Abstract

We present a convenient method to generate vector beams of light having polarization singularities on their axis, via partial spin-to-orbital angular momentum conversion in a suitably patterned liquid crystal cell. The resulting polarization patterns exhibit a C-point on the beam axis and an L-line loop around it, and may have different geometrical structures such as “lemon”, “star”, and “spiral”. Our generation method allows us to control the radius of L-line loop around the central C-point. Moreover, we investigate the free-air propagation of these fields across a Rayleigh range.

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  1. J. F. Nye and M. V. Berry, “Dislocations in Wave Trains,” Proc. R. Soc. Lond. A336, 165–190 (1974).
    [CrossRef]
  2. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Optical Vortices and Polarization Singularities,” Prog. Opt.53, 293–363 (2009).
    [CrossRef]
  3. M. S. Soskin and M. V. Vasnetsov, (Ed. E. Wolf), “Singular optics,” Prog. Opt.42, 219–276 (2001).
    [CrossRef]
  4. S. Franke-Arnold, L. Allen, and M. J. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev.2, 299–313 (2008).
    [CrossRef]
  5. J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. Roy. Soc. Lond. A387, 105–132 (1983).
    [CrossRef]
  6. J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. Roy. Soc. Lond. A389, 279–290 (1983).
    [CrossRef]
  7. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun.213, 201–221 (2002).
    [CrossRef]
  8. I. Freund, “Polarization flowers,” Opt. Commun.199, 47–63 (2001).
    [CrossRef]
  9. O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun.207, 57–65 (2002).
    [CrossRef]
  10. M. V. Berry, M. R. Dennis, and R. L. Lee, “Polarization singularities in the clear sky,” New J. Phys.6, 162 (2004).
    [CrossRef]
  11. M. S. Soskin, V. Denisenko, and I. Freund, “Optical polarization singularities and elliptic stationary points,” Opt. Lett.28, 1475–1477 (2003).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  13. 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).
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    [CrossRef] [PubMed]
  16. L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media,” Phys. Rev. Lett.96, 163905 (2006).
    [CrossRef] [PubMed]
  17. F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt.51, C1–C6 (2012).
    [CrossRef] [PubMed]
  18. L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt.13, 064001 (2011).
    [CrossRef]
  19. E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett.94, 231124 (2009).
    [CrossRef]
  20. S. Slussarenko, A. Murauski, T. Du, V. Chigrinov, L. Marrucci, and E. Santamato, “Tunable liquid crystal q-plates with arbitrary topological charge,” Opt. Express19, 4085–4090 (2011).
    [CrossRef] [PubMed]
  21. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett.32, 3053–3055 (2007).
    [CrossRef] [PubMed]
  22. E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Light propagation in a birefringent plate with topological charge,” Opt. Lett.34, 1225–1227 (2009).
    [CrossRef] [PubMed]
  23. C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing Local Field Structure of Focused Ultrashort Pulses,” Phys. Rev. Lett.106, 123901 (2011).
    [CrossRef] [PubMed]
  24. A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat. Commun.3, 989 (2012).
    [CrossRef] [PubMed]

2012 (5)

2011 (3)

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing Local Field Structure of Focused Ultrashort Pulses,” Phys. Rev. Lett.106, 123901 (2011).
[CrossRef] [PubMed]

S. Slussarenko, A. Murauski, T. Du, V. Chigrinov, L. Marrucci, and E. Santamato, “Tunable liquid crystal q-plates with arbitrary topological charge,” Opt. Express19, 4085–4090 (2011).
[CrossRef] [PubMed]

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt.13, 064001 (2011).
[CrossRef]

2010 (1)

2009 (3)

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Light propagation in a birefringent plate with topological charge,” Opt. Lett.34, 1225–1227 (2009).
[CrossRef] [PubMed]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett.94, 231124 (2009).
[CrossRef]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Optical Vortices and Polarization Singularities,” Prog. Opt.53, 293–363 (2009).
[CrossRef]

2008 (1)

S. Franke-Arnold, L. Allen, and M. J. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev.2, 299–313 (2008).
[CrossRef]

2007 (1)

2006 (1)

L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media,” Phys. Rev. Lett.96, 163905 (2006).
[CrossRef] [PubMed]

2004 (1)

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

2003 (1)

2002 (2)

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun.213, 201–221 (2002).
[CrossRef]

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun.207, 57–65 (2002).
[CrossRef]

2001 (1)

I. Freund, “Polarization flowers,” Opt. Commun.199, 47–63 (2001).
[CrossRef]

1983 (2)

J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. Roy. Soc. Lond. A387, 105–132 (1983).
[CrossRef]

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. Roy. Soc. Lond. A389, 279–290 (1983).
[CrossRef]

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in Wave Trains,” Proc. R. Soc. Lond. A336, 165–190 (1974).
[CrossRef]

Alexeyev, C. N.

Allen, L.

S. Franke-Arnold, L. Allen, and M. J. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev.2, 299–313 (2008).
[CrossRef]

Alonso, M. A.

