, P.

Shanor, C.

Sharma, A.

Shen, Y.

Y. Shen, Z. Wan, Y. Meng, X. Fu, and M. Gong, “Polygonal vortex beams,” IEEE Photonics J. 10, 28588452018.

Siemens, M. E.

S. N. Alperin and M. E. Siemens, “Angular Momentum of Topologically Structured Darkness,” Phys. Rev. Lett. 119, 203902 (2017).
[Crossref] [PubMed]

Soares, W. C.

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topological charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8, 6370 (2018).
[Crossref] [PubMed]

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[Crossref]

Spezzani, C.

D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8, 14971 (2017).
[Crossref]

Stupar, M.

D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8, 14971 (2017).
[Crossref]

Su, K. W.

Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, “Devil’s Staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces,” Phys. Rev. Lett. 96, 213902 (2006).
[Crossref]

Tuan, P. H.

P. H. Tuan, H. C. Liang, K. F. Huang, and Y. F. Chen, “Realizing high-pulse-energy large-angular-momentum beams by astigmatic transformation of geometric modes in an Nd:YAG/Cr4+:YAG laser,” IEEE J. Sel. Top. Quantum Electron. 24, 1600809 (2018)..
[Crossref]

P. H. Tuan, Y. H. Hsieh, Y. H. Lai, K. F. Huang, and Y. F. Chen, “Characterization and generation of high-power multi-axis vortex beams by using off-axis pumped degenerate cavities with external astigmatic mode converter,” Opt. Express 26, 20481–20491(2018).
[Crossref] [PubMed]

Y. F. Chen, J. C. Tung, P. H. Tuan, and K. F. Huang, “symmetry breaking induced geometric surfaces with topological curves in quantum and classical dynamics of the SU(2) coupled oscillators,” Ann. Phys. 529, 1600253 (2017).
[Crossref]

Tung, J. C.

Y. F. Chen, J. C. Tung, P. H. Tuan, and K. F. Huang, “symmetry breaking induced geometric surfaces with topological curves in quantum and classical dynamics of the SU(2) coupled oscillators,” Ann. Phys. 529, 1600253 (2017).
[Crossref]

J. C. Tung, T. Omatsu, H. C. Liang, K. F. Huang, and K. F. Huang, “Exploring the self-mode locking and vortex structures of nonplanar elliptical modes in selectively end-pumped Nd:YVO4 lasers: manifestation of large fractional orbital angular momentum,” Opt. Express. 25, 22769–22779 (2017).
[Crossref] [PubMed]

J. C. Tung, H. C. Liang, T. H. Lu, K. F. Huang, and Y. F. Chen, “Exploring vortex structures in orbital-angular-momentum beams generated from planar geometric modes with a mode converter,” Opt. Express 24, 22796–22805 (2016).
[Crossref] [PubMed]

Y. F. Chen, J. C. Tung, P. Y. Chiang, H. C. Liang, and K. F. 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]

Tur, M.

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

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

van Exter, M. P.

J. Dingjan, M. P. van Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188, 345–351 (2001).
[Crossref]

Wan, Z.

Y. Shen, Z. Wan, Y. Meng, X. Fu, and M. Gong, “Polygonal vortex beams,” IEEE Photonics J. 10, 28588452018.

Wang, J.

J. Wang, “Advances in communications using optical vortices,” Photonics Res. 4, B14–B28 (2016).
[Crossref]

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

Willner, A. E.

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

Woerdemann, M.

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

Woerdman, J. P.

J. Dingjan, M. P. van Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188, 345–351 (2001).
[Crossref]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Wright, E. M.

Yan, Y.

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

Yang, J.

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

Yue, Y.

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

Zeilinger, A.

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light: Sci. Appl. 7, 17146 (2018).
[Crossref]

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science. 388, 640–643 (2012).
[Crossref]

Zhang, C.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8, 14970 (2017).
[Crossref] [PubMed]

Ann. Phys. (1)

Y. F. Chen, J. C. Tung, P. H. Tuan, and K. F. Huang, “symmetry breaking induced geometric surfaces with topological curves in quantum and classical dynamics of the SU(2) coupled oscillators,” Ann. Phys. 529, 1600253 (2017).
[Crossref]

Appl. Phys. B (2)

T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Generation of multi-axis Laguerre–Gaussian beams from geometric modes of a hemiconfocal cavity,” Appl. Phys. B 103, 991–999 (2011).
[Crossref]

T. H. Lu and L. H. 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]

IEEE J. Sel. Top. Quantum Electron. (1)

P. H. Tuan, H. C. Liang, K. F. Huang, and Y. F. Chen, “Realizing high-pulse-energy large-angular-momentum beams by astigmatic transformation of geometric modes in an Nd:YAG/Cr4+:YAG laser,” IEEE J. Sel. Top. Quantum Electron. 24, 1600809 (2018)..
[Crossref]

IEEE Photonics J. (1)

Y. Shen, Z. Wan, Y. Meng, X. Fu, and M. Gong, “Polygonal vortex beams,” IEEE Photonics J. 10, 28588452018.

J. Phys. A: Math. Theor. (1)

M. S. Kumar and B. Dutta-Roy, “Commensurate anisotropic oscillator, SU(2) coherent states and the classical limit,” J. Phys. A: Math. Theor. 41, 075306 (2008).
[Crossref]

Laser Photonics Rev. (1)

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

Light: Sci. Appl. (1)

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light: Sci. Appl. 7, 17146 (2018).
[Crossref]

Nat. Commun. (2)

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8, 14970 (2017).
[Crossref] [PubMed]

D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8, 14971 (2017).
[Crossref]

Nat. Photonics (2)

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

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

Opt. Commun. (2)

J. Dingjan, M. P. van Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188, 345–351 (2001).
[Crossref]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Opt. Express (4)

Opt. Express. (2)

J. C. Tung, T. Omatsu, H. C. Liang, K. F. Huang, and K. F. Huang, “Exploring the self-mode locking and vortex structures of nonplanar elliptical modes in selectively end-pumped Nd:YVO4 lasers: manifestation of large fractional orbital angular momentum,” Opt. Express. 25, 22769–22779 (2017).
[Crossref] [PubMed]

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

Opt. Lett. (1)

Optica (1)

Photonics Res. (1)

J. Wang, “Advances in communications using optical vortices,” Photonics Res. 4, B14–B28 (2016).
[Crossref]

Phys. Rev. A (4)

Y. F. Chen, C. H. Jiang, Y. P. Lan, and K. F. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A 69, 053870 (2004).
[Crossref]

Y. F. Chen, J. C. Tung, P. Y. Chiang, H. C. Liang, and K. F. 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]

Y. F. Chen, Y. C. Lin, K. F. Huang, and T. H. Lu, “Spatial transformation of coherent optical waves with orbital morphologies,” Phys. Rev. A 82, 043801 (2010).
[Crossref]

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

Phys. Rev. E (1)

Y. C. Lin, T. H. Lu, K. F. Huang, and Y. F. Chen, “Model of commensurate harmonic oscillators with SU(2) coupling interactions: Analogous observation in laser transverse modes,” Phys. Rev. E 85, 046217 (2012).
[Crossref]

Phys. Rev. Lett. (5)

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[Crossref]

Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, “Devil’s Staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces,” Phys. Rev. Lett. 96, 213902 (2006).
[Crossref]

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

S. N. Alperin and M. E. Siemens, “Angular Momentum of Topologically Structured Darkness,” Phys. Rev. Lett. 119, 203902 (2017).
[Crossref] [PubMed]

T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Three-dimensional coherent optical waves localized on trochoidal parametric surfaces,” Phys. Rev. Lett. 101, 233901 (2008).
[Crossref] [PubMed]

Sci. Rep. (1)

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topological charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8, 6370 (2018).
[Crossref] [PubMed]

Science. (1)

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science. 388, 640–643 (2012).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 The experimental setup: LD, laser diode; DM, dichroic mirror; Xtal, crystal; OC, output coupler; BS, beam splitter; PM, power meter; AMC, astigmatic mode converter; TA, triangular aperture; CCD, charge coupled device. Inset: (I) the controlling of pump spot on crystal; (II) the classical periodic ray trajectories in resonator under SU(2) degenerate states of (a) |Ω = 1/4〉 and (b) Ω = 1/3〉, the classical ray trajectories in AMC for (c) |Ω = 1/4〉 and (d) |Ω = 1/3〉, and the theoretical intensity-phase profiles of SU(2) wave-packets versus z-axis for (e) |Ω = 1/4〉 and (f) |Ω = 1/3〉; (III) the details of the triangular aperture.
Fig. 2
Fig. 2 The experimental output power curves under different pump currents off-axis displacements, associated with the corresponding transverse patterns.
Fig. 3
Fig. 3 The experimental nonplanar SU(2) mode patterns evolution with different cavity length from |Ω = 1/4〉 to |Ω = 1/3〉 [from (I)-(IV)] at various propagation distances (a) z = zR, (b) z = 1/3 · zR, (c) z = 2/3 · zR, (d) z = zR, (e) z = 4/3 · zR.
Fig. 4
Fig. 4 The theoretical and experimental results of far-field truncated diffraction for (I) a |Ω = 1/4⟩ SU(2) mode, (II) a |Ω = 1/3⟩ SU(2) mode and (III) an LG mode [(a): intensity; (b): phase], including the conditions of TA (left columns), simulated patterns (middle columns), and experimental patterns (right columns). The symbols of + mark the center origin of coordinates and the dashed circles mark the partial OAM regions.
Fig. 5
Fig. 5 The OAM detection of SU(2) wave-packets (I) Φ 4 4 | Ω = 1 / 4, (II) Φ 7 7 | Ω = 1 / 4, and (III) Φ 10 9 | Ω = 1 / 4, including (a) the schematics of center and partial OAM detection, (b,c) the simulated and experimental intensity field, (d,e) the simulated and experimental truncated diffraction lattices for partial OAM detection, (f,g) the simulated and experimental truncated diffraction lattices for center OAM detection.

