Abstract

Polarization singularities are shown to emerge spontaneously from the incoherent addition of uncorrelated optical fields that individually need not contain singularities. Examples of this phenomenon are given for both vector and ellipse fields. The incoherent addition of vector fields whose singularities have integer winding numbers is shown to yield fields whose singularities have half-integer winding numbers. These findings are used to make predictions about the singularities of the polarized component of the cosmic microwave background.

© 2004 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. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), pp. 219–276.
    [CrossRef]
  3. E. Wolf, Opt. Lett. 28, 5 (2003).
    [CrossRef] [PubMed]
  4. A. Dogariu and G. Popescu, Phys. Rev. Lett. 89, 243902 (2003).
    [CrossRef]
  5. J. F. Nye, Natural Focusing and the Fine Structure of Light (Institute of Physics, Bristol, UK, 1999).
  6. D. A. Kessler and I. Freund, Opt. Lett. 28, 111 (2003).
    [CrossRef] [PubMed]
  7. E. Wolf, Phys. Lett. A 312, 263 (2003).
    [CrossRef]
  8. I. Freund, Opt. Lett. 28, 2150 (2003).
    [CrossRef] [PubMed]
  9. M. Born and E. W. Wolf, Principles of Optics (Pergamon, Oxford, 1959).
  10. I. Freund, Opt. Commun. 196, 63 (2001).
    [CrossRef]
  11. J. M. Kovak, E. M. Leitch, C. Pryke, J. E. Carlstrom, M. W. Halverson, and W. L. Holtzapfel, Nature 420, 772 (2002).
    [CrossRef]
  12. T. Vachaspati and A. Lue, Phys. Rev. D 67, 121302(R) (2003).
    [CrossRef]
  13. M. V. Berry and J. H. Hannay, J. Phys. A 10, 2083 (1977).
    [CrossRef]
  14. M. R. Dennis, Opt. Commun. 213, 201 (2002).
    [CrossRef]
  15. A. D. Dolgov, A. G. Doroshkevich, I. D. Novikov, and D. I. Novikov, JETP Lett. 69, 427 (1999).
    [CrossRef]
  16. I. Freund, M. S. Soskin, and A. I. Mokhun, Opt. Commun. 208, 223 (2002).
    [CrossRef]
  17. S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, Mass., 1994), Chap. 6.

2003 (6)

E. Wolf, Opt. Lett. 28, 5 (2003).
[CrossRef] [PubMed]

A. Dogariu and G. Popescu, Phys. Rev. Lett. 89, 243902 (2003).
[CrossRef]

D. A. Kessler and I. Freund, Opt. Lett. 28, 111 (2003).
[CrossRef] [PubMed]

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

I. Freund, Opt. Lett. 28, 2150 (2003).
[CrossRef] [PubMed]

T. Vachaspati and A. Lue, Phys. Rev. D 67, 121302(R) (2003).
[CrossRef]

2002 (3)

J. M. Kovak, E. M. Leitch, C. Pryke, J. E. Carlstrom, M. W. Halverson, and W. L. Holtzapfel, Nature 420, 772 (2002).
[CrossRef]

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

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

2001 (1)

I. Freund, Opt. Commun. 196, 63 (2001).
[CrossRef]

1999 (1)

A. D. Dolgov, A. G. Doroshkevich, I. D. Novikov, and D. I. Novikov, JETP Lett. 69, 427 (1999).
[CrossRef]

1977 (1)

M. V. Berry and J. H. Hannay, J. Phys. A 10, 2083 (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 and J. H. Hannay, J. Phys. A 10, 2083 (1977).
[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).

Carlstrom, J. E.

J. M. Kovak, E. M. Leitch, C. Pryke, J. E. Carlstrom, M. W. Halverson, and W. L. Holtzapfel, Nature 420, 772 (2002).
[CrossRef]

Dennis, M. R.

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

Dogariu, A.

A. Dogariu and G. Popescu, Phys. Rev. Lett. 89, 243902 (2003).
[CrossRef]

Dolgov, A. D.

A. D. Dolgov, A. G. Doroshkevich, I. D. Novikov, and D. I. Novikov, JETP Lett. 69, 427 (1999).
[CrossRef]

Doroshkevich, A. G.

A. D. Dolgov, A. G. Doroshkevich, I. D. Novikov, and D. I. Novikov, JETP Lett. 69, 427 (1999).
[CrossRef]

Freund, I.

D. A. Kessler and I. Freund, Opt. Lett. 28, 111 (2003).
[CrossRef] [PubMed]

I. Freund, Opt. Lett. 28, 2150 (2003).
[CrossRef] [PubMed]

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

I. Freund, Opt. Commun. 196, 63 (2001).
[CrossRef]

Halverson, M. W.

J. M. Kovak, E. M. Leitch, C. Pryke, J. E. Carlstrom, M. W. Halverson, and W. L. Holtzapfel, Nature 420, 772 (2002).
[CrossRef]

Hannay, J. H.

M. V. Berry and J. H. Hannay, J. Phys. A 10, 2083 (1977).
[CrossRef]

Holtzapfel, W. L.

J. M. Kovak, E. M. Leitch, C. Pryke, J. E. Carlstrom, M. W. Halverson, and W. L. Holtzapfel, Nature 420, 772 (2002).
[CrossRef]

Kessler, D. A.

