Abstract

We have theoretically designed and experimentally observed free-space propagation of topological singular lines of cylindrical vector optical fields with non-integer topological charges. The polarization singular lines are due to the orientation uncertainty of the polarization states, caused by non-integer topological charges. The results reveal that during propagation, evolution of the polarization singular lines results in the special intensity pattern, distribution of polarization states, and chains of polarization singularities. We have also proposed a method to generate triple straight and spiral singular lines, which may contribute to the research of complex optical fields.

© 2019 Chinese Laser Press

Full Article  |  PDF Article
OSA Recommended Articles
Pseudo-topological property of Julia fractal vector optical fields

Guan-Lin Zhang, Meng-Qiang Cai, Xin-Ling He, Xu-Zhen Gao, Meng-Dan Zhao, Dan Wang, Yongnan Li, Chenghou Tu, and Hui-Tian Wangrmark
Opt. Express 27(9) 13263-13279 (2019)

Generation and dynamics of optical beams with polarization singularities

Filippo Cardano, Ebrahim Karimi, Lorenzo Marrucci, Corrado de Lisio, and Enrico Santamato
Opt. Express 21(7) 8815-8820 (2013)

Controlling the polarization singularities of the focused azimuthally polarized beams

Wei Zhang, Sheng Liu, Peng Li, Xiangyang Jiao, and Jianlin Zhao
Opt. Express 21(1) 974-983 (2013)

References

  • View by:
  • |
  • |
  • |

  1. G. J. Gbur, Singular Optics (CRC Press, 2016).
  2. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
    [Crossref]
  3. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
    [Crossref]
  4. G. Gbur, “Fractional vortex Hilbert’s hotel,” Optica 3, 222–225 (2016).
    [Crossref]
  5. D. H. Goldstein, Polarized Light, Revised and Expanded (CRC Press, 2003).
  6. M. V. Berry and J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809–1821 (1977).
    [Crossref]
  7. J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. Lond. A 389, 279–290 (1983).
    [Crossref]
  8. J. F. Nye, “Line singularities in wave fields,” Philos. Trans. R. Soc. Lond. A 355, 2065–2069 (1997).
    [Crossref]
  9. I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201, 251–270 (2002).
    [Crossref]
  10. E. J. Galvez, S. Khadka, W. H. Schubert, and S. Nomoto, “Poincaré-beam patterns produced by nonseparable superpositions of Laguerre-Gauss and polarization modes of light,” Appl. Opt. 51, 2925–2934 (2012).
    [Crossref]
  11. E. J. Galvez and B. Khajavi, “High-order disclinations in the polarization of light,” Proc. SPIE 9764, 97640R (2016).
    [Crossref]
  12. B. Khajavi and E. J. Galvez, “High-order disclinations in space-variant polarization,” J. Opt. 18, 084003 (2016).
    [Crossref]
  13. E. Otte, C. Alpmann, and C. Denz, “Polarization singularity explosions in tailored light fields,” Laser Photon. Rev. 12, 1700200 (2018).
    [Crossref]
  14. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Manipulation of the Pancharatnam phase in vectorial vortices,” Opt. Express 14, 4208–4220 (2006).
    [Crossref]
  15. X. L. Wang, J. P. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007).
    [Crossref]
  16. P. H. Jones, M. Rashid, M. Makita, and O. M. Maragò, “Sagnac interferometer method for synthesis of fractional polarization vortices,” Opt. Lett. 34, 2560–2562 (2009).
    [Crossref]
  17. Y. Wang and G. Gbur, “Hilbert’s hotel in polarization singularities,” Opt. Lett. 42, 5154–5157 (2017).
    [Crossref]
  18. X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
    [Crossref]
  19. J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
    [Crossref]

2018 (1)

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

2017 (1)

2016 (3)

E. J. Galvez and B. Khajavi, “High-order disclinations in the polarization of light,” Proc. SPIE 9764, 97640R (2016).
[Crossref]

B. Khajavi and E. J. Galvez, “High-order disclinations in space-variant polarization,” J. Opt. 18, 084003 (2016).
[Crossref]

G. Gbur, “Fractional vortex Hilbert’s hotel,” Optica 3, 222–225 (2016).
[Crossref]

2012 (1)

2010 (1)

2009 (1)

2007 (1)

2006 (1)

2004 (2)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
[Crossref]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

2002 (1)

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

1997 (1)

J. F. Nye, “Line singularities in wave fields,” Philos. Trans. R. Soc. Lond. A 355, 2065–2069 (1997).
[Crossref]

1983 (1)

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. Lond. A 389, 279–290 (1983).
[Crossref]

1977 (1)

M. V. Berry and J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809–1821 (1977).
[Crossref]

1976 (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Alpmann, C.

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

Berry, M. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
[Crossref]

M. V. Berry and J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809–1821 (1977).
[Crossref]

Biener, G.

Chen, J.

Denz, C.

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

Ding, J. P.

Feit, M. D.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Fleck, J. A.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Freund, I.

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

Galvez, E. J.

E. J. Galvez and B. Khajavi, “High-order disclinations in the polarization of light,” Proc. SPIE 9764, 97640R (2016).
[Crossref]

B. Khajavi and E. J. Galvez, “High-order disclinations in space-variant polarization,” J. Opt. 18, 084003 (2016).
[Crossref]

E. J. Galvez, S. Khadka, W. H. Schubert, and S. Nomoto, “Poincaré-beam patterns produced by nonseparable superpositions of Laguerre-Gauss and polarization modes of light,” Appl. Opt. 51, 2925–2934 (2012).
[Crossref]

Gbur, G.

Gbur, G. J.

G. J. Gbur, Singular Optics (CRC Press, 2016).

Goldstein, D. H.

D. H. Goldstein, Polarized Light, Revised and Expanded (CRC Press, 2003).

Guo, C. S.

Hannay, J. H.

M. V. Berry and J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809–1821 (1977).
[Crossref]

Hasman, E.

Jones, P. H.

Khadka, S.

Khajavi, B.

E. J. Galvez and B. Khajavi, “High-order disclinations in the polarization of light,” Proc. SPIE 9764, 97640R (2016).
[Crossref]

B. Khajavi and E. J. Galvez, “High-order disclinations in space-variant polarization,” J. Opt. 18, 084003 (2016).
[Crossref]

Kleiner, V.

Leach, J.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

Li, Y. N.

Makita, M.

Maragò, O. M.

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Ni, W. J.

Niv, A.

Nomoto, S.

Nye, J. F.

J. F. Nye, “Line singularities in wave fields,” Philos. Trans. R. Soc. Lond. A 355, 2065–2069 (1997).
[Crossref]

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. Lond. A 389, 279–290 (1983).
[Crossref]

Otte, E.

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

Padgett, M. J.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

Rashid, M.

Schubert, W. H.

Wang, H. T.

Wang, X. L.

Wang, Y.

Yao, E.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

Appl. Opt. (1)

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

J. Opt. (1)

B. Khajavi and E. J. Galvez, “High-order disclinations in space-variant polarization,” J. Opt. 18, 084003 (2016).
[Crossref]

J. Opt. A (1)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
[Crossref]

J. Phys. A (1)

M. V. Berry and J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809–1821 (1977).
[Crossref]

Laser Photon. Rev. (1)

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

New J. Phys. (1)

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

Opt. Commun. (1)

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

Opt. Express (2)

Opt. Lett. (3)

Optica (1)

Philos. Trans. R. Soc. Lond. A (1)

J. F. Nye, “Line singularities in wave fields,” Philos. Trans. R. Soc. Lond. A 355, 2065–2069 (1997).
[Crossref]

Proc. R. Soc. Lond. A (1)

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. Lond. A 389, 279–290 (1983).
[Crossref]

Proc. SPIE (1)

E. J. Galvez and B. Khajavi, “High-order disclinations in the polarization of light,” Proc. SPIE 9764, 97640R (2016).
[Crossref]

Other (2)

G. J. Gbur, Singular Optics (CRC Press, 2016).

D. H. Goldstein, Polarized Light, Revised and Expanded (CRC Press, 2003).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1.
Fig. 1. Distributions of polarization states of four polarization singularities. (a) Lemon C-point with m=0.5 and τ=r, (b) star C-point with m=0.5 and τ=r, (c) lemon V-point with m=0.5 and τ=1, and (d) star V-point with m=0.5 and τ=1. (e)–(h) Corresponding distributions of the orientation angles of the major axes of polarization states to (a)–(d), respectively. (i)–(l) Corresponding distributions of the angles of ellipticity of the polarization states to (a)–(d), respectively.
Fig. 2.
Fig. 2. Schematic of experimental setup. SLM, spatial light modulator; L1, plano-concave lens; L2, plano-convex lens; L3 and L4, bi-convex lenses; SF, spatial filter; QWP, quarter-wave plate; G, Ronchi grating; CCD, charge coupled device.
Fig. 3.
Fig. 3. Measured intensity distributions around a lemon V-point (m=0.5) at different propagation distances as z=0.1, 0.4, 0.7, and 1.0 m.
Fig. 4.
Fig. 4. Distributions of the intensity and polarization states around a V-point at a propagation distance z=0.5  m. First, second, and third rows show the intensity, orientation angles of the major axes of polarization states, and angle of ellipticity, respectively. First (third) column shows the simulated results with m=0.5 (2.5), and second (fourth) column shows the corresponding measured results.
Fig. 5.
Fig. 5. Distributions of the intensity and polarization states around a C-point at a propagation distance z=0.5  m. First, second, and third rows show the intensity, orientation angles of the major axes of polarization states, and angle of ellipticity, respectively. First (third) column shows the simulated results with m=0.5 (2.5), and second (fourth) column shows the corresponding measured results.
Fig. 6.
Fig. 6. Any polarization state can be seen as a superposition of right- and left-handed circularly polarized components carrying the phases. (a) Polarization states at all points surrounding the V-point are linearly polarized but different in orientation, which are located on the equator of the Poincaré sphere. (b) Polarization states at all points surrounding the C-point are elliptically polarized but different in orientation, which are located on a certain latitude in the northern (or southern) hemisphere of the Poincaré sphere.
Fig. 7.
Fig. 7. Comparison of the polarization topological index T and algebraic sum of the singularities of vector optical fields with different topological charge m.
Fig. 8.
Fig. 8. Distributions of the orientation angles 2ϕ of the major axes of the polarization states for different singular lines. (a) Single straight singular line case, (b) triple straight singular case (n=3 and η=0), and (c) triple spiral singular case (n=3 and η=r).
Fig. 9.
Fig. 9. Illustration of the experimentally measured intensity and polarization distribution of the ternary case [(a)–(c)] and the ternary spiral case [(d)–(f)] at a propagation distance z=0.5  m. (a), (d) Intensity; (b), (e) orientation angles of the major axes of the polarization states; and (c), (f) angle of ellipticity.

Equations (2)

Equations on this page are rendered with MathJax. Learn more.

E(r,φ)=2A01+τ2[τexp(jmφ)e^R+exp(+jmφ)e^L],
H(x,y)=12+241+τ2[τcos(2πf0xδ)+cos(2πf0y+δ)],