Ambrosio, A.

A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat. Commun.3, 989 (2012).
[CrossRef] [PubMed]

Angelsky, O.

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun.207, 57–65 (2002).
[CrossRef]

Anischenko, P. M.

Beckley, A. M.

Berry, M. V.

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

J. F. Nye and M. V. Berry, “Dislocations in Wave Trains,” Proc. R. Soc. Lond. A336, 165–190 (1974).
[CrossRef]

Borbone, F.

A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat. Commun.3, 989 (2012).
[CrossRef] [PubMed]

Borwinska, M.

Brown, T. G.

Cardano, F.

Chigrinov, V.

de Lisio, C.

Denisenko, V.

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Optical Vortices and Polarization Singularities,” Prog. Opt.53, 293–363 (2009).
[CrossRef]

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

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun.213, 201–221 (2002).
[CrossRef]

Du, T.

Fadeyeva, T. A.

Franke-Arnold, S.

S. Franke-Arnold, L. Allen, and M. J. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev.2, 299–313 (2008).
[CrossRef]

Freund, I.

Galvez, E. J.

Hnatovsky, C.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing Local Field Structure of Focused Ultrashort Pulses,” Phys. Rev. Lett.106, 123901 (2011).
[CrossRef] [PubMed]

Karimi, E.

F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt.51, C1–C6 (2012).
[CrossRef] [PubMed]

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt.13, 064001 (2011).
[CrossRef]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett.94, 231124 (2009).
[CrossRef]

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Light propagation in a birefringent plate with topological charge,” Opt. Lett.34, 1225–1227 (2009).
[CrossRef] [PubMed]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett.32, 3053–3055 (2007).
[CrossRef] [PubMed]

Khadka, S.

Krolikowski, W.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing Local Field Structure of Focused Ultrashort Pulses,” Phys. Rev. Lett.106, 123901 (2011).
[CrossRef] [PubMed]

Kurzynowski, P.

Lee, R. L.

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

Maddalena, P.

A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat. Commun.3, 989 (2012).
[CrossRef] [PubMed]

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media,” Phys. Rev. Lett.96, 163905 (2006).
[CrossRef] [PubMed]

Marrucci, L.

F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt.51, C1–C6 (2012).
[CrossRef] [PubMed]

A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat. Commun.3, 989 (2012).
[CrossRef] [PubMed]

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt.13, 064001 (2011).
[CrossRef]

S. Slussarenko, A. Murauski, T. Du, V. Chigrinov, L. Marrucci, and E. Santamato, “Tunable liquid crystal q-plates with arbitrary topological charge,” Opt. Express19, 4085–4090 (2011).
[CrossRef] [PubMed]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett.94, 231124 (2009).
[CrossRef]

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Light propagation in a birefringent plate with topological charge,” Opt. Lett.34, 1225–1227 (2009).
[CrossRef] [PubMed]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett.32, 3053–3055 (2007).
[CrossRef] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media,” Phys. Rev. Lett.96, 163905 (2006).
[CrossRef] [PubMed]

Mokhun, A.

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun.207, 57–65 (2002).
[CrossRef]

Mokhun, I.

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun.207, 57–65 (2002).
[CrossRef]

Murauski, A.

Nagali, E.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt.13, 064001 (2011).
[CrossRef]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett.94, 231124 (2009).
[CrossRef]

Nomoto, S.

Nye, J. F.

J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. Roy. Soc. Lond. A387, 105–132 (1983).
[CrossRef]

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. Roy. Soc. Lond. A389, 279–290 (1983).
[CrossRef]

J. F. Nye and M. V. Berry, “Dislocations in Wave Trains,” Proc. R. Soc. Lond. A336, 165–190 (1974).
[CrossRef]

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Optical Vortices and Polarization Singularities,” Prog. Opt.53, 293–363 (2009).
[CrossRef]

Padgett, M. J.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Optical Vortices and Polarization Singularities,” Prog. Opt.53, 293–363 (2009).
[CrossRef]

S. Franke-Arnold, L. Allen, and M. J. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev.2, 299–313 (2008).
[CrossRef]

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media,” Phys. Rev. Lett.96, 163905 (2006).
[CrossRef] [PubMed]

Piccirillo, B.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt.13, 064001 (2011).
[CrossRef]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett.94, 231124 (2009).
[CrossRef]

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Light propagation in a birefringent plate with topological charge,” Opt. Lett.34, 1225–1227 (2009).
[CrossRef] [PubMed]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett.32, 3053–3055 (2007).
[CrossRef] [PubMed]

Rode, A.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing Local Field Structure of Focused Ultrashort Pulses,” Phys. Rev. Lett.106, 123901 (2011).
[CrossRef] [PubMed]

Roviello, A.

A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat. Commun.3, 989 (2012).
[CrossRef] [PubMed]

Santamato, E.

Schubert, W. H.

