Abstract

C points, that is, isolated points of circular polarization in transverse fields of varying polarization, are classified morphologically into three distinct types known as lemons, stars, and monstars. These morphologies are interpreted here according to two natural parameters associated with the singularity, namely, the anisotropy of the C point and the polarization azimuth on the anisotropy axis. In addition to providing insight into singularity morphology, this observation applies to the densities of the various morphologies in isotropic random polarization speckle fields.

© 2008 Optical Society of America

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  1. J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
    [CrossRef]
  2. J. F. Nye, Natural Focusing and Fine Structure of Light (IoPP, 1999).
  3. J. F. Nye, Proc. R. Soc. London Ser. A 389, 279 (1983).
    [CrossRef]
  4. M. V. Berry and J. H. Hannay, J. Phys. A 10, 1809 (1977).
    [CrossRef]
  5. G. Darboux, Leçons sur la Théorie Générale des Surfaces Note VII (Gauthier-Villars, 1896), Vol. 4.
  6. I. R. Porteous, Geometric Differentiation (Cambridge U. Press, 2001).
  7. M. R. Dennis, Opt. Commun. 213, 201 (2002).
    [CrossRef]
  8. H. Poincaré, Théorie Mathématique de la Lumière (Éditions Jacques Gabay, 1995), Vol. 2.
  9. A. Konukhov and L. Melnikov, J. Opt. B 3, S139 (2001).
    [CrossRef]
  10. I. Freund, M. Soskin, and A. Mokhun, Opt. Commun. 208, 223 (2002).
    [CrossRef]
  11. M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., New J. Phys. 6, 162 (2004).
    [CrossRef]
  12. M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 459, 1261 (2003).
    [CrossRef]
  13. M. R. Dennis, J. Opt. A 6, S217 (2004).
    [CrossRef]
  14. Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, J. Opt. A 6, S217 (2004).
    [CrossRef]
  15. F. S. Roux, J. Opt. Soc. Am. B 21, 664 (2004).
    [CrossRef]
  16. F. Flossmann, K. O'Holleran, M. R. Dennis, and M. J. Padgett, Phys. Rev. Lett. 100, 203902 (2008).
    [CrossRef] [PubMed]
  17. V. G. Denisenko, R. I. Egorov, and M. S. Soskin, JETP Lett. 31, 17 (2004).
    [CrossRef]
  18. M. S. Soskin, V. G. Denisenko, and R. I. Egorov, J. Opt. A 6, S281 (2004).
    [CrossRef]
  19. V. Vasil'ev and M. Soskin, Opt. Commun. 281, 5527 (2008).
    [CrossRef]
  20. M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 456, 2059 (2000).
    [CrossRef]
  21. D. R. Nelson, Defects and Geometry in Condensed Matter Physics (Cambridge U. Press, 2002).
  22. R. I. Egorov, M. S. Soskin, and I. Freund, Opt. Lett. 31, 2048 (2006).
    [CrossRef] [PubMed]

2008 (2)

F. Flossmann, K. O'Holleran, M. R. Dennis, and M. J. Padgett, Phys. Rev. Lett. 100, 203902 (2008).
[CrossRef] [PubMed]

V. Vasil'ev and M. Soskin, Opt. Commun. 281, 5527 (2008).
[CrossRef]

2006 (1)

2004 (6)

F. S. Roux, J. Opt. Soc. Am. B 21, 664 (2004).
[CrossRef]

V. G. Denisenko, R. I. Egorov, and M. S. Soskin, JETP Lett. 31, 17 (2004).
[CrossRef]

M. S. Soskin, V. G. Denisenko, and R. I. Egorov, J. Opt. A 6, S281 (2004).
[CrossRef]

M. R. Dennis, J. Opt. A 6, S217 (2004).
[CrossRef]

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, J. Opt. A 6, S217 (2004).
[CrossRef]

M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., New J. Phys. 6, 162 (2004).
[CrossRef]

2003 (1)

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 459, 1261 (2003).
[CrossRef]

2002 (2)

I. Freund, M. Soskin, and A. Mokhun, Opt. Commun. 208, 223 (2002).
[CrossRef]

M. R. Dennis, Opt. Commun. 213, 201 (2002).
[CrossRef]

2001 (1)

A. Konukhov and L. Melnikov, J. Opt. B 3, S139 (2001).
[CrossRef]

2000 (1)

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 456, 2059 (2000).
[CrossRef]

1983 (1)

J. F. Nye, Proc. R. Soc. London Ser. A 389, 279 (1983).
[CrossRef]

1977 (1)

M. V. Berry and J. H. Hannay, J. Phys. A 10, 1809 (1977).
[CrossRef]

1974 (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Berry, M. V.

