Abstract

Polarization singularities in paraxial vector optical fields are analyzed in terms of the phase singularities of complex Stokes scalar fields. Six independent relationships are obtained that connect the topological charges of these singularities on special closed contours with the charges of singularities that are enclosed by these contours. These relationships, which have been confirmed by experimental data and computer simulations, imply topological polarization correlations of an infinite range.

© 2002 Optical Society of America

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References

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  1. J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
    [CrossRef]
  2. M. Soskin and M. Vasnetsov, Photon. Sci. News 4(4), 21 (1999).
  3. Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
    [CrossRef]
  4. J. F. Nye, Natural Focusing and the Fine Structure of Light (IOP, Bristol, UK, 1999).
  5. M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 457, 141 (2001).
    [CrossRef]
  6. O. V. Angelsky, A. Mokhun, I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of components’ vortices and polarization singularities” (submitted to Opt. Commun.).
  7. A. I. Konukhov and L. A. Melnikov, J. Opt. B 3, S139 (2001).
    [CrossRef]
  8. I. Freund, Opt. Lett. 26, 1996 (2001).
    [CrossRef]
  9. I. Freund, Waves Random Media 8, 119 (1998).
  10. M. Born and E. W. Wolf, Principles of Optics (Pergamon, Oxford, 1959), Sec. 1.4.2.
  11. I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
    [CrossRef] [PubMed]
  12. I. Freund, Opt. Commun. 159, 99 (1999).
    [CrossRef]
  13. S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, Mass., 1994), Chap. 6.
  14. O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, Phys. Rev. E 65, 036602(5) (2002).
    [CrossRef]

2002 (1)

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, Phys. Rev. E 65, 036602(5) (2002).
[CrossRef]

2001 (3)

I. Freund, Opt. Lett. 26, 1996 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 457, 141 (2001).
[CrossRef]

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

1999 (2)

M. Soskin and M. Vasnetsov, Photon. Sci. News 4(4), 21 (1999).

I. Freund, Opt. Commun. 159, 99 (1999).
[CrossRef]

1998 (2)

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

I. Freund, Waves Random Media 8, 119 (1998).

1994 (1)

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef] [PubMed]

1974 (1)

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

Angelsky, O. V.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, Phys. Rev. E 65, 036602(5) (2002).
[CrossRef]

O. V. Angelsky, A. Mokhun, I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of components’ vortices and polarization singularities” (submitted to Opt. Commun.).

Berry, M. V.

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 457, 141 (2001).
[CrossRef]

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

Born, M.

M. Born and E. W. Wolf, Principles of Optics (Pergamon, Oxford, 1959), Sec. 1.4.2.

Dennis, M. R.

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 457, 141 (2001).
[CrossRef]

Freund, I.

I. Freund, Opt. Lett. 26, 1996 (2001).
[CrossRef]

I. Freund, Opt. Commun. 159, 99 (1999).
[CrossRef]

I. Freund, Waves Random Media 8, 119 (1998).

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef] [PubMed]

Kivshar, Y. S.

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Konukhov, A. I.

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

Luther-Davies, B.

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Melnikov, L. A.

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

Mokhun, A.

O. V. Angelsky, A. Mokhun, I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of components’ vortices and polarization singularities” (submitted to Opt. Commun.).

Mokhun, A. I.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, Phys. Rev. E 65, 036602(5) (2002).
[CrossRef]

Mokhun, I.

O. V. Angelsky, A. Mokhun, I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of components’ vortices and polarization singularities” (submitted to Opt. Commun.).

Mokhun, I. I.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, Phys. Rev. E 65, 036602(5) (2002).
[CrossRef]

Nye, J. F.

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

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

Shvartsman, N.

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef] [PubMed]

Soskin, M.

M. Soskin and M. Vasnetsov, Photon. Sci. News 4(4), 21 (1999).

Soskin, M. S.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, Phys. Rev. E 65, 036602(5) (2002).
[CrossRef]

O. V. Angelsky, A. Mokhun, I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of components’ vortices and polarization singularities” (submitted to Opt. Commun.).

Strogatz, S. H.

S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, Mass., 1994), Chap. 6.

Vasnetsov, M.

M. Soskin and M. Vasnetsov, Photon. Sci. News 4(4), 21 (1999).

Wolf, E. W.

M. Born and E. W. Wolf, Principles of Optics (Pergamon, Oxford, 1959), Sec. 1.4.2.

J. Opt. B (1)

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

Opt. Commun. (1)

I. Freund, Opt. Commun. 159, 99 (1999).
[CrossRef]

Opt. Lett. (1)

Photon. Sci. News (1)

M. Soskin and M. Vasnetsov, Photon. Sci. News 4(4), 21 (1999).

Phys. Rep. (1)

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Phys. Rev. A (1)

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef] [PubMed]

Phys. Rev. E (1)

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, Phys. Rev. E 65, 036602(5) (2002).
[CrossRef]

Proc. R. Soc. London Ser. A (2)

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 457, 141 (2001).
[CrossRef]

Waves Random Media (1)

I. Freund, Waves Random Media 8, 119 (1998).

Other (4)

M. Born and E. W. Wolf, Principles of Optics (Pergamon, Oxford, 1959), Sec. 1.4.2.

O. V. Angelsky, A. Mokhun, I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of components’ vortices and polarization singularities” (submitted to Opt. Commun.).

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

S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, Mass., 1994), Chap. 6.

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Figures (3)

Fig. 1
Fig. 1

Stokes vortices. Zero crossings Zm of Stokes parameters Sm are shown by solid curves; the side of Zm where Sm is negative is shown by a dashed curve, and positive (negative) Smn vortices are shown by filled (open) circles m,n=i,j,k. (a) Illustration of Eq. (5). (b), (c) Zero-crossing structures (phase ratchet) of (b) a positive and (c) a negative vortex. For the arrangement in (a), Aσjjqki=-1-1+1-1=+1, Bjσiqjk=+1+1+-1-1=+2, and Cjσkqij=+1+1++1-1++1+1+-1-1+-1+1+-1-1=+2, so, in accord with Eq. (5), 2A=B=C. Note that, without the factors σm, A=-1, and as required by the sign rule, B=C=0.

Fig. 2
Fig. 2

Experimental data6,14 for Ex vortices (circles) on an L line that encloses C points (squares). Positive (negative) charges are shown by filled (open) symbols. As the coordinate axis rotates from θ=80° to θ=160°, the C points and the L line remain stationary, but the vortices move along the L line [Eq. (3)] and collide and annihilate each other at θ170°. As can be seen, these data satisfy Eq. (5), albeit with zero net charge.

Fig. 3
Fig. 3

C points (squares) on a closed Z1 contour (dashed curve) that encloses Ex vortices (circles). Positive (negative) singularities are shown by filled (open) symbols, and the signs ± of σ are shown on both sides of all zero crossings. Where the Z1 contour is crossed by Z3, S31 vortices appear that are left as an exercise for the interested reader. These data are taken from the computer simulation described in Ref. 9.

Equations (5)

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s0=Ex2+Ey2, S1=s0-1Ex2-Ey2, S2=2s0-1 ReEx*Ey, S3=2s0-1 ImEx*Ey.
Ex=Ex cos θ-Ey sin θ, Ey=Ex sin θ+Ey cos θ,
S1=S1 cos 2θ-S2 sin 2θ, S2=S1 sin 2θ+S2 cos 2θ.
S12=S1+iS2, S23=S2+iS3, S31=S3+iS1.
2σkkqij=kσiqjk=kσjqik.

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