Abstract

We present a study of Poincaré-beam polarization patterns produced by collinear superposition of two Laguerre–Gauss spatial modes in orthogonal polarization eigenstates (circular or linear). We explore theoretically and experimentally the combinations that are possible. We find that the resulting patterns can be explained in terms of mappings of points on the Poincaré sphere onto points in the transverse plane of the beam mode. The modes that we produced yielded many types of polarization singularities.

© 2012 Optical Society of America

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    [CrossRef]
  5. K. S. Youngsworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000).
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  6. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
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  9. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
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  10. S. C. McEldowney, D. M. Shemo, and R. A. Chipman, “Vortex retarders produced from photo-aligned liquid crystals polymers,” Opt. Express 16, 7295–7308 (2008).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  20. J. F. Nye, Natural Focusing and Fine Structure of Light (IOP, 1999).
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  22. M. S. Soskin, V. Denisenko, and I. Freund, “Optical polarization singularities and elliptic stationary points,” Opt. Lett. 28, 1475–1477 (2003).
    [CrossRef]
  23. F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
    [CrossRef]
  24. R. I. Egorov, M. S. Soskin, D. A. Kessler, and I. Freund, “Experimental measurements of topological singularity screening in random paraxial scalar and vector optical fields,” Phys. Rev. Lett. 100, 103901 (2008).
    [CrossRef]
  25. F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
    [CrossRef]
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    [CrossRef]
  27. I. O. Buinyi, V. G. Denisenko, and M. S. Soskin, “Topological structure in polarization resolved conoscopic patterns for nematic liquid crystal cells,” Opt. Commun. 282, 143–155 (2009).
    [CrossRef]
  28. Y. V. Jayasurya, V. V. G. Krishna Inavalli, and N. K. Viswanathan, “Polarization singularities in the two-mode optical fiber output,” Appl. Opt. 50, E131–E137 (2011).
    [CrossRef]
  29. I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201, 251–270 (2002).
    [CrossRef]
  30. 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]
  31. E. J. Galvez and S. Khadka, “Poincaré modes of light,” Proc. SPIE 8274, 83740Y (2012).
  32. I. Freund, “Polarization flowers,” Opt. Commun. 199, 47–63 (2001).
    [CrossRef]
  33. E. J. Galvez, “Vector beams in free space,” in The Angular Momentum of Light, D. L. Andrews and M. Babiker, eds. (Cambridge, to be published).
  34. E. J. Galvez, N. Smiley, and N. Fernandes, “Composite optical vortices formed by collinear Laguerre–Gauss beams,” Proc. SPIE 6131, 613105 (2006).
  35. S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17, 9818–9827 (2009).
    [CrossRef]
  36. I. Freund, “Poincaré vortices,” Opt. Lett. 26, 1996–1998 (2001).
    [CrossRef]
  37. I. Freund, A. I. Mokhum, M. S. Soskin, O. V. Angelsky, and I. I. Mokhum, “Stokes singularity relations,” Opt. Lett. 27, 545–547 (2002).
    [CrossRef]
  38. I. Freund, M. S. Soskin, and A. I. Mokhum, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253 (2002).
    [CrossRef]
  39. I. Freund, “Möbius strips and twisted ribbons in intersecting Gauss–Laguerre beams,” Opt. Commun. 284, 3816–3845 (2011).
    [CrossRef]

2012 (1)

E. J. Galvez and S. Khadka, “Poincaré modes of light,” Proc. SPIE 8274, 83740Y (2012).

2011 (2)

I. Freund, “Möbius strips and twisted ribbons in intersecting Gauss–Laguerre beams,” Opt. Commun. 284, 3816–3845 (2011).
[CrossRef]

Y. V. Jayasurya, V. V. G. Krishna Inavalli, and N. K. Viswanathan, “Polarization singularities in the two-mode optical fiber output,” Appl. Opt. 50, E131–E137 (2011).
[CrossRef]

2010 (2)

2009 (6)

G. Milione, H. I. Sztul, R. R. Alfano, and D. A. Nolan, “Stokes polarimetry of a hybrid vector beam from a spun elliptical core optical fiber,” Proc. SPIE 7613, 761305 (2009).

