Abstract

The twist numbers of circular optical Möbius strips and twisted ribbons are shown to obey the index theorem under rotation of the plane of observation, and under change in the radius of the path.

© 2011 Optical Society of America

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References

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  1. J. F. Nye, Natural Focusing and Fine Structure of Light (IOP, 1999).
  2. J. F. Nye, Proc. R. Soc. A 389, 279 (1983).
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  4. M. V. Berry and M. R. Dennis, Proc. R. Soc. A 457, 141 (2001).
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  5. M. V. Berry, Proc. SPIE 4403, 1 (2001).
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  6. M. R. Dennis, Opt. Commun. 213, 201 (2002).
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  7. A. I. Mokhun, M. S. Soskin, and I. Freund, Opt. Lett. 27, 995 (2002).
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  8. M. V. Berry, J. Opt. A 6, 675 (2004).
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  9. V. G. Denisenko, R. I. Egorov, and M. S. Soskin, JETP Lett. 80, 17 (2004).
    [CrossRef]
  10. Y. F. Chen, T. H. Lu, and K. F. Huang, Phys. Rev. Lett. 96, 033901 (2006).
    [CrossRef] [PubMed]
  11. R. W. Schoonover and T. D. Visser, Opt. Express 14, 5733 (2006).
    [CrossRef] [PubMed]
  12. Y. F. Chen, T. H. Lu, and K. F. Huang, Phys. Rev. Lett. 97, 233903 (2006).
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  13. S. Zhang, B. Hu, Y. Lockerman, P. Sebbah, and A. Z. Genack, J. Opt. Soc. Am. A 24, A33 (2007).
    [CrossRef]
  14. M. R. Dennis, Opt. Lett. 33, 2572 (2008).
    [CrossRef] [PubMed]
  15. F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, Phys. Rev. Lett. 100, 203902 (2008).
    [CrossRef] [PubMed]
  16. M. Burresi, R. J. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 033902 (2009).
    [CrossRef] [PubMed]
  17. M. R. Dennis, K. O’Holleran, and M. J. Padgett, Prog. Opt. 53, 293 (2009).
    [CrossRef]
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    [CrossRef] [PubMed]
  19. Y. Luo and B. Lü, J. Opt. Soc. Am. A 27, 578 (2010).
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  20. G. H. Yuan, S. B. Wei, and X.-C. Yuan, Opt. Lett. 36, 3479 (2011).
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  21. I. Freund, Opt. Commun. 283, 1 (2010).
    [CrossRef]
  22. M. R. Dennis, Opt. Lett. 36, 3765 (2011).
    [CrossRef] [PubMed]
  23. I. Freund, Opt. Commun. 249, 7 (2005).
    [CrossRef]
  24. I. Freund, Opt. Commun. 256, 220 (2005).
    [CrossRef]
  25. I. Freund, Opt. Commun. 283, 16 (2010).
    [CrossRef]
  26. S. H. Strogatz, in Nonlinear Dynamics and Chaos(Addison-Wesley, 1994), pp. 174–180.
  27. I. Freund, Opt. Commun. 242, 65 (2004).
    [CrossRef]
  28. I. Freund, Opt. Lett. 35, 148 (2010).
    [CrossRef] [PubMed]
  29. I. Freund, Opt. Commun. 284, 3816 (2011).
    [CrossRef]

2011 (3)

2010 (4)

2009 (3)

M. Burresi, R. J. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef] [PubMed]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Prog. Opt. 53, 293 (2009).
[CrossRef]

J. F. Wheeldon and H. Schriemer, Opt. Express 17, 2111 (2009).
[CrossRef] [PubMed]

2008 (2)

M. R. Dennis, Opt. Lett. 33, 2572 (2008).
[CrossRef] [PubMed]

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

2007 (1)

2006 (3)

Y. F. Chen, T. H. Lu, and K. F. Huang, Phys. Rev. Lett. 96, 033901 (2006).
[CrossRef] [PubMed]

R. W. Schoonover and T. D. Visser, Opt. Express 14, 5733 (2006).
[CrossRef] [PubMed]

Y. F. Chen, T. H. Lu, and K. F. Huang, Phys. Rev. Lett. 97, 233903 (2006).
[CrossRef]

2005 (2)

I. Freund, Opt. Commun. 249, 7 (2005).
[CrossRef]

I. Freund, Opt. Commun. 256, 220 (2005).
[CrossRef]

2004 (3)

I. Freund, Opt. Commun. 242, 65 (2004).
[CrossRef]

M. V. Berry, J. Opt. A 6, 675 (2004).
[CrossRef]

V. G. Denisenko, R. I. Egorov, and M. S. Soskin, JETP Lett. 80, 17 (2004).
[CrossRef]

2002 (2)

2001 (2)

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

M. V. Berry, Proc. SPIE 4403, 1 (2001).
[CrossRef]

1990 (1)

J. V. Hajnal, Proc. R. Soc. A 430, 413 (1990).
[CrossRef]

1983 (1)

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

Baba, T.