M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., New J. Phys. 6, 162 (2004).
[CrossRef]

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 459, 1261 (2003).
[CrossRef]

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 456, 2059 (2000).
[CrossRef]

M. V. Berry and J. H. Hannay, J. Phys. A 10, 1809 (1977).
[CrossRef]

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Darboux, G.

G. Darboux, Leçons sur la Théorie Générale des Surfaces Note VII (Gauthier-Villars, 1896), Vol. 4.

Denisenko, V. G.

M. S. Soskin, V. G. Denisenko, and R. I. Egorov, J. Opt. A 6, S281 (2004).
[CrossRef]

V. G. Denisenko, R. I. Egorov, and M. S. Soskin, JETP Lett. 31, 17 (2004).
[CrossRef]

Dennis, M. R.

F. Flossmann, K. O'Holleran, M. R. Dennis, and M. J. Padgett, Phys. Rev. Lett. 100, 203902 (2008).
[CrossRef] [PubMed]

M. R. Dennis, J. Opt. A 6, S217 (2004).
[CrossRef]

M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., New J. Phys. 6, 162 (2004).
[CrossRef]

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 459, 1261 (2003).
[CrossRef]

M. R. Dennis, Opt. Commun. 213, 201 (2002).
[CrossRef]

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 456, 2059 (2000).
[CrossRef]

Egorov, R. I.

R. I. Egorov, M. S. Soskin, and I. Freund, Opt. Lett. 31, 2048 (2006).
[CrossRef] [PubMed]

M. S. Soskin, V. G. Denisenko, and R. I. Egorov, J. Opt. A 6, S281 (2004).
[CrossRef]

V. G. Denisenko, R. I. Egorov, and M. S. Soskin, JETP Lett. 31, 17 (2004).
[CrossRef]

Egorov, Y. A.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, J. Opt. A 6, S217 (2004).
[CrossRef]

Fadeyeva, T. A.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, J. Opt. A 6, S217 (2004).
[CrossRef]

Flossmann, F.

F. Flossmann, K. O'Holleran, M. R. Dennis, and M. J. Padgett, Phys. Rev. Lett. 100, 203902 (2008).
[CrossRef] [PubMed]

Freund, I.

R. I. Egorov, M. S. Soskin, and I. Freund, Opt. Lett. 31, 2048 (2006).
[CrossRef] [PubMed]

I. Freund, M. Soskin, and A. Mokhun, Opt. Commun. 208, 223 (2002).
[CrossRef]

Hannay, J. H.

M. V. Berry and J. H. Hannay, J. Phys. A 10, 1809 (1977).
[CrossRef]

Konukhov, A.

A. Konukhov and L. Melnikov, J. Opt. B 3, S139 (2001).
[CrossRef]

Lee, R. L.

M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., New J. Phys. 6, 162 (2004).
[CrossRef]

Melnikov, L.

A. Konukhov and L. Melnikov, J. Opt. B 3, S139 (2001).
[CrossRef]

Mokhun, A.

I. Freund, M. Soskin, and A. Mokhun, Opt. Commun. 208, 223 (2002).
[CrossRef]

Nelson, D. R.

D. R. Nelson, Defects and Geometry in Condensed Matter Physics (Cambridge U. Press, 2002).

Nye, J. F.

J. F. Nye, Proc. R. Soc. London Ser. A 389, 279 (1983).
[CrossRef]

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

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

O'Holleran, K.

F. Flossmann, K. O'Holleran, M. R. Dennis, and M. J. Padgett, Phys. Rev. Lett. 100, 203902 (2008).
[CrossRef] [PubMed]

Padgett, M. J.

F. Flossmann, K. O'Holleran, M. R. Dennis, and M. J. Padgett, Phys. Rev. Lett. 100, 203902 (2008).
[CrossRef] [PubMed]

Poincaré, H.

H. Poincaré, Théorie Mathématique de la Lumière (Éditions Jacques Gabay, 1995), Vol. 2.

Porteous, I. R.

I. R. Porteous, Geometric Differentiation (Cambridge U. Press, 2001).

Roux, F. S.

Soskin, M.

V. Vasil'ev and M. Soskin, Opt. Commun. 281, 5527 (2008).
[CrossRef]

I. Freund, M. Soskin, and A. Mokhun, Opt. Commun. 208, 223 (2002).
[CrossRef]

Soskin, M. S.

R. I. Egorov, M. S. Soskin, and I. Freund, Opt. Lett. 31, 2048 (2006).
[CrossRef] [PubMed]

M. S. Soskin, V. G. Denisenko, and R. I. Egorov, J. Opt. A 6, S281 (2004).
[CrossRef]

V. G. Denisenko, R. I. Egorov, and M. S. Soskin, JETP Lett. 31, 17 (2004).
[CrossRef]

Vasil'ev, V.

