Abstract

Polarization singularities are shown to be unavoidable features of three-dimensional optical lattices. These singularities take the form of lines of circular polarization, C lines, and lines of linear polarization, L lines. The polarization figures surrounding a C line (L line) rotate about the line with winding number ±1/2 (±1). C and L lines permeate the lattice, meander throughout the unit cell, and form closed loops. Surprisingly, every point in a linearly polarized optical lattice is found to be a singularity about which the surrounding polarization vectors rotate with an integer winding number.

© 2004 Optical Society of America

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References

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  1. L. Yuan, G. P. Wang, and X. Huang, Opt. Lett. 28, 1769 (2003).
    [CrossRef] [PubMed]
  2. J. K. Pachos and P. L. Knight, Phys. Rev. Lett. 91, 107902 (2003).
    [CrossRef]
  3. H. P. Buchler and G. Blatter, Phys. Rev. Lett. 91, 130404 (2003).
    [CrossRef]
  4. L. M. Duan, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 91, 090402 (2003).
    [CrossRef]
  5. X. L. Yang and L. Z. Cai, J. Mod. Opt. 50, 1445 (2003).
    [CrossRef]
  6. M. Born and E. W. Wolf, Principles of Optics (Pergamon, Oxford, England, 1959).
  7. J. F. Nye, Natural Focusing and the Fine Structure of Light (Institute of Physics, Bristol, England, 1999).
  8. I. Freund, “Coherency matrix description of optical polarization singularities,” J. Opt. A (to be published).
  9. I. Freund, Opt. Commun. 199, 47 (2001).
    [CrossRef]
  10. F. Gori, J. Opt. Soc. Am. A 18, 1612 (2001).
    [CrossRef]
  11. A. I. Konukhov and L. A. Melnikov, J. Opt. B 3, S139 (2001).
    [CrossRef]
  12. I. Freund, Opt. Lett. 26, 199 (2001).
    [CrossRef]
  13. I. Freund, Opt. Commun. 201, 251 (2002).
    [CrossRef]

2003

J. K. Pachos and P. L. Knight, Phys. Rev. Lett. 91, 107902 (2003).
[CrossRef]

H. P. Buchler and G. Blatter, Phys. Rev. Lett. 91, 130404 (2003).
[CrossRef]

L. M. Duan, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 91, 090402 (2003).
[CrossRef]

X. L. Yang and L. Z. Cai, J. Mod. Opt. 50, 1445 (2003).
[CrossRef]

L. Yuan, G. P. Wang, and X. Huang, Opt. Lett. 28, 1769 (2003).
[CrossRef] [PubMed]

2002

I. Freund, Opt. Commun. 201, 251 (2002).
[CrossRef]

2001

I. Freund, Opt. Lett. 26, 199 (2001).
[CrossRef]

F. Gori, J. Opt. Soc. Am. A 18, 1612 (2001).
[CrossRef]

I. Freund, Opt. Commun. 199, 47 (2001).
[CrossRef]

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

1999

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

1959

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

Blatter, G.

H. P. Buchler and G. Blatter, Phys. Rev. Lett. 91, 130404 (2003).
[CrossRef]

Born, M.

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

Buchler, H. P.

H. P. Buchler and G. Blatter, Phys. Rev. Lett. 91, 130404 (2003).
[CrossRef]

Cai, L. Z.

X. L. Yang and L. Z. Cai, J. Mod. Opt. 50, 1445 (2003).
[CrossRef]

Demler, E.

L. M. Duan, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 91, 090402 (2003).
[CrossRef]

Duan, L. M.

L. M. Duan, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 91, 090402 (2003).
[CrossRef]

Freund, I.

I. Freund, Opt. Commun. 201, 251 (2002).
[CrossRef]

I. Freund, Opt. Commun. 199, 47 (2001).
[CrossRef]

I. Freund, Opt. Lett. 26, 199 (2001).
[CrossRef]

I. Freund, “Coherency matrix description of optical polarization singularities,” J. Opt. A (to be published).

Gori, F.

