Abstract

Optical Möbius strips that surround points of circular polarization, C points, in a generic three-dimensional optical field are cloaked by lines of twisted ribbons attached to the C points. When cloaking occurs, the observable signed twist index that counts the number of half-twists (one or three), and also measures the handedness (right or left), of a generic Möbius strip is determined by the twisted ribbon cloaks. Although some cloaks can be detached, they can never all be removed.

© 2014 Optical Society of America

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2013 (1)

2012 (1)

2011 (4)

G. H. Yuan, S. B. Wei, and X.-C. Yuan, Opt. Lett. 36, 3499 (2011).

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

I. Freund, Opt. Lett. 36, 4506 (2011).
[CrossRef]

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

2010 (3)

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]

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]

2008 (2)

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

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

2007 (1)

2006 (3)

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

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

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]

2000 (1)

I. Freund, Opt. Commun. 173, 435 (2000).
[CrossRef]

1999 (1)

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

1979 (1)

N. D. Mermin, Rev. Mod. Phys. 159, 99 (1979).

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]

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]

Cardano, F.

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]

de Lisio, C.

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]

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]

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

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]

Flossmann, F.

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

Freund, I.

I. Freund, Opt. Lett. 36, 4506 (2011).
[CrossRef]

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

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

I. Freund, Opt. Lett. 35, 148 (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]

I. Freund, Opt. Commun. 173, 435 (2000).
[CrossRef]

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

Galvez, E. J.

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]

Karimi, E.

Khadka, S.

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]

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]

Lü, B.

Luo, Y.

Marrucci, L.

Mermin, N. D.

N. D. Mermin, Rev. Mod. Phys. 159, 99 (1979).

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]

Nomoto, S.

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]

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]

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]

Santamato, E.

Schoonover, R. W.

Schriemer, H.

Schubert, W. 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]

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]

Visser, T. D.

Wei, S. B.

G. H. Yuan, S. B. Wei, and X.-C. Yuan, Opt. Lett. 36, 3499 (2011).

Wheeldon, J. F.

Yuan, G. H.

G. H. Yuan, S. B. Wei, and X.-C. Yuan, Opt. Lett. 36, 3499 (2011).

Yuan, X.-C.

G. H. Yuan, S. B. Wei, and X.-C. Yuan, Opt. Lett. 36, 3499 (2011).

Zhang, S.

Appl. Opt. (1)

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. (8)

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

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

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

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

I. Freund, Opt. Commun. 173, 435 (2000).
[CrossRef]

Opt. Express (3)

Opt. Lett. (6)

Phys. Rev. Lett. (4)

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

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

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

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]

Rev. Mod. Phys. (1)

N. D. Mermin, Rev. Mod. Phys. 159, 99 (1979).

Other (1)

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

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Figures (3)

Fig. 1.
Fig. 1.

(a) τ=1/2 Möbius strip in a simulated random 3D optical field. (b) A τ=+1 twisted ribbon. In both figures the short straight black lines are axis α of the surrounding ellipses whose centers (red) lie on a circle Σ centered in (a) on the C point and in (b) on the origin of the twisted ribbon (specified later). An observer walking along the “top” edge (green) of the Möbius strip finds herself on the “bottom” edge (blue) after one 2π circuit; she returns to her starting point only after a second 2π circuit, showing that the Möbius strip is periodic in 4π, not 2π. For clarity, only half-axes are shown for the twisted ribbon in (b). In contrast to the Mobius strip in (a), a walk along the edge (green) of this ribbon returns to its starting point after a single 2π circuit.

Fig. 2.
Fig. 2.

The six physically possible negative index configurations. Lines of twisted ribbons labeled +1/1 by their twist index are colored red/blue. These red/blue lines are the ZZ that form the cloaks. The dotted lines are the corresponding Zρ. The small circle is ΣM, and is labeled by the twist index of the Möbius strip it contains. The text explains how the pattern of the ZZ and Zρ determines the twist index.

Fig. 3.
Fig. 3.

Cloak detachment under a small rotation of the observation plane P. Here (a) corresponds to Fig. 2(b) and (b) to Fig. 2(d). In both cases the two negative cloaks in Fig. 2 detach from the central C point, leaving attached a single positive cloak. (During detachment the positive index cloak in (b) rotates through 180°.) On the small, innermost circles in both (a) and (b) the signs of the twist indices of the Möbius strips switch from negative in Fig. 2 to positive here, as expected for a single positive cloak. The ZCs on the outermost circle in (a) are isosequenced with those in the parent Fig. 2(b), and so the twist index on this path returns to 1/2, the index in that figure. This is also the case on the outermost circle in (b), where here the ZCs are isosequenced with those in the parent Fig. 2(d). