Abstract

We present a generalized Poincaré sphere (G sphere) and generalized Stokes parameters (G parameters), as a geometric representation, which unifies the descriptors of a variety of vector fields. Unlike the standard Poincaré sphere, the radial dimension in the G sphere is not used to describe the partially polarized field. The G sphere is constructed by extending the basic Jones vector bases to the general vector bases with the continuously changeable ellipticity (spin angular momentum, SAM) and the higher dimensional orbital angular momentum (OAM). The north and south poles of different spherical shells in the G sphere represent the pair of different orthogonal vector basis with different ellipticity (SAM) and the opposite OAM. The higher-order Poincaré spheres are just the two special spherical shells of the G sphere. We present a quite flexible scheme, which can generate all the vector fields described in the G sphere.

© 2015 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
    [Crossref]
  2. L. A. Boyle, P. J. Steinhardt, and N. Turok, “Inflationary predictions for scalar and tensor fluctuations reconsidered,” Phys. Rev. Lett. 96, 111301 (2006).
    [Crossref] [PubMed]
  3. S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).
  4. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. A 392, 45–57 (1984).
    [Crossref]
  5. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1, 1–57 (2009).
    [Crossref]
  6. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [Crossref] [PubMed]
  7. W. D. Kimura, G. H. Kim, R. D. Romea, L. C. Steinhauer, I. V. Pogorelsky, K. P. Kusche, R. C. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546–549 (1995).
    [Crossref] [PubMed]
  8. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
    [Crossref] [PubMed]
  9. A. Ciattoni, B. Crosignani, P. Di Porto, and A. Yariv, ‘Azimuthally polarized spatial dark solitons: exact solutions of Maxwell’s equations in a Kerr medium,” Phys. Rev. Lett. 94, 073902 (2005).
    [Crossref]
  10. A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
    [Crossref] [PubMed]
  11. X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
    [Crossref]
  12. F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
    [Crossref]
  13. G. Milione, T. A. Nguyen, D. A. Nolan, E. Karimi, S. Slussarenko, L. Marrucci, and R. R. Alfano, “Superdense coding with vector vortex beams: a classical analogy of entanglement,” Frountiers in Optics FM3F (2013).
  14. 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]
  15. A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Poincaré sphere representation for classical inseparable Bell-like states of the electromagnetic field,” arXiv: 1007.2528 (2010).
  16. A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Classical and quantum properties of cylindrically polarized states of light,” Opt. Express 19, 9714–9736 (2011).
    [Crossref] [PubMed]
  17. G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
    [Crossref]
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    [Crossref] [PubMed]
  19. 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] [PubMed]
  20. G. Milione, H. I. Sztul, and R. R. Alfano, “Stokes polarimetry of a hybrid vector beam from a spun elliptical core optical fiber,” Proc. SPIE 7613, 761305 (2010).
    [Crossref]
  21. S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep. 2, 1007 (2012).
    [PubMed]
  22. M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24, 430–432 (1999).
    [Crossref]
  23. E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
    [Crossref] [PubMed]
  24. S. M. Li, S. X. Qian, L. J. Kong, Z. C. Ren, Y. N. Li, C. H. Tu, and H. T. Wang, “An efficient and robust scheme for controlling the states of polarization in a Sagnac interferometric configuration,” Europhys. Lett. 105, 64006 (2014).
    [Crossref]

2015 (1)

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]

2014 (2)

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

S. M. Li, S. X. Qian, L. J. Kong, Z. C. Ren, Y. N. Li, C. H. Tu, and H. T. Wang, “An efficient and robust scheme for controlling the states of polarization in a Sagnac interferometric configuration,” Europhys. Lett. 105, 64006 (2014).
[Crossref]

2012 (2)

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref] [PubMed]

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep. 2, 1007 (2012).
[PubMed]

2011 (2)

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

A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Classical and quantum properties of cylindrically polarized states of light,” Opt. Express 19, 9714–9736 (2011).
[Crossref] [PubMed]

2010 (3)

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] [PubMed]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