Sciarrino, F.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt.13, 064001 (2011).
[CrossRef]

Shvedov, V.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing Local Field Structure of Focused Ultrashort Pulses,” Phys. Rev. Lett.106, 123901 (2011).
[CrossRef] [PubMed]

Slussarenko, S.

Soskin, M.

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun.207, 57–65 (2002).
[CrossRef]

Soskin, M. S.

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, (Ed. E. Wolf), “Singular optics,” Prog. Opt.42, 219–276 (2001).
[CrossRef]

Volyar, A. V.

WoŸniak, W. A.

Zdunek, M.

Zito, G.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett.94, 231124 (2009).
[CrossRef]

J. Opt. (1)

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt.13, 064001 (2011).
[CrossRef]

Laser Photonics Rev. (1)

S. Franke-Arnold, L. Allen, and M. J. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev.2, 299–313 (2008).
[CrossRef]

Nat. Commun. (1)

A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat. Commun.3, 989 (2012).
[CrossRef] [PubMed]

New J. Phys. (1)

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

Opt. Commun. (3)

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun.213, 201–221 (2002).
[CrossRef]

I. Freund, “Polarization flowers,” Opt. Commun.199, 47–63 (2001).
[CrossRef]

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun.207, 57–65 (2002).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. Lett. (2)

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing Local Field Structure of Focused Ultrashort Pulses,” Phys. Rev. Lett.106, 123901 (2011).
[CrossRef] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media,” Phys. Rev. Lett.96, 163905 (2006).
[CrossRef] [PubMed]

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

J. F. Nye and M. V. Berry, “Dislocations in Wave Trains,” Proc. R. Soc. Lond. A336, 165–190 (1974).
[CrossRef]

Proc. Roy. Soc. Lond. A (2)

J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. Roy. Soc. Lond. A387, 105–132 (1983).
[CrossRef]

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. Roy. Soc. Lond. A389, 279–290 (1983).
[CrossRef]

Prog. Opt. (1)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Optical Vortices and Polarization Singularities,” Prog. Opt.53, 293–363 (2009).
[CrossRef]

Other (1)

M. S. Soskin and M. V. Vasnetsov, (Ed. E. Wolf), “Singular optics,” Prog. Opt.42, 219–276 (2001).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Polarization patterns of three different polarization singular beams: “star” (index η = −1/2), “lemon” (η = 1/2) and “spiral” (η = 1). The central point at the origin indicates the C-point, while dashed blue lines show the L-line circles with radius ρ0. The handednesses of the polarization ellipses in the outer region, ρ > ρ0, and in the inner one, ρ < ρ0, are opposite.

Fig. 2
Fig. 2

Experimental setup. A He-Ne laser beam is first spatial-mode filtered by focusing via a microscope objective (4X) into a 50 μm pinhole and recollimating by a lens (f). Its polarization is then prepared in a left (right) circular state by a polarizing beam-splitter (PBS) and a quarter-wave plate (Q). An untuned q-plate then transforms the beam into a PSB. The PSB polarization pattern was analyzed by a quarter-wave plate, a half-wave-plate (H) and another polarizer, followed by imaging on a CCD camera. The δ parameter of the PSB was adjusted by electrically tuning the q-plate. In order to study the propagation dynamics of the PSB, the CCD camera was mounted on a translation stage and moved around the focal plane of an output lens. Upper insets show the intensity and reconstructed polarization patterns, in the near field, of the lemon, star, and spiral beams generated by q-plates with q = 1/2, −1/2, 1, respectively.

Fig. 3
Fig. 3

Intensity distribution and reconstructed polarization patterns of beam generated by a q-plate with q = 1/2 for seven different optical retardations: (a) δ = 0 (or 2π), (b) δ = π/4, (c) δ = π/2, (d) δ = 3π/4, (e) δ = π, (f) δ = 5π/4, and (g) δ = 3π/2. The corresponding L-line radii relative to the beam waist w0 are the following: (a) undefined (b) ρ0 = 2.0w0, (c) ρ0 = 1.4w0, (d) ρ0 = 1.1w0, (e) undefined, (f) ρ0 = 1.0w0, and (g) ρ0 = 1.5w0.

Fig. 4
Fig. 4

Reconstructed experimental polarization patterns of different PSB beams. (a) η = m/2 = +1/2, (b) η = m/2 = −1/2 and (c) η = m/2 = +1. Patterns have been reconstructed by measuring the maps of reduced Stokes parameters in six different longitudinal planes within the beam Rayleigh range, from −zR to +zR. The corresponding rotation of polarization patterns for (a) and (b) are 90° and 88° = (30 + 30 + 28) °, respectively.

Equations (2)

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

| PSB = cos ( δ 2 ) LG 0 , 0 ( ρ , ϕ , z ) | L + e i α sin ( δ 2 ) LG 0 , m ( ρ , ϕ , z ) | R
U ^ | L = HyGG | q | 2 , q 2 ( ρ , d / n ¯ ) [ cos ( δ 2 ) | L + e i α sin ( δ 2 ) | R e 2 i q ϕ ] ,

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