S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. 34, 2525–2527 (2009).
[CrossRef]

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

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef]

I. O. Buinyi, V. G. Denisenko, and M. S. Soskin, “Topological structure in polarization resolved conoscopic patterns for nematic liquid crystal cells,” Opt. Commun. 282, 143–155 (2009).
[CrossRef]

S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17, 9818–9827 (2009).
[CrossRef]

2008 (3)

R. I. Egorov, M. S. Soskin, D. A. Kessler, and I. Freund, “Experimental measurements of topological singularity screening in random paraxial scalar and vector optical fields,” Phys. Rev. Lett. 100, 103901 (2008).
[CrossRef]

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[CrossRef]

S. C. McEldowney, D. M. Shemo, and R. A. Chipman, “Vortex retarders produced from photo-aligned liquid crystals polymers,” Opt. Express 16, 7295–7308 (2008).
[CrossRef]

2007 (1)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

2006 (1)

E. J. Galvez, N. Smiley, and N. Fernandes, “Composite optical vortices formed by collinear Laguerre–Gauss beams,” Proc. SPIE 6131, 613105 (2006).

2005 (2)

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81, 597–600 (2005).
[CrossRef]

2004 (1)

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre–Gaussian beams,” Opt. Commun. 237, 89–95 (2004).
[CrossRef]

2003 (1)

2002 (4)

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

I. Freund, A. I. Mokhum, M. S. Soskin, O. V. Angelsky, and I. I. Mokhum, “Stokes singularity relations,” Opt. Lett. 27, 545–547 (2002).
[CrossRef]

I. Freund, M. S. Soskin, and A. I. Mokhum, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253 (2002).
[CrossRef]

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space variant dielectric subwavelength gratings,” Opt. Lett. 27, 285–287 (2002).
[CrossRef]

2001 (3)

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. A 457, 141–155 (2001).
[CrossRef]

I. Freund, “Poincaré vortices,” Opt. Lett. 26, 1996–1998 (2001).
[CrossRef]

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

2000 (3)

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal and radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

K. S. Youngsworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

1996 (1)

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]

1990 (1)

1983 (3)

J. F. Nye, “Line singularities in wave fields,” Proc. R. Soc. A 387, 105–132 (1983).
[CrossRef]

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

J. R. Fontana and R. H. Pantell, “A high-energy laser accelerator for using the inverse Cherenkov effect,” J. Appl. Phys. 54, 4285–4288 (1983).
[CrossRef]

1972 (1)

Y. Mushiake, K. Matsumura, and N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60, 1107–1109 (1972).
[CrossRef]

Alfano, R. R.

G. Milione, H. I. Sztul, R. R. Alfano, and D. A. Nolan, “Stokes polarimetry of a hybrid vector beam from a spun elliptical core optical fiber,” Proc. SPIE 7613, 761305 (2009).

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]

Alonso, M. A.

Angelsky, O. V.

Baba, T.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef]

Baumann, S. M.

Beckley, A. M.

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]

Bernet, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Berry, M. V.

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. A 457, 141–155 (2001).
[CrossRef]

Biener, G.

Blit, S.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal and radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Bomzon, Z.

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space variant dielectric subwavelength gratings,” Opt. Lett. 27, 285–287 (2002).
[CrossRef]

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal and radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Brasselet, E.

Brown, T. G.

Buinyi, I. O.

I. O. Buinyi, V. G. Denisenko, and M. S. Soskin, “Topological structure in polarization resolved conoscopic patterns for nematic liquid crystal cells,” Opt. Commun. 282, 143–155 (2009).
[CrossRef]

Burresi, M.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef]

Chipman, R. A.

Davidson, N.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal and radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Denisenko, V.

Denisenko, V. G.