M. Burresi, R. J. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry, J. Opt. A 6, 675 (2004).
[CrossRef]

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

M. V. Berry, Proc. SPIE 4403, 1 (2001).
[CrossRef]

Burresi, M.

M. Burresi, R. J. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef] [PubMed]

Chen, Y. F.

Y. F. Chen, T. H. Lu, and K. F. Huang, Phys. Rev. Lett. 97, 233903 (2006).
[CrossRef]

Y. F. Chen, T. H. Lu, and K. F. Huang, Phys. Rev. Lett. 96, 033901 (2006).
[CrossRef] [PubMed]

Denisenko, V. G.

V. G. Denisenko, R. I. Egorov, and M. S. Soskin, JETP Lett. 80, 17 (2004).
[CrossRef]

Dennis, M. R.

M. R. Dennis, Opt. Lett. 36, 3765 (2011).
[CrossRef] [PubMed]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Prog. Opt. 53, 293 (2009).
[CrossRef]

M. R. Dennis, Opt. Lett. 33, 2572 (2008).
[CrossRef] [PubMed]

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

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

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

Egorov, R. I.

V. G. Denisenko, R. I. Egorov, and M. S. Soskin, JETP Lett. 80, 17 (2004).
[CrossRef]

Engelen, R. J.

M. Burresi, R. J. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef] [PubMed]

Flossmann, F.

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

Freund, I.

I. Freund, Opt. Commun. 284, 3816 (2011).
[CrossRef]

I. Freund, Opt. Lett. 35, 148 (2010).
[CrossRef] [PubMed]

I. Freund, Opt. Commun. 283, 16 (2010).
[CrossRef]

I. Freund, Opt. Commun. 283, 1 (2010).
[CrossRef]

I. Freund, Opt. Commun. 249, 7 (2005).
[CrossRef]

I. Freund, Opt. Commun. 256, 220 (2005).
[CrossRef]

I. Freund, Opt. Commun. 242, 65 (2004).
[CrossRef]

A. I. Mokhun, M. S. Soskin, and I. Freund, Opt. Lett. 27, 995 (2002).
[CrossRef]

Genack, A. Z.

Hajnal, J. V.

J. V. Hajnal, Proc. R. Soc. A 430, 413 (1990).
[CrossRef]

Hu, B.

Huang, K. F.

Y. F. Chen, T. H. Lu, and K. F. Huang, Phys. Rev. Lett. 97, 233903 (2006).
[CrossRef]

Y. F. Chen, T. H. Lu, and K. F. Huang, Phys. Rev. Lett. 96, 033901 (2006).
[CrossRef] [PubMed]

Kuipers, L.

M. Burresi, R. J. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef] [PubMed]

Lockerman, Y.

Lu, T. H.

Y. F. Chen, T. H. Lu, and K. F. Huang, Phys. Rev. Lett. 97, 233903 (2006).
[CrossRef]

Y. F. Chen, T. H. Lu, and K. F. Huang, Phys. Rev. Lett. 96, 033901 (2006).
[CrossRef] [PubMed]

Lü, B.

Luo, Y.

Mokhun, A. I.

Mori, D.

M. Burresi, R. J. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef] [PubMed]

Nye, J. F.

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

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

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Prog. Opt. 53, 293 (2009).
[CrossRef]

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

Opheij, A.

M. Burresi, R. J. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef] [PubMed]

Padgett, M. J.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Prog. Opt. 53, 293 (2009).
[CrossRef]

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

Schoonover, R. W.

Schriemer, H.

Sebbah, P.

Soskin, M. S.

V. G. Denisenko, R. I. Egorov, and M. S. Soskin, JETP Lett. 80, 17 (2004).
[CrossRef]

A. I. Mokhun, M. S. Soskin, and I. Freund, Opt. Lett. 27, 995 (2002).
[CrossRef]

Strogatz, S. H.

S. H. Strogatz, in Nonlinear Dynamics and Chaos(Addison-Wesley, 1994), pp. 174–180.

van Oosten, D.

M. Burresi, R. J. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef] [PubMed]

Visser, T. D.

Wei, S. B.

Wheeldon, J. F.

Yuan, G. H.

Yuan, X.-C.

Zhang, S.