G. Milione, H. I. Sztul, and R. R. Alfano, “Stokes polarimetry of a hybrid vector beam from a spun elliptical core optical fiber,” Proc. SPIE 7613, 761305 (2010).
[Crossref]

2009 (1)

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

2006 (1)

L. A. Boyle, P. J. Steinhardt, and N. Turok, “Inflationary predictions for scalar and tensor fluctuations reconsidered,” Phys. Rev. Lett. 96, 111301 (2006).
[Crossref] [PubMed]

2005 (1)

A. Ciattoni, B. Crosignani, P. Di Porto, and A. Yariv, ‘Azimuthally polarized spatial dark solitons: exact solutions of Maxwell’s equations in a Kerr medium,” Phys. Rev. Lett. 94, 073902 (2005).
[Crossref]

2003 (3)

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref] [PubMed]

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

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref] [PubMed]

2001 (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

1999 (1)

1995 (1)

W. D. Kimura, G. H. Kim, R. D. Romea, L. C. Steinhauer, I. V. Pogorelsky, K. P. Kusche, R. C. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546–549 (1995).
[Crossref] [PubMed]

1984 (1)

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. A 392, 45–57 (1984).
[Crossref]

1956 (1)

S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).

Aiello, A.

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]

A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Classical and quantum properties of cylindrically polarized states of light,” Opt. Express 19, 9714–9736 (2011).
[Crossref] [PubMed]

A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Poincaré sphere representation for classical inseparable Bell-like states of the electromagnetic field,” arXiv: 1007.2528 (2010).

Alfano, R. R.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref] [PubMed]

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

G. Milione, H. I. Sztul, and R. R. Alfano, “Stokes polarimetry of a hybrid vector beam from a spun elliptical core optical fiber,” Proc. SPIE 7613, 761305 (2010).
[Crossref]

G. Milione, T. A. Nguyen, D. A. Nolan, E. Karimi, S. Slussarenko, L. Marrucci, and R. R. Alfano, “Superdense coding with vector vortex beams: a classical analogy of entanglement,” Frountiers in Optics FM3F (2013).

Berry, M. V.

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. A 392, 45–57 (1984).
[Crossref]

Beversluis, M.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref] [PubMed]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
[Crossref]

Bouhelier, A.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref] [PubMed]

Boyle, L. A.

L. A. Boyle, P. J. Steinhardt, and N. Turok, “Inflationary predictions for scalar and tensor fluctuations reconsidered,” Phys. Rev. Lett. 96, 111301 (2006).
[Crossref] [PubMed]

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

Chen, J.

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

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] [PubMed]

Ciattoni, A.

A. Ciattoni, B. Crosignani, P. Di Porto, and A. Yariv, ‘Azimuthally polarized spatial dark solitons: exact solutions of Maxwell’s equations in a Kerr medium,” Phys. Rev. Lett. 94, 073902 (2005).
[Crossref]

Courtial, J.

Crawford, P. R.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref] [PubMed]

Crosignani, B.

A. Ciattoni, B. Crosignani, P. Di Porto, and A. Yariv, ‘Azimuthally polarized spatial dark solitons: exact solutions of Maxwell’s equations in a Kerr medium,” Phys. Rev. Lett. 94, 073902 (2005).
[Crossref]

Di Porto, P.

A. Ciattoni, B. Crosignani, P. Di Porto, and A. Yariv, ‘Azimuthally polarized spatial dark solitons: exact solutions of Maxwell’s equations in a Kerr medium,” Phys. Rev. Lett. 94, 073902 (2005).
[Crossref]

Ding, J. P.

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] [PubMed]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Dorn, R.

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

Evans, S.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref] [PubMed]

Fernow, R. C.

W. D. Kimura, G. H. Kim, R. D. Romea, L. C. Steinhauer, I. V. Pogorelsky, K. P. Kusche, R. C. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546–549 (1995).
[Crossref] [PubMed]

Gabriel, C.

A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Classical and quantum properties of cylindrically polarized states of light,” Opt. Express 19, 9714–9736 (2011).
[Crossref] [PubMed]

A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Poincaré sphere representation for classical inseparable Bell-like states of the electromagnetic field,” arXiv: 1007.2528 (2010).