I. O. Buinyi, V. G. Denisenko, and M. S. Soskin, “Topological structure in polarization resolved conoscopic patterns for nematic liquid crystal cells,” Opt. Commun. 282, 143–155 (2009).
[CrossRef]

Dennis, M. R.

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[CrossRef]

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. A 457, 141–155 (2001).
[CrossRef]

Desyatnikov, A. S.

Dorn, R.

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81, 597–600 (2005).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Egorov, R. I.

R. I. Egorov, M. S. Soskin, D. A. Kessler, and I. Freund, “Experimental measurements of topological singularity screening in random paraxial scalar and vector optical fields,” Phys. Rev. Lett. 100, 103901 (2008).
[CrossRef]

Engelen, R. J. P.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef]

Fadeyeva, T. A.

Fernandes, N.

E. J. Galvez, N. Smiley, and N. Fernandes, “Composite optical vortices formed by collinear Laguerre–Gauss beams,” Proc. SPIE 6131, 613105 (2006).

Flossmann, F.

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[CrossRef]

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

Fontana, J. R.

J. R. Fontana and R. H. Pantell, “A high-energy laser accelerator for using the inverse Cherenkov effect,” J. Appl. Phys. 54, 4285–4288 (1983).
[CrossRef]

Ford, D. H.

Freund, I.

I. Freund, “Möbius strips and twisted ribbons in intersecting Gauss–Laguerre beams,” Opt. Commun. 284, 3816–3845 (2011).
[CrossRef]

R. I. Egorov, M. S. Soskin, D. A. Kessler, and I. Freund, “Experimental measurements of topological singularity screening in random paraxial scalar and vector optical fields,” Phys. Rev. Lett. 100, 103901 (2008).
[CrossRef]

M. S. Soskin, V. Denisenko, and I. Freund, “Optical polarization singularities and elliptic stationary points,” Opt. Lett. 28, 1475–1477 (2003).
[CrossRef]

I. Freund, A. I. Mokhum, M. S. Soskin, O. V. Angelsky, and I. I. Mokhum, “Stokes singularity relations,” Opt. Lett. 27, 545–547 (2002).
[CrossRef]

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

I. Freund, M. S. Soskin, and A. I. Mokhum, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253 (2002).
[CrossRef]

I. Freund, “Poincaré vortices,” Opt. Lett. 26, 1996–1998 (2001).
[CrossRef]

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

Friesem, A. A.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal and radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Fürhapter, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Galvez, E. J.

E. J. Galvez and S. Khadka, “Poincaré modes of light,” Proc. SPIE 8274, 83740Y (2012).

S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17, 9818–9827 (2009).
[CrossRef]

E. J. Galvez, N. Smiley, and N. Fernandes, “Composite optical vortices formed by collinear Laguerre–Gauss beams,” Proc. SPIE 6131, 613105 (2006).

E. J. Galvez, “Vector beams in free space,” in The Angular Momentum of Light, D. L. Andrews and M. Babiker, eds. (Cambridge, to be published).

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Hasman, E.

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space variant dielectric subwavelength gratings,” Opt. Lett. 27, 285–287 (2002).
[CrossRef]

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal and radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Izdebskaya, Y. V.

Jayasurya, Y. V.

Jesacher, A.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Kalb, D. M.

Kessler, D. A.

R. I. Egorov, M. S. Soskin, D. A. Kessler, and I. Freund, “Experimental measurements of topological singularity screening in random paraxial scalar and vector optical fields,” Phys. Rev. Lett. 100, 103901 (2008).
[CrossRef]

Khadka, S.

E. J. Galvez and S. Khadka, “Poincaré modes of light,” Proc. SPIE 8274, 83740Y (2012).

Kimura, W. D.

Kivshar, Y. S.

Kleiner, V.

Krishna Inavalli, V. V. G.

Kristensen, P.

Krolikowski, W.

Kuipers, L.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef]

Leuchs, G.

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81, 597–600 (2005).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

MacMillan, L. H.

Maier, M.

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

Matsumura, K.