V. Vasil'ev and M. Soskin, Opt. Commun. 281, 5527 (2008).
[CrossRef]

Volyar, A. V.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, J. Opt. A 6, S217 (2004).
[CrossRef]

J. Opt. A (3)

M. R. Dennis, J. Opt. A 6, S217 (2004).
[CrossRef]

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, J. Opt. A 6, S217 (2004).
[CrossRef]

M. S. Soskin, V. G. Denisenko, and R. I. Egorov, J. Opt. A 6, S281 (2004).
[CrossRef]

J. Opt. B (1)

A. Konukhov and L. Melnikov, J. Opt. B 3, S139 (2001).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. A (1)

M. V. Berry and J. H. Hannay, J. Phys. A 10, 1809 (1977).
[CrossRef]

JETP Lett. (1)

V. G. Denisenko, R. I. Egorov, and M. S. Soskin, JETP Lett. 31, 17 (2004).
[CrossRef]

New J. Phys. (1)

M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., New J. Phys. 6, 162 (2004).
[CrossRef]

Opt. Commun. (3)

I. Freund, M. Soskin, and A. Mokhun, Opt. Commun. 208, 223 (2002).
[CrossRef]

M. R. Dennis, Opt. Commun. 213, 201 (2002).
[CrossRef]

V. Vasil'ev and M. Soskin, Opt. Commun. 281, 5527 (2008).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

F. Flossmann, K. O'Holleran, M. R. Dennis, and M. J. Padgett, Phys. Rev. Lett. 100, 203902 (2008).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A (4)

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 459, 1261 (2003).
[CrossRef]

J. F. Nye, Proc. R. Soc. London Ser. A 389, 279 (1983).
[CrossRef]

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 456, 2059 (2000).
[CrossRef]

Other (5)

D. R. Nelson, Defects and Geometry in Condensed Matter Physics (Cambridge U. Press, 2002).

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

G. Darboux, Leçons sur la Théorie Générale des Surfaces Note VII (Gauthier-Villars, 1896), Vol. 4.

I. R. Porteous, Geometric Differentiation (Cambridge U. Press, 2001).

H. Poincaré, Théorie Mathématique de la Lumière (Éditions Jacques Gabay, 1995), Vol. 2.

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Figures (2)

Fig. 1
Fig. 1

Figures showing the three different line types of C point morphology with typical shape parameters and corresponding plots of θ ( ϕ ) : a, star ( Υ = 0.95 , β = π 6 ); b, monstar ( Υ = 0.31 , β = π 15 ); c, lemon ( Υ = 0.87 , β = π 10 ). The plots on the left-hand side show grayscale contour plots of σ , showing the elliptic anisotropy and the polarization lines (black) with colored separatrices. The straight lines in the plots on the right-hand side correspond to the lines θ = ϕ and ϕ = ± π , which intersect the curve at the straight line separatrices ending at the singularity. θ ( ϕ ) , being periodic, multiply wraps around the plot.

Fig. 2
Fig. 2

Υ , β plane parameterizing C point morphology ( 1 Υ 1 , π 4 β π 4 ) , together with several choices of singularity parameters, plotted as Fig. 1. The plane is partitioned into three distinct regions corresponding to star (S, blue), monstar (M, red) and lemon (L, yellow). C point geometries at different points of this area are also shown. The points labeled a–c correspond to those parts of Fig. 1, and the other choices, from maximum to minimum β, have coordinates ( Υ , β ) : ( ± 0.10 , π 4 ) , ( ± 0.31 , π 10 ) , ( ± 0.05 , π 10 ) , and ( ± 0.5 , π 5 ) .

Equations (8)

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σ S 1 + i S 2 = E R * E L ,
D L = [ ( 2 S 1 y + S 2 x ) 2 3 S 2 y ( 2 S 1 x S 2 y ) ] [ ( 2 S 1 x S 2 y ) 2 + 3 S 2 x ( 2 S 1 y + S 2 x ) ] ( 2 S 1 x S 1 y + S 1 x S 2 x S 1 y S 2 y + 4 S 2 x S 2 y ) 2
Υ Im σ * × σ σ 2 = 2 ( S 1 x S 2 y S 2 x S 1 y ) S 1 x 2 + S 1 y 2 + S 2 x 2 + S 2 y 2 .
θ ( ϕ ) = 1 2 arctan S 2 x cos ϕ + S 2 y sin ϕ S 1 x cos ϕ + S 1 y sin ϕ .
σ = exp ( 2 i β ) { T X , i T Y } ,
θ ( ϕ ) = 1 2 arctan [ Υ sin ϕ ( ( 1 1 Υ 2 ) ) cos ϕ ] + β .
D L = 3 4 σ 4 ( 2 + Υ ( 11 Υ 14 ) + 2 ( 1 + Υ ) 1 Υ 2 cos 4 β ) .
cos 4 β crit = 2 + Υ ( 11 Υ 14 ) 2 ( 1 + Υ ) 3 2 ( 1 Υ ) 1 2 ,

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