Galvez, E. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref] [PubMed]

Giacobino, E.

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]

Guo, C. S.

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

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] [PubMed]

Haglin, P. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref] [PubMed]

Hartschuh, A.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref] [PubMed]

Holleczek, A.

A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Classical and quantum properties of cylindrically polarized states of light,” Opt. Express 19, 9714–9736 (2011).
[Crossref] [PubMed]

A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Poincaré sphere representation for classical inseparable Bell-like states of the electromagnetic field,” arXiv: 1007.2528 (2010).

Karimi, E.

G. Milione, T. A. Nguyen, D. A. Nolan, E. Karimi, S. Slussarenko, L. Marrucci, and R. R. Alfano, “Superdense coding with vector vortex beams: a classical analogy of entanglement,” Frountiers in Optics FM3F (2013).

Kim, G. H.

W. D. Kimura, G. H. Kim, R. D. Romea, L. C. Steinhauer, I. V. Pogorelsky, K. P. Kusche, R. C. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546–549 (1995).
[Crossref] [PubMed]

Kimura, W. D.

W. D. Kimura, G. H. Kim, R. D. Romea, L. C. Steinhauer, I. V. Pogorelsky, K. P. Kusche, R. C. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546–549 (1995).
[Crossref] [PubMed]

Kong, L. J.

S. M. Li, S. X. Qian, L. J. Kong, Z. C. Ren, Y. N. Li, C. H. Tu, and H. T. Wang, “An efficient and robust scheme for controlling the states of polarization in a Sagnac interferometric configuration,” Europhys. Lett. 105, 64006 (2014).
[Crossref]

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep. 2, 1007 (2012).
[PubMed]

Kusche, K. P.

W. D. Kimura, G. H. Kim, R. D. Romea, L. C. Steinhauer, I. V. Pogorelsky, K. P. Kusche, R. C. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546–549 (1995).
[Crossref] [PubMed]

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]

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

A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Classical and quantum properties of cylindrically polarized states of light,” Opt. Express 19, 9714–9736 (2011).
[Crossref] [PubMed]

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

A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Poincaré sphere representation for classical inseparable Bell-like states of the electromagnetic field,” arXiv: 1007.2528 (2010).

Li, S. M.

S. M. Li, S. X. Qian, L. J. Kong, Z. C. Ren, Y. N. Li, C. H. Tu, and H. T. Wang, “An efficient and robust scheme for controlling the states of polarization in a Sagnac interferometric configuration,” Europhys. Lett. 105, 64006 (2014).
[Crossref]

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep. 2, 1007 (2012).
[PubMed]

Li, Y. N.

S. M. Li, S. X. Qian, L. J. Kong, Z. C. Ren, Y. N. Li, C. H. Tu, and H. T. Wang, “An efficient and robust scheme for controlling the states of polarization in a Sagnac interferometric configuration,” Europhys. Lett. 105, 64006 (2014).
[Crossref]

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep. 2, 1007 (2012).
[PubMed]

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] [PubMed]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Liu, Y.

W. D. Kimura, G. H. Kim, R. D. Romea, L. C. Steinhauer, I. V. Pogorelsky, K. P. Kusche, R. C. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546–549 (1995).
[Crossref] [PubMed]

Lou, K.

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep. 2, 1007 (2012).
[PubMed]

Marquardt, C.

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]

A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Classical and quantum properties of cylindrically polarized states of light,” Opt. Express 19, 9714–9736 (2011).
[Crossref] [PubMed]

A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Poincaré sphere representation for classical inseparable Bell-like states of the electromagnetic field,” arXiv: 1007.2528 (2010).

Marrucci, L.

G. Milione, T. A. Nguyen, D. A. Nolan, E. Karimi, S. Slussarenko, L. Marrucci, and R. R. Alfano, “Superdense coding with vector vortex beams: a classical analogy of entanglement,” Frountiers in Optics FM3F (2013).