J. Opt. A (1)

M. V. Berry, J. Opt. A 6, 675 (2004).
[CrossRef]

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

JETP Lett. (1)

V. G. Denisenko, R. I. Egorov, and M. S. Soskin, JETP Lett. 80, 17 (2004).
[CrossRef]

Opt. Commun. (7)

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

I. Freund, Opt. Commun. 283, 1 (2010).
[CrossRef]

I. Freund, Opt. Commun. 249, 7 (2005).
[CrossRef]

I. Freund, Opt. Commun. 256, 220 (2005).
[CrossRef]

I. Freund, Opt. Commun. 283, 16 (2010).
[CrossRef]

I. Freund, Opt. Commun. 242, 65 (2004).
[CrossRef]

I. Freund, Opt. Commun. 284, 3816 (2011).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Phys. Rev. Lett. (4)

Y. F. Chen, T. H. Lu, and K. F. Huang, Phys. Rev. Lett. 96, 033901 (2006).
[CrossRef] [PubMed]

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

M. Burresi, R. J. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef] [PubMed]

Y. F. Chen, T. H. Lu, and K. F. Huang, Phys. Rev. Lett. 97, 233903 (2006).
[CrossRef]

Proc. R. Soc. A (3)

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

J. V. Hajnal, Proc. R. Soc. A 430, 413 (1990).
[CrossRef]

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

Proc. SPIE (1)

M. V. Berry, Proc. SPIE 4403, 1 (2001).
[CrossRef]

Prog. Opt. (1)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Prog. Opt. 53, 293 (2009).
[CrossRef]

Other (2)

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

S. H. Strogatz, in Nonlinear Dynamics and Chaos(Addison-Wesley, 1994), pp. 174–180.

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Figures (3)

Fig. 1
Fig. 1

(a) Möbius strip and (b) twisted ribbon generated by axis α in a random field. The corresponding axis β structures are similar, but not identical. For clarity, here and throughout, only half axes are shown. The 180 ° discontinuity in (a) after a single 2 π circuit is the hallmark of a Möbius strip, reflecting the fact that Möbius strips have a single edge and are periodic in 4 π , not 2 π . Both the Möbius strip in (a) and the twisted ribbon in (b) form right-handed screws (as indicated by the arrows): the Möbius strip has one-half turn and twist index τ C = 1 / 2 ; the twisted ribbon has one full turn and twist index τ tr = 1 .

Fig. 2
Fig. 2

Conservation of twist index: rotational transformation of an axis α , τ C ( P 0 ) = 3 / 2 Möbius strip. (a) The Möbius strip in P 0 viewed from above. Throughout, axis endpoints lying above (below) the circle of ellipse centers, gray ring, are colored white (black); here these endpoints form a right-handed helix with one and a half turns. (b) Vortex mapping of the Möbius strip in (a): vortex phase Φ ( x , y ) coded 0 π black to white. On a closed circuit around the central singularity (C point), Φ winds through three 0 π cycles in the negative, clockwise direction, and the vortex topological charge is 3 / 2 . As the central C point is approached the three contour spikes narrow and appear to end when the widths of their tips become less than one pixel. However, on expanded scales (not shown) the contours approach the C point arbitrarily closely. (c) Φ after a 10   deg rotation away from P 0 to a final plane P f . The vortex (Möbius strip) in (b) splits into three vortices corresponding to a central τ C ( P f ) = + 1 / 2 Möbius strip, and two τ tr ( P f ) = 1 twisted ribbons. The τ ( P f ) = τ C ( P f ) + 2 τ tr ( P f ) = 3 / 2 Möbius strip on the large circle is nearly identical to the τ ( P 0 ) = 3 / 2 Möbius strip in (a). (d) Upper right-hand-side vortex in (c) enlarged. Φ winds through two 0 π cycles in the negative direction, and the vortex charge is 1 . (e)  τ tr ( P f ) = 1 twisted ribbon on the circular path in (d). Results similar to (d) and (e) are obtained for the upper left-hand-side vortex in (c). (f) Central vortex (C point) in (c) enlarged. Φ winds positively and negatively on the right-hand side, making no net contribution to the vortex charge of + 1 / 2 , which is due to the single 0 π positive, counterclockwise cycle on the left-hand side. (g)  τ C ( P f ) = + 1 / 2 Möbius strip on the circle in (f).

Fig. 3
Fig. 3

Multitwist Möbius strips and the index theorem. Shown is a short, three-step segment of an ascending τ staircase that begins at the central C point. (a)  Φ ( x , y ) coded as in Fig. 2. The numbers along the top list the vortex charges. (b) Phase map in (a) with twist indices τ (path labels) measured on the circular paths shown. As can be seen, the index theorem is obeyed for all paths.

Equations (6)

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

M ± 1 / 2 M ± 1 / 2 [ 0.147 / 0.148 ] ;
M ± 1 / 2 M 1 / 2 + T ± 1 [ 0.162 / 0.166 ] ;
M ± 1 / 2 M ± 1 / 2 + T 1 + T ± 1 [ 0.063 / 0.063 ] ;
M ± 1 / 2 M 1 / 2 + 2 T ± 1 + T 1 [ 0.024 / 0.025 ] ;
M ± 3 / 2 M ± 1 / 2 + T ± 1 [ 0.058 / 0.056 ] ;
M ± 3 / 2 M 1 / 2 + 2 T ± 1 [ 0.042 / 0.044 ] .

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