Huang, X.

Knight, P. L.

J. K. Pachos and P. L. Knight, Phys. Rev. Lett. 91, 107902 (2003).
[CrossRef]

Konukhov, A. I.

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

Lukin, M. D.

L. M. Duan, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 91, 090402 (2003).
[CrossRef]

Melnikov, L. A.

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

Nye, J. F.

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

Pachos, J. K.

J. K. Pachos and P. L. Knight, Phys. Rev. Lett. 91, 107902 (2003).
[CrossRef]

Wang, G. P.

Wolf, E. W.

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

Yang, X. L.

X. L. Yang and L. Z. Cai, J. Mod. Opt. 50, 1445 (2003).
[CrossRef]

Yuan, L.

J. Mod. Opt.

X. L. Yang and L. Z. Cai, J. Mod. Opt. 50, 1445 (2003).
[CrossRef]

J. Opt. A

I. Freund, “Coherency matrix description of optical polarization singularities,” J. Opt. A (to be published).

J. Opt. B

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

J. Opt. Soc. Am. A

Opt. Commun.

I. Freund, Opt. Commun. 201, 251 (2002).
[CrossRef]

I. Freund, Opt. Commun. 199, 47 (2001).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

J. K. Pachos and P. L. Knight, Phys. Rev. Lett. 91, 107902 (2003).
[CrossRef]

H. P. Buchler and G. Blatter, Phys. Rev. Lett. 91, 130404 (2003).
[CrossRef]

L. M. Duan, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 91, 090402 (2003).
[CrossRef]

Other

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

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

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Figures (4)

Fig. 1
Fig. 1

Four-beam lattice at Z=0. The region covered is Δx,Δy=±3π,±3π. (a) Electric field energy density. (b) Discriminant DC. The unit cell contains four C points (black areas) at x,y=π/3,π,-π/3,π,π/3+δx,π+δy,-π/3-δx,π-δy, δx=1.248, δy=1.448. (c) Discriminant DL. Here the unit cell contains four L points (black areas) located at the corners of the square x,y=0,0,π,0,π,π,0,π.

Fig. 2
Fig. 2

(a) IC=+1/2 C point (filled black circle) at x,y,Z=π/3,π,0. The C point in Fig. 1(b) at x,y,Z=-π/3,π,0 also has index IC=+1/2, and the other two C points in the unit cell have index IC=-1/2. (b) IL=+1 L point in Fig. 1(c) at x,y,Z=0,0,0. IL of the four L points in Fig. 1(c) alternates ±1 around the square 0,0,0,π,0,0,π,π,0,0,π,0. The net index of the unit cell (C and L points) is zero, as it must be for a periodic structure. (c) Closed C line loop corresponding to the Fig. 1 C points at x,y=±π/3,π. (d) Meandering C lines corresponding to the Fig. 1 C points at x,y=π/3+δx,π+δy,-π/3-δx,π-δy.

Fig. 3
Fig. 3

Six-beam cubic lattice. (a) Polarization vectors in the plane y=0. (b) IL=-1 singularity at the arbitrary point x,y,Z=-0.844,-0.476,0.647.

Fig. 4
Fig. 4

IL=+2 singularity in a six-beam hexagonal optical lattice. (a) Vector field at x,y,Z=0,0,π. (b), (c) Argument Φ12, coded -π to +π black to white, of complex Stokes field11,12 S1+iS2 calculated in the plane of the figure. Φ12=2α provides a continuous, high-resolution representation of vector and ellipse fields, where α is the azimuthal orientation angle of the polarization figures in the plane. (b) Stokes singularity with winding number +4 corresponding to the +2 vector field singularity in (a). (c) x,y,Z=0,0,π+0.0025. The singularity splits into four singularities, three with Stokes winding number +1 (white circles with black rims), one with Stokes winding number -1 (black circle with white rim), together with three Stokes saddle points (white dots), thereby conserving both the winding number and the Poincaré–Hopf index.13

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