Milione, G.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref] [PubMed]

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

G. Milione, H. I. Sztul, and R. R. Alfano, “Stokes polarimetry of a hybrid vector beam from a spun elliptical core optical fiber,” Proc. SPIE 7613, 761305 (2010).
[Crossref]

G. Milione, T. A. Nguyen, D. A. Nolan, E. Karimi, S. Slussarenko, L. Marrucci, and R. R. Alfano, “Superdense coding with vector vortex beams: a classical analogy of entanglement,” Frountiers in Optics FM3F (2013).

Nguyen, T. A.

G. Milione, T. A. Nguyen, D. A. Nolan, E. Karimi, S. Slussarenko, L. Marrucci, and R. R. Alfano, “Superdense coding with vector vortex beams: a classical analogy of entanglement,” Frountiers in Optics FM3F (2013).

Nolan, D. A.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref] [PubMed]

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

G. Milione, T. A. Nguyen, D. A. Nolan, E. Karimi, S. Slussarenko, L. Marrucci, and R. R. Alfano, “Superdense coding with vector vortex beams: a classical analogy of entanglement,” Frountiers in Optics FM3F (2013).

Novotny, L.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

Padgett, M. J.

Pancharatnam, S.

S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).

Pogorelsky, I. V.

W. D. Kimura, G. H. Kim, R. D. Romea, L. C. Steinhauer, I. V. Pogorelsky, K. P. Kusche, R. C. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546–549 (1995).
[Crossref] [PubMed]

Pysher, M. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref] [PubMed]

Qian, S. X.

S. M. Li, S. X. Qian, L. J. Kong, Z. C. Ren, Y. N. Li, C. H. Tu, and H. T. Wang, “An efficient and robust scheme for controlling the states of polarization in a Sagnac interferometric configuration,” Europhys. Lett. 105, 64006 (2014).
[Crossref]

Quabis, S.

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

Ren, Z. C.

S. M. Li, S. X. Qian, L. J. Kong, Z. C. Ren, Y. N. Li, C. H. Tu, and H. T. Wang, “An efficient and robust scheme for controlling the states of polarization in a Sagnac interferometric configuration,” Europhys. Lett. 105, 64006 (2014).
[Crossref]

Romea, R. D.

W. D. Kimura, G. H. Kim, R. D. Romea, L. C. Steinhauer, I. V. Pogorelsky, K. P. Kusche, R. C. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546–549 (1995).
[Crossref] [PubMed]

Slussarenko, S.

G. Milione, T. A. Nguyen, D. A. Nolan, E. Karimi, S. Slussarenko, L. Marrucci, and R. R. Alfano, “Superdense coding with vector vortex beams: a classical analogy of entanglement,” Frountiers in Optics FM3F (2013).

Steinhardt, P. J.

L. A. Boyle, P. J. Steinhardt, and N. Turok, “Inflationary predictions for scalar and tensor fluctuations reconsidered,” Phys. Rev. Lett. 96, 111301 (2006).
[Crossref] [PubMed]

Steinhauer, L. C.

W. D. Kimura, G. H. Kim, R. D. Romea, L. C. Steinhauer, I. V. Pogorelsky, K. P. Kusche, R. C. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546–549 (1995).
[Crossref] [PubMed]

Sztul, H. I.

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

G. Milione, H. I. Sztul, and R. R. Alfano, “Stokes polarimetry of a hybrid vector beam from a spun elliptical core optical fiber,” Proc. SPIE 7613, 761305 (2010).
[Crossref]

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref] [PubMed]

Tian, Y. J.

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep. 2, 1007 (2012).
[PubMed]

Töppel, F.

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]

Tu, C. H.

S. M. Li, S. X. Qian, L. J. Kong, Z. C. Ren, Y. N. Li, C. H. Tu, and H. T. Wang, “An efficient and robust scheme for controlling the states of polarization in a Sagnac interferometric configuration,” Europhys. Lett. 105, 64006 (2014).
[Crossref]

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep. 2, 1007 (2012).
[PubMed]

Turok, N.