Y. Mushiake, K. Matsumura, and N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60, 1107–1109 (1972).
[CrossRef]

Maurer, C.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

McEldowney, S. C.

Milione, G.

G. Milione, H. I. Sztul, R. R. Alfano, and D. A. Nolan, “Stokes polarimetry of a hybrid vector beam from a spun elliptical core optical fiber,” Proc. SPIE 7613, 761305 (2009).

Mokhum, A. I.

I. Freund, M. S. Soskin, and A. I. Mokhum, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253 (2002).
[CrossRef]

I. Freund, A. I. Mokhum, M. S. Soskin, O. V. Angelsky, and I. I. Mokhum, “Stokes singularity relations,” Opt. Lett. 27, 545–547 (2002).
[CrossRef]

Mokhum, I. I.

Mori, D.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef]

Mushiake, Y.

Y. Mushiake, K. Matsumura, and N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60, 1107–1109 (1972).
[CrossRef]

Nakajima, N.

Y. Mushiake, K. Matsumura, and N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60, 1107–1109 (1972).
[CrossRef]

Neshev, D. N.

Nolan, D. A.

G. Milione, H. I. Sztul, R. R. Alfano, and D. A. Nolan, “Stokes polarimetry of a hybrid vector beam from a spun elliptical core optical fiber,” Proc. SPIE 7613, 761305 (2009).

Nye, J. F.

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

J. F. Nye, “Line singularities in wave fields,” Proc. R. Soc. A 387, 105–132 (1983).
[CrossRef]

J. F. Nye, Natural Focusing and Fine Structure of Light (IOP, 1999).

O’Holleran, K.

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[CrossRef]

Opheij, A.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef]

Oron, R.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal and radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Padgett, M. J.

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[CrossRef]

Pantell, R. H.

J. R. Fontana and R. H. Pantell, “A high-energy laser accelerator for using the inverse Cherenkov effect,” J. Appl. Phys. 54, 4285–4288 (1983).
[CrossRef]

Petrov, D.

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre–Gaussian beams,” Opt. Commun. 237, 89–95 (2004).
[CrossRef]

Quabis, S.

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81, 597–600 (2005).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Ramachandran, S.

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Schadt, M.

Schwarz, U. T.

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

Shemo, D. M.

Shvedov, V. G.

Smiley, N.

E. J. Galvez, N. Smiley, and N. Fernandes, “Composite optical vortices formed by collinear Laguerre–Gauss beams,” Proc. SPIE 6131, 613105 (2006).

Soskin, M. S.

I. O. Buinyi, V. G. Denisenko, and M. S. Soskin, “Topological structure in polarization resolved conoscopic patterns for nematic liquid crystal cells,” Opt. Commun. 282, 143–155 (2009).
[CrossRef]

R. I. Egorov, M. S. Soskin, D. A. Kessler, and I. Freund, “Experimental measurements of topological singularity screening in random paraxial scalar and vector optical fields,” Phys. Rev. Lett. 100, 103901 (2008).
[CrossRef]

M. S. Soskin, V. Denisenko, and I. Freund, “Optical polarization singularities and elliptic stationary points,” Opt. Lett. 28, 1475–1477 (2003).
[CrossRef]

I. Freund, A. I. Mokhum, M. S. Soskin, O. V. Angelsky, and I. I. Mokhum, “Stokes singularity relations,” Opt. Lett. 27, 545–547 (2002).
[CrossRef]

I. Freund, M. S. Soskin, and A. I. Mokhum, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253 (2002).
[CrossRef]

Stalder, M.

Sztul, H. I.

G. Milione, H. I. Sztul, R. R. Alfano, and D. A. Nolan, “Stokes polarimetry of a hybrid vector beam from a spun elliptical core optical fiber,” Proc. SPIE 7613, 761305 (2009).

Tidwell, S. C.

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 Oosten, D.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef]

Viswanathan, N. K.

Volpe, G.

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre–Gaussian beams,” Opt. Commun. 237, 89–95 (2004).
[CrossRef]

Volyar, A. V.