Equations (8)

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E ( x , y , 0 ) = [ x + i sgn ( ) y ] | | E ( r , θ , 0 ) = r | | exp ( i θ ) ,
E ( ξ , η , f ) = t ( x , y ) [ x + i sgn ( ) y ] | | exp [ 2 π i λ f ( ξ x η y ) ] d x d y = 1 2 π [ t ( x , y ) ] { [ x + i sgn ( ) y ] | | } = 1 2 π T ( ω x , ω y ) { k = 0 | | ( | | k ) x | | k y k [ i sgn ( ) ] k } = 1 2 π T ( ω x , ω y ) k = 0 | | ( | | k ) [ i sgn ( ) ] k [ 1 2 π ( x | | k ) ( y k ) ] = 1 4 π 2 T ( ω x , ω y ) k = 0 | | ( | | k ) [ i sgn ( ) ] k [ 2 π i | | k δ ( | | k ) ( ω x ) ] [ 2 π i k δ ( k ) ( ω y ) ] = i | | k = 0 | | ( | | k ) [ i sgn ( ) ] k T ( ω x , ω y ) δ ( | | k ) ( ω x ) δ ( k ) ( ω y ) = ( i ) | | k = 0 | | ( | | k ) [ i sgn ( ) ] k | | ω x | | k ω y k T ( ω x , ω y ) ,
Ψ n 0 M ( x , y , z ; ϕ 0 | Ω ) = 1 2 M / 2 K = 0 M M ! K ! ( M K ) ! e i K ϕ 0 ψ n 0 + Q K , 0 , s 0 P K ( H G ) ( x , y , z ) ,
ψ n , m , s ( HG ) ( x , y , z ) = 2 π m ! n ! 2 ( m + n ) / 2 w ( z ) H n [ 2 x w ( z ) ] H m [ 2 x w ( z ) ] exp [ x 2 + y 2 w 2 ( z ) ] × exp [ i k n , m , s z + i k n , m , s x 2 + y 2 2 R ( z ) i ( m + n + 1 ) ϑ ( z ) ] ,
Φ n 0 M ( ρ , θ , z ; ϕ 0 | Ω ) = 1 2 M / 2 K = 0 M M ! K ! ( M K ) ! e i K ϕ 0 φ 0 , ± ( n 0 + Q K ) , s 0 P K ( LG ) ( ρ , θ , z ) ,
φ p , , s ( LG ) ( r , θ , z ) = 2 p ! π ( p + | l | ) ! 1 w ( z ) [ 2 r w ( z ) ] | | exp [ r 2 w 2 ( z ) ] L p | | [ 2 r 2 w 2 ( z ) ] exp ( i θ ) × exp [ i k n , m , s z + i k n , m , s r 2 2 R ( z ) i ( 2 p + | | + 1 ) ϑ ( z ) ] φ p , , s ( LG ) ( x , y , z ) = 2 | | p ! π ( p + | l | ) ! 1 w | | + 1 ( z ) exp [ x 2 + y 2 w 2 ( z ) ] L p | | [ 2 x 2 + y 2 w 2 ( z ) ] [ x + i sgn ( ) y ] | | × exp [ i k n , m , s z + i k n , m , s x 2 + y 2 2 R ( z ) i ( 2 p + | | + 1 ) ϑ ( z ) ] ,
E φ ( ξ , η ) = t ( x , y ) Φ n 0 M ( ρ , θ , z ; ϕ 0 | Ω ) exp [ 2 π i λ f ( ξ x + η y ) ] d x d y = 1 2 M / 2 K = 0 M ( M K ) 1 / 2 e i K ϕ 0 [ t ( x , y ) φ 0 , n 0 + Q K , s 0 P K ( LG ) ( x , y , z ) ] = 1 4 π 2 e i k n 0 , 0 , s 0 z 2 M / 2 K = 0 M 2 n 0 + Q K p ! π ( p + n 0 + Q K ) ! 1 w n 0 + Q K + 1 ( M K ) 1 / 2 e i K ϕ 0 e i ( n 0 + Q K + 1 ) ϑ × { t ( x , y ) ( x ± i y ) n 0 + Q K } [ exp ( x 2 + y 2 w 2 ) ] [ L 0 n 0 + Q K ( 2 x 2 + y 2 w 2 ) ] ,
E φ ( ξ , η ) K = 0 M 2 Q K / 2 e i K ϕ 0 e i Q K ϑ w Q K ( p + n 0 + Q K ) ! ( M K ) 1 / 2 { t ( x , y ) ( x ± i y ) n 0 + Q K } [ exp ( x 2 + y 2 w 2 ) ] K = 0 M k = 0 n 0 + Q K 2 Q K / 2 e i K ϕ 0 e i Q K ϑ w Q K ( p + n 0 + Q K ) ! ( M K ) 1 / 2 ( n 0 + Q K k ) ( ± i ) k × [ n 0 + Q K ω x n 0 + Q K k ω y k T ( ω x , ω y ) ] exp ( w 2 ω x 2 + ω y 2 4 ) ,

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