Kovak, J. M.

J. M. Kovak, E. M. Leitch, C. Pryke, J. E. Carlstrom, M. W. Halverson, and W. L. Holtzapfel, Nature 420, 772 (2002).
[CrossRef]

Leitch, E. M.

J. M. Kovak, E. M. Leitch, C. Pryke, J. E. Carlstrom, M. W. Halverson, and W. L. Holtzapfel, Nature 420, 772 (2002).
[CrossRef]

Lue, A.

T. Vachaspati and A. Lue, Phys. Rev. D 67, 121302(R) (2003).
[CrossRef]

Mokhun, A. I.

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

Novikov, D. I.

A. D. Dolgov, A. G. Doroshkevich, I. D. Novikov, and D. I. Novikov, JETP Lett. 69, 427 (1999).
[CrossRef]

Novikov, I. D.

A. D. Dolgov, A. G. Doroshkevich, I. D. Novikov, and D. I. Novikov, JETP Lett. 69, 427 (1999).
[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 (Institute of Physics, Bristol, UK, 1999).

Popescu, G.

A. Dogariu and G. Popescu, Phys. Rev. Lett. 89, 243902 (2003).
[CrossRef]

Pryke, C.

J. M. Kovak, E. M. Leitch, C. Pryke, J. E. Carlstrom, M. W. Halverson, and W. L. Holtzapfel, Nature 420, 772 (2002).
[CrossRef]

Soskin, M. S.

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

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), pp. 219–276.
[CrossRef]

Strogatz, S. H.

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

Vachaspati, T.

T. Vachaspati and A. Lue, Phys. Rev. D 67, 121302(R) (2003).
[CrossRef]

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), pp. 219–276.
[CrossRef]

Wolf, E.

Wolf, E. W.

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

J. Phys. A (1)

M. V. Berry and J. H. Hannay, J. Phys. A 10, 2083 (1977).
[CrossRef]

JETP Lett. (1)

A. D. Dolgov, A. G. Doroshkevich, I. D. Novikov, and D. I. Novikov, JETP Lett. 69, 427 (1999).
[CrossRef]

Nature (1)

J. M. Kovak, E. M. Leitch, C. Pryke, J. E. Carlstrom, M. W. Halverson, and W. L. Holtzapfel, Nature 420, 772 (2002).
[CrossRef]

Opt. Commun. (3)

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

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

I. Freund, Opt. Commun. 196, 63 (2001).
[CrossRef]

Opt. Lett. (3)

Phys. Lett. A (1)

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

Phys. Rev. D (1)

T. Vachaspati and A. Lue, Phys. Rev. D 67, 121302(R) (2003).
[CrossRef]

Phys. Rev. Lett. (1)

A. Dogariu and G. Popescu, Phys. Rev. Lett. 89, 243902 (2003).
[CrossRef]

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

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

Other (4)

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), pp. 219–276.
[CrossRef]

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

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

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

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Figures (3)

Fig. 1
Fig. 1

Vector field singularities. (a) Monochromatic vector field containing a central singularity with winding number +1. (b) Bichromatic Lissajous field composed of two randomly phased, incommensurate vector fields, shown in (c) and (d), that do not contain singularities. The bichromatic sum field is averaged over 50 optical cycles, and the lines that compose the Lissajous figures crowd together to form visually solid figures. The lines show the direction of the principal eigenvector of M that determines the orientation of the figures; the central figure, the singularity, is isotropic (square) and has no preferred axis. The winding number of this singularity is +1/2, reflecting the generic winding number of a positive singularity in the underlying field of coherency ellipses. (e), (f) Stokes phase Φ12 coded -π to +π, black to white, for the fields in (a) and (b). As expected, the winding number of the central Stokes singularity is twice that of the corresponding vector, or Lissajous, singularity.

Fig. 2
Fig. 2

Ellipse field singularities. (a) Monochromatic ellipse field containing a central singularity with winding number +1/2. (b) Lissajous field containing an emergent singularity with winding number +1/2 generated from Eqs. (1) with V=x+iy. Because the component fields are uncorrelated, the end point of E does not repetitively retrace the polarization ellipse as it does in (a). Instead, the end point of E tends to wander randomly8 and over the 50 optical cycles shown produces a visually solid figure composed of closely spaced, unresolved lines. The light-gray lines show the direction of the principal eigenvector of M. The central, singular, figure is circular and has no preferred axis. (c) Stokes phase Φ12 coded -π to +π, black to white. This phase portrait, which contains a central singularity with winding number +1, is the same for both (a) and (b). (d) Stokes six-petal flower with winding number +3 generated from Eqs. (1); V is given in Ref. 10. The corresponding Lissajous field is too complicated to be displayed on a small scale.

Fig. 3
Fig. 3

Stokes phase Φ12 of random fields coded -π to +π, black to white. Positive (negative) singularities are shown by filled white circles with black rims (filled black circles with white rims). (a) Random monochromatic vector field. The Stokes winding number of all singularities is ±2, as expected for a vector field. (b) Incoherent superposition of the field in (a) and 49 other, qualitatively similar but quantitatively different, random vector fields. All Stokes winding numbers in the superposition are ±1, as expected.

Equations (2)

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ExA=1+f/2,    EyA=i1-f/2,
ExB=1+ig/2,    EyB=i+g/2

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