Abstract

Using a recent method able to characterize the polarimetry of a random field with high polarimetric and spatial accuracy even near places of destructive interference, we study polarized optical vortices at a scale below the transverse correlation width of a speckle field. We perform high accuracy polarimetric measurements of known singularities described with an half-integer topological index and we study rare integer index singularities which have, to our knowledge, never been observed in a speckle field.

© 2015 Optical Society of America

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References

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  1. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London A 336, 165–190 (1974).
    [Crossref]
  2. W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, and M. Takeda, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express 14, 120–127 (2006).
    [Crossref] [PubMed]
  3. X. Wang, Y. Liu, L. Guo, and H. Li, 11 Potential of vortex beams with orbital angular momentum modulation for deep-space optical communication,” Opt. Eng. 53, 056107 (2014).
    [Crossref]
  4. J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. London A. 387, 105–132 (1983).
    [Crossref]
  5. J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A 389, 279–290 (1983).
    [Crossref]
  6. X. Shi and Y. Ma, “Topological structure dynamics revealing collective evolution in active nematics,” Nat. Commun. 4, 3013 (2013).
    [Crossref] [PubMed]
  7. T. Machon and G. P. Alexander, “Knotted defects in nematic liquid crystals,” Phys. Rev. Lett. 113, 027801 (2014).
    [Crossref] [PubMed]
  8. V. V. Isaeva, N. V. Kasyanov, and E. V. Presnov, “Topological singularities and symmetry breaking in development,” Biosyst. 109, 280–298 (2012).
    [Crossref]
  9. V. Vitelli, B. Jain, and R. D. Kamien, “Topological defects in gravitational lensing shear fields,” J. Cosmol. Astropart. Phys. 09, 1475–7516 (2009).
  10. J. Dupont, X. Orlik, A. Gabbach, M. Zerrad, G. Soriano, and C. Amra, “Polarization analysis of speckle field below its transverse correlation width : application to surface and bulk scattering,” Opt. Express 22, 24133–24141 (2014).
    [Crossref] [PubMed]
  11. A. S. Thorndike, C. R. Cooley, and J. F. Nye, 11 The structure and evolution of flow fields and other vector fields,” J. Phys. A: Math. Gen. 11, 1455–1490 (1978).
    [Crossref]
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    [Crossref] [PubMed]
  13. M. S. Soskin and V. I. Vasil’ev, “Topological singular chain reactions in dynamic speckle fields,” J. Opt. Soc. Am. B 31, A56–A61 (2014).
    [Crossref]
  14. V. Kumar and N. K. Viswanathan, “Topological structures in vector-vortex beam fields,” J. Opt. Soc. Am. B 31, A40–A45 (2014).
    [Crossref]
  15. 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] [PubMed]
  16. J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. II. observations on the electric field,” Proc. R. Soc. London A 414, 447–468 (1987).
    [Crossref]
  17. I. Freund, “Polarization flowers,” Opt. Commun. 199, 47–63 (2001).
    [Crossref]
  18. I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
    [Crossref]
  19. M. R. Dennis, “Polarization singularity anisotropy: determining monstardom,” Opt. Lett. 33, 2572–2574 (2008).
    [Crossref] [PubMed]
  20. S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Ind. Acad. Sci. A 44, 247–262 (1956).
  21. G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27, 1875–1877 (2002).
    [Crossref]

2014 (5)

2013 (1)

X. Shi and Y. Ma, “Topological structure dynamics revealing collective evolution in active nematics,” Nat. Commun. 4, 3013 (2013).
[Crossref] [PubMed]

2012 (1)

V. V. Isaeva, N. V. Kasyanov, and E. V. Presnov, “Topological singularities and symmetry breaking in development,” Biosyst. 109, 280–298 (2012).
[Crossref]

2009 (1)

V. Vitelli, B. Jain, and R. D. Kamien, “Topological defects in gravitational lensing shear fields,” J. Cosmol. Astropart. Phys. 09, 1475–7516 (2009).

2008 (2)

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] [PubMed]

M. R. Dennis, “Polarization singularity anisotropy: determining monstardom,” Opt. Lett. 33, 2572–2574 (2008).
[Crossref] [PubMed]

2006 (1)

2003 (1)

2002 (1)

2001 (1)

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

1993 (1)

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[Crossref]

1987 (1)

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. II. observations on the electric field,” Proc. R. Soc. London A 414, 447–468 (1987).
[Crossref]

1983 (2)

J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. London A. 387, 105–132 (1983).
[Crossref]

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

1978 (1)

A. S. Thorndike, C. R. Cooley, and J. F. Nye, 11 The structure and evolution of flow fields and other vector fields,” J. Phys. A: Math. Gen. 11, 1455–1490 (1978).
[Crossref]

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London A 336, 165–190 (1974).
[Crossref]

1956 (1)

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

Alexander, G. P.

T. Machon and G. P. Alexander, “Knotted defects in nematic liquid crystals,” Phys. Rev. Lett. 113, 027801 (2014).
[Crossref] [PubMed]

Amra, C.