Woerdman, J. P.

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]

Yan, M. F.

Youngsworth, K. S.

Zhan, Q.

Adv. Opt. Photon. (1)

Appl. Opt. (2)

Appl. Phys. B (1)

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81, 597–600 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal and radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

J. Appl. Phys. (1)

J. R. Fontana and R. H. Pantell, “A high-energy laser accelerator for using the inverse Cherenkov effect,” J. Appl. Phys. 54, 4285–4288 (1983).
[CrossRef]

New J. Phys. (1)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Opt. Commun. (8)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

I. O. Buinyi, V. G. Denisenko, and M. S. Soskin, “Topological structure in polarization resolved conoscopic patterns for nematic liquid crystal cells,” Opt. Commun. 282, 143–155 (2009).
[CrossRef]

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

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

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre–Gaussian beams,” Opt. Commun. 237, 89–95 (2004).
[CrossRef]

I. Freund, M. S. Soskin, and A. I. Mokhum, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253 (2002).
[CrossRef]

I. Freund, “Möbius strips and twisted ribbons in intersecting Gauss–Laguerre beams,” Opt. Commun. 284, 3816–3845 (2011).
[CrossRef]

Opt. Express (5)

Opt. Lett. (6)

Phys. Rev. Lett. (4)

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

R. I. Egorov, M. S. Soskin, D. A. Kessler, and I. Freund, “Experimental measurements of topological singularity screening in random paraxial scalar and vector optical fields,” Phys. Rev. Lett. 100, 103901 (2008).
[CrossRef]

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[CrossRef]

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef]

Proc. IEEE (1)

Y. Mushiake, K. Matsumura, and N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60, 1107–1109 (1972).
[CrossRef]

Proc. R. Soc. A (3)

J. F. Nye, “Line singularities in wave fields,” Proc. R. Soc. A 387, 105–132 (1983).
[CrossRef]

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

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. A 457, 141–155 (2001).
[CrossRef]

Proc. SPIE (3)

G. Milione, H. I. Sztul, R. R. Alfano, and D. A. Nolan, “Stokes polarimetry of a hybrid vector beam from a spun elliptical core optical fiber,” Proc. SPIE 7613, 761305 (2009).

E. J. Galvez, N. Smiley, and N. Fernandes, “Composite optical vortices formed by collinear Laguerre–Gauss beams,” Proc. SPIE 6131, 613105 (2006).

E. J. Galvez and S. Khadka, “Poincaré modes of light,” Proc. SPIE 8274, 83740Y (2012).

Other (2)

J. F. Nye, Natural Focusing and Fine Structure of Light (IOP, 1999).

E. J. Galvez, “Vector beams in free space,” in The Angular Momentum of Light, D. L. Andrews and M. Babiker, eds. (Cambridge, to be published).

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Figures (7)

Fig. 1.
Fig. 1.

(a) Poincaré sphere showing how the states of polarization are represented on the surface of the sphere. (b) Matrix of polarization states obtained by varying the amplitude (vertical) and phase (horizontal) between component circular-state modes [see Eq. (7)]. Red and blue colors represent right-handed and left-handed polarization states, respectively, whereas black represents linear polarization.

Fig. 2.
Fig. 2.

Schematic of the apparatus. (a) General setup to produce and diagnose vector modes and (b) arrangement to exclude specific polarization states. Optical components include forked gratings (G1 and G2), glass blank (B), polarizers (P), half-wave plate (H), quarter-wave plate (Q), and digital camera (C).

Fig. 3.
Fig. 3.

Polarization-state maps of the Poincaré mode obtained by combining spatial modes with (a) 12=1 and (b) 12=+1. In both cases, the spatial modes have p1=p2=0, and the polarization of the component modes is circular. (c) Poincaré sphere paths showing the sequence of states obtained by either increasing r with fixed ϕ or increasing ϕ with fixed r. (d) Images of the Poincaré mode with 1=0 and 2=1 after passage through a polarization-state analyzer, with settings specified by χ and θ.