L. A. Boyle, P. J. Steinhardt, and N. Turok, “Inflationary predictions for scalar and tensor fluctuations reconsidered,” Phys. Rev. Lett. 96, 111301 (2006).
[Crossref] [PubMed]

Wang, H. T.

S. M. Li, S. X. Qian, L. J. Kong, Z. C. Ren, Y. N. Li, C. H. Tu, and H. T. Wang, “An efficient and robust scheme for controlling the states of polarization in a Sagnac interferometric configuration,” Europhys. Lett. 105, 64006 (2014).
[Crossref]

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep. 2, 1007 (2012).
[PubMed]

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] [PubMed]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Wang, X.

W. D. Kimura, G. H. Kim, R. D. Romea, L. C. Steinhauer, I. V. Pogorelsky, K. P. Kusche, R. C. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546–549 (1995).
[Crossref] [PubMed]

Wang, X. L.

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep. 2, 1007 (2012).
[PubMed]

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] [PubMed]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Williams, R. E.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
[Crossref]

Yariv, A.

A. Ciattoni, B. Crosignani, P. Di Porto, and A. Yariv, ‘Azimuthally polarized spatial dark solitons: exact solutions of Maxwell’s equations in a Kerr medium,” Phys. Rev. Lett. 94, 073902 (2005).
[Crossref]

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

Zhan, Q.

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

Adv. Opt. Photonics (1)

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

Europhys. Lett. (1)

S. M. Li, S. X. Qian, L. J. Kong, Z. C. Ren, Y. N. Li, C. H. Tu, and H. T. Wang, “An efficient and robust scheme for controlling the states of polarization in a Sagnac interferometric configuration,” Europhys. Lett. 105, 64006 (2014).
[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. (1)

Phys. Rev. Lett. (10)

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref] [PubMed]

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

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref] [PubMed]

L. A. Boyle, P. J. Steinhardt, and N. Turok, “Inflationary predictions for scalar and tensor fluctuations reconsidered,” Phys. Rev. Lett. 96, 111301 (2006).
[Crossref] [PubMed]

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

W. D. Kimura, G. H. Kim, R. D. Romea, L. C. Steinhauer, I. V. Pogorelsky, K. P. Kusche, R. C. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74, 546–549 (1995).
[Crossref] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

A. Ciattoni, B. Crosignani, P. Di Porto, and A. Yariv, ‘Azimuthally polarized spatial dark solitons: exact solutions of Maxwell’s equations in a Kerr medium,” Phys. Rev. Lett. 94, 073902 (2005).
[Crossref]

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref] [PubMed]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Proc. Indian Acad. Sci. A (1)

S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).

Proc. R. Soc. A (1)

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. A 392, 45–57 (1984).
[Crossref]

Proc. SPIE (1)

G. Milione, H. I. Sztul, and R. R. Alfano, “Stokes polarimetry of a hybrid vector beam from a spun elliptical core optical fiber,” Proc. SPIE 7613, 761305 (2010).
[Crossref]

Sci. Rep. (1)

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep. 2, 1007 (2012).
[PubMed]

Other (3)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
[Crossref]

A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Poincaré sphere representation for classical inseparable Bell-like states of the electromagnetic field,” arXiv: 1007.2528 (2010).

G. Milione, T. A. Nguyen, D. A. Nolan, E. Karimi, S. Slussarenko, L. Marrucci, and R. R. Alfano, “Superdense coding with vector vortex beams: a classical analogy of entanglement,” Frountiers in Optics FM3F (2013).

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Figures (3)

Fig. 1
Fig. 1

G sphere for m = 1. The antipodal points in the sphere’s axis G 3 R m stand for continuously varying basis. The green shell (R = 0.5) and the yellow shell (R = 1) are equivalent to the H spheres with m = 1 and m = −1, respectively. The dark red shell (R = 0.75) represents the vector fields with the orthogonal linearly polarized basis carrying the opposite OAMs and its equator represents the hybridly polarized vector fields.