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London A 336, 165–190 (1974).
[Crossref]

Biener, G.

Cooley, C. R.

A. S. Thorndike, C. R. Cooley, and J. F. Nye, 11 The structure and evolution of flow fields and other vector fields,” J. Phys. A: Math. Gen. 11, 1455–1490 (1978).
[Crossref]

Denisenko, V.

Dennis, M. R.

M. R. Dennis, “Polarization singularity anisotropy: determining monstardom,” Opt. Lett. 33, 2572–2574 (2008).
[Crossref] [PubMed]

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] [PubMed]

Dupont, J.

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] [PubMed]

Freilikher, V.

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[Crossref]

Freund, I.

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

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

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[Crossref]

Gabbach, A.

Guo, L.

X. Wang, Y. Liu, L. Guo, and H. Li, 11 Potential of vortex beams with orbital angular momentum modulation for deep-space optical communication,” Opt. Eng. 53, 056107 (2014).
[Crossref]

Hajnal, J. V.

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. II. observations on the electric field,” Proc. R. Soc. London A 414, 447–468 (1987).
[Crossref]

Hasman, E.

Isaeva, V. V.

V. V. Isaeva, N. V. Kasyanov, and E. V. Presnov, “Topological singularities and symmetry breaking in development,” Biosyst. 109, 280–298 (2012).
[Crossref]

Ishijima, R.

Jain, B.

V. Vitelli, B. Jain, and R. D. Kamien, “Topological defects in gravitational lensing shear fields,” J. Cosmol. Astropart. Phys. 09, 1475–7516 (2009).

Kamien, R. D.

V. Vitelli, B. Jain, and R. D. Kamien, “Topological defects in gravitational lensing shear fields,” J. Cosmol. Astropart. Phys. 09, 1475–7516 (2009).

Kasyanov, N. V.

V. V. Isaeva, N. V. Kasyanov, and E. V. Presnov, “Topological singularities and symmetry breaking in development,” Biosyst. 109, 280–298 (2012).
[Crossref]

Kleiner, V.

Kumar, V.

Li, H.

X. Wang, Y. Liu, L. Guo, and H. Li, 11 Potential of vortex beams with orbital angular momentum modulation for deep-space optical communication,” Opt. Eng. 53, 056107 (2014).
[Crossref]

Liu, Y.

X. Wang, Y. Liu, L. Guo, and H. Li, 11 Potential of vortex beams with orbital angular momentum modulation for deep-space optical communication,” Opt. Eng. 53, 056107 (2014).
[Crossref]

Ma, Y.

X. Shi and Y. Ma, “Topological structure dynamics revealing collective evolution in active nematics,” Nat. Commun. 4, 3013 (2013).
[Crossref] [PubMed]

Machon, T.

T. Machon and G. P. Alexander, “Knotted defects in nematic liquid crystals,” Phys. Rev. Lett. 113, 027801 (2014).
[Crossref] [PubMed]

Miyamoto, Y.

Niv, A.

Nye, J. F.

J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. London A. 387, 105–132 (1983).
[Crossref]

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

A. S. Thorndike, C. R. Cooley, and J. F. Nye, 11 The structure and evolution of flow fields and other vector fields,” J. Phys. A: Math. Gen. 11, 1455–1490 (1978).
[Crossref]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London A 336, 165–190 (1974).
[Crossref]

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] [PubMed]

Orlik, X.

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] [PubMed]

Pancharatnam, S.

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

Presnov, E. V.

V. V. Isaeva, N. V. Kasyanov, and E. V. Presnov, “Topological singularities and symmetry breaking in development,” Biosyst. 109, 280–298 (2012).
[Crossref]

Shi, X.

X. Shi and Y. Ma, “Topological structure dynamics revealing collective evolution in active nematics,” Nat. Commun. 4, 3013 (2013).
[Crossref] [PubMed]

Shvartsman, N.

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[Crossref]

Soriano, G.

Soskin, M. S.

Takeda, M.

Thorndike, A. S.

A. S. Thorndike, C. R. Cooley, and J. F. Nye, 11 The structure and evolution of flow fields and other vector fields,” J. Phys. A: Math. Gen. 11, 1455–1490 (1978).
[Crossref]

Vasil’ev, V. I.

Viswanathan, N. K.

Vitelli, V.

V. Vitelli, B. Jain, and R. D. Kamien, “Topological defects in gravitational lensing shear fields,” J. Cosmol. Astropart. Phys. 09, 1475–7516 (2009).

Wada, A.

Wang, W.

Wang, X.

X. Wang, Y. Liu, L. Guo, and H. Li, 11 Potential of vortex beams with orbital angular momentum modulation for deep-space optical communication,” Opt. Eng. 53, 056107 (2014).
[Crossref]

Yokozeki, T.

Zerrad, M.