Fig. 4.
Fig. 4.

Vector mode obtained by combining 1=+3 and 2=3 spatial modes in the circular polarization basis: (a) polarization-state map, (b) path that the state of polarization follows on the Poincaré sphere for a circular path about the center of the beam (i.e., increasing ϕ at fixed r), and (c) images of the vector mode obtained after passage through a polarizer in several orientations β.

Fig. 5.
Fig. 5.

Polarization-state maps of the Poincaré mode obtained by combining spatial modes with (a) 12=1 and (b) 12=2. In both cases the spatial modes have p1=p2=0 and |1|<|2| with the polarization of the component modes linear. (c) Poincaré sphere paths for states with increasing r with fixed ϕ and increasing ϕ with fixed r. (d) Measurements of the Poincaré mode involving the superposition of spatial modes with 1=0 and 2=2 in the linear basis. Images show the mode that passed a linear polarization analyzer oriented an angle β. The phase between the two modes was 2α, with α22°.

Fig. 6.
Fig. 6.

(a) Polarization-state map obtained by combining 1=2 and 2=2 spatial modes in the linear polarization basis, (b) path on the Poincaré sphere followed by the state of polarization of points of increasing transverse angle ϕ and fixed radius r, and (c) images of the vector mode obtained after passing it through a polarizer in several orientations β.

Fig. 7.
Fig. 7.

(a) Polarization map of modes with p1=0, 1=1 and p2=1, 2=0 in the circular basis. The circle drawn has radius r. (b) Path taken on the Poincaré sphere when following a radial trajectory.

Equations (29)

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LGp=Ap,r||eiϕGLp||Wp,,
Ap,=(p!2||+1π(||+p)!)1/21w||+1
G=er2/w2
Up,=ei(x2+y2)/(2R)iφ
φ=(2p+||+1)tan1(z/zR)
r=(2)1/2.
e^θ,χ=e+iθcosχe^R+eiθsinχe^L,
ϵ=ba=tan(π/4χ),
Vp1,1,p2,2=12(LGp11eiαe^R+LGp22eiαe^L)
Wp1,1,p2,2=12(LGp11eiαe^X+LGp22eiαe^Y)
V0,1,0,2=G2(A0,1r|1|ei(1ϕ+α)e^R+A0,2r|2|ei(2ϕα)e^L).
V0,1,0,2=NGei(1+2)ϕ/2(eiθcosχe^R+eiθsinχe^L),
χ=tan1A0,2r|2|A0,1r|1|,
θ=(12)ϕ/2+α,
rv=(A0,2A0,1)1/(|1||2|),
V0,,0,=A0,r||G2(ei(ϕ+α)e^R+ei(ϕ+α)e^L),
=A0,r||Ge^(ϕ+α),π/4,
ϕm=(2m1)π2α+β,
W0,1,0,2=Gei(1+2)ϕ/22(A0,1r|1|ei(12)ϕ/2+αe^X+A0,2r|2|ei(12)ϕ/2αe^Y).
W0,1,0,2=NGei(1+2)ϕ/2(eiθcosχe^X+eiθsinχe^Y),
W0,1,0,2=(LGp11eiαcosβ+LGp22eiαsinβ)e^β,
W0,,0,=A0,r||G2(ei(ϕ+α)e^X+ei(ϕ+α)e^Y).
W0,,0,=A0,r|Gi[eiπ/4cos(ϕ+απ/4)e^R+eiπ/4sin(ϕ+απ/4)e^L],
=A0,r||Ge^π/4,ϕ+απ/4.
χ=ϕ+απ/4.
W0,1,1,0=Geiϕ/22[A0,1rei(ϕ/2+α)e^R+A1,0(2r2+1)ei(ϕ/2+α)e^L],
W0,1,1,0=NGeiϕ/2(cosχeiθe^R+sinχeiθe^L),
tanχ=A1,0(2r2+1)A0,1r,
θ=ϕ/2+α.

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