Fig. 2
Fig. 2

Scheme for generating all the vector fields in the G sphere. P is a fixed polarizer. PBS is a polarization beam splitter; SVPP is a spiral vortex phase plate to generate vortex field. HWPs and QWPs are rotatable λ/2 and λ/4 wave plates, respectively. L11, L12, L21, and L22 are four lenses with different focal lengths. HWP2, QWP1, and QWP2 constitute a geometric phase adjuster to control the relative phase. The common-path two orthogonally polarized beams outputted from this Sagnac interferometer carry the opposite topological phases, and then combine to generate the vector fields which can be represented by the G sphere. HWP1 and QWP3 control the three spherical coordinates in the G sphere, respectively.

Fig. 3
Fig. 3

Experimentally generated vector fields described in the G sphere. First row shows the distributions of polarization states, which correspond to the spherical coordinates on the G sphere (Fig. 1) as (R, 2θ, 2φ) = (0.5, 0, 0), (0.5, π/4, 0), (0.75, 0, 0), (1, 0, 0), (0.5, π/2, 0), respectively. Second row shows the intensity distributions of the corresponding vector fields generated. Third and fourth rows indicate the intensity patterns behind a horizontal and vertical polarizers, respectively.

Equations (26)

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

| A = a R | R + a L | L , with | a R | 2 + | a L | 2 = 1 .
[ S 1 S 2 S 3 ] = [ A | σ 1 | A A | σ 2 | A A | σ 3 | A ] = [ 2 Re ( a R * a L ) 2 Im ( a R * a L ) | a R | 2 | a L | 2 ] ,
[ S 1 S 2 S 3 ] = [ cos 2 θ cos 2 φ cos 2 θ sin 2 φ sin 2 θ ] .
| N R m = 1 2 e j m ϕ ( e j R π e ^ x j e j R π e ^ y ) ,
| S R m = 1 2 e + j m ϕ ( e j R π e ^ x + j e j R π e ^ y ) .
| ψ m = ψ N m | N R m + ψ S m | S R m ,
ψ N m = sin β e j ϕ 0 ,
ψ S m = cos β e + j ϕ 0 ,
E S 0 R m = | N R m | ψ m | 2 + | S R m | ψ m | 2 = 1 ,
E S 1 R m = 2 Re ( N R m | ψ m * S R m | ψ m ) = sin 2 β cos 2 ϕ 0 ,
E S 2 R m = 2 Im ( N R m | ψ m * S R m | ψ m ) = sin 2 β sin 2 ϕ 0 ,
E S 3 R m = | N R m | ψ m | 2 | S R m | ψ m | 2 = cos 2 β .
G 0 R m = E S 0 R m R = R ,
G 1 R m = E S 1 R m R = R sin 2 β cos 2 ϕ 0 ,
G 2 R m = E S 2 R m R = R sin 2 β sin 2 ϕ 0 ,
G 3 R m = E S 3 R m R = R cos 2 β .
R = G 0 R m ,
sin 2 θ = G 3 R m / G 0 R m ,
tan 2 φ = G 2 R m / G 1 R m .
J 1 = [ e j m ϕ 0 0 e j m ϕ ] .
J 2 = [ e j ( 2 θ 2 + π / 2 ) 0 0 e j ( 2 θ 2 + π / 2 ) ] .
J 3 = 1 2 [ 1 j cos ( 2 θ 3 ) j sin ( 2 θ 3 ) j sin ( 2 θ 3 ) 1 + j cos ( 2 θ 3 ) ] .
P ^ in = cos ( 2 α ) e ^ x + sin ( 2 α ) e ^ y ,
P ^ out = J 3 J 2 J 1 P ^ in = sin ( 2 α ) exp [ + j ( 2 θ 2 + π / 2 ) ] e ^ N + cos ( 2 α ) exp [ j ( 2 θ 2 + π / 2 ) ] e ^ S ,
e ^ N = 1 2 e j m ϕ { [ 1 + j cos ( 2 θ 3 ) ] e ^ y j sin ( 2 θ 3 ) e ^ x } ,
e ^ S = 1 2 e + j m ϕ { [ 1 j cos ( 2 θ 3 ) ] e ^ x j sin ( 2 θ 3 ) e ^ y } .

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