Biosyst. (1)

V. V. Isaeva, N. V. Kasyanov, and E. V. Presnov, “Topological singularities and symmetry breaking in development,” Biosyst. 109, 280–298 (2012).
[Crossref]

J. Cosmol. Astropart. Phys. (1)

V. Vitelli, B. Jain, and R. D. Kamien, “Topological defects in gravitational lensing shear fields,” J. Cosmol. Astropart. Phys. 09, 1475–7516 (2009).

J. Opt. Soc. Am. B (2)

J. Phys. A: Math. Gen. (1)

A. S. Thorndike, C. R. Cooley, and J. F. Nye, 11 The structure and evolution of flow fields and other vector fields,” J. Phys. A: Math. Gen. 11, 1455–1490 (1978).
[Crossref]

Nat. Commun. (1)

X. Shi and Y. Ma, “Topological structure dynamics revealing collective evolution in active nematics,” Nat. Commun. 4, 3013 (2013).
[Crossref] [PubMed]

Opt. Commun. (2)

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

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[Crossref]

Opt. Eng. (1)

X. Wang, Y. Liu, L. Guo, and H. Li, 11 Potential of vortex beams with orbital angular momentum modulation for deep-space optical communication,” Opt. Eng. 53, 056107 (2014).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. Lett. (2)

T. Machon and G. P. Alexander, “Knotted defects in nematic liquid crystals,” Phys. Rev. Lett. 113, 027801 (2014).
[Crossref] [PubMed]

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] [PubMed]

Proc. Ind. Acad. Sci. A (1)

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

Proc. R. Soc. London A (3)

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. II. observations on the electric field,” Proc. R. Soc. London A 414, 447–468 (1987).
[Crossref]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London A 336, 165–190 (1974).
[Crossref]

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

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

J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. London A. 387, 105–132 (1983).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Experimental setup used to compute the 2D SOP map of a speckle field produced by bulk scattering. A laser diode of wavelength 532nm illuminates a sheet of paper with right handed circular SOP, obtained with a linear polarizer (Plin) oriented at 45° with respect to the fast axis of the quarter-wave plate (λ/4). The scattered field is projected on 300 different SOP by two NLCs (CL1, CL2) and a polarizer (Plin), and the resulting intensity is recorded by a 103 × 103 pixels CCD camera. The pupil diameter is reduced by a pinhole (P) in order to increase the number of pixels per speckle grain.
Fig. 2
Fig. 2 (a) Typical normalized intensity (Iin(i, j) in Eq. (5)) of a speckle field produced by bulk scattering measured by SOPAFP method on a 250×250 pixels area, (b) Corresponding polarization streamlines. In the following of the document, we will focus on the 20 × 20 pixels areas designated by A, B and C.
Fig. 3
Fig. 3 (a) SOP variations on a 20 × 20 pixels area with 2 singularities, n°1 is a star and n°2 is a lemon. The ellipse colour is representative of the DOP. Polarization streamlines are superimposed to the polarimetric ellipses. The black stripes represent the areas of left handed ellipticity, the other states are right handed ones. There are L lines at each demarcation between areas of opposite handedness. The inset in the upper-left corner is the error map varying from 3.5% to 17%. (b) The same representation is used for the SOP variations in the area designated by A in Fig. 2, we can see a monstar type singularity.
Fig. 4
Fig. 4 (a) SOP variations on the 20 × 20 pixels area designated by B in Fig. 2. The ellipse colour is representative of the DOP. Polarization streamlines are superimposed to the polarimetric ellipses. The black stripes represent the areas of left handed ellipticity, the other states are right handed ones. We can see that the ellipses major axis are oriented along lines crossing at the singularity. (b) SOP variations on the 20 × 20 pixels area marked C in Fig. 2. The same representation is used, the ellipses major axis are oriented along a spiral centered on the singularity. Two L lines are crossing close to the center of the singularity, demarcating two areas of opposite handedness.
Fig. 5
Fig. 5 SOP variations on a 20 × 20 pixels area, around a saddle type singularity. The ellipses colour is representative of the DOP. Polarization streamlines are superimposed to the polarimetric ellipses. The black stripes distinguish areas of left handed ellipticity from the right handed ones.

Equations (9)

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

S = { I = I V + I H Q = I V I H U = I 45 I + 45 V = I L I R
DOP = Q 2 + U 2 + V 2 I
tan ( 2 α ) = U Q
sin ( 2 ε ) = V I
I ( i , j , k ) = 1 2 S in ( i , j ) . S k out = 1 2 [ I in ( i , j ) I k out + Q in ( i , j ) Q k out + U in ( i , j ) U k out + V in ( i , j ) V k out ]
D I = Q x U y Q y U x
D L = [ ( 2 Q y + U x ) 2 3 U y ( 2 Q x U y ) ] . [ ( 2 Q x U y ) 2 + 3 U x ( 2 Q y + U x ) ] ( 2 Q x Q y + Q x U x Q y U y + 4 U x U y ) 2
D C = ( Q x U y Q y U x ) 2 ( Q x I y Q y I x ) 2 ( I x U y I y U x ) 2
ϒ = 2 ( Q x U y Q y U x ) Q x 2 + Q y 2 + U x 2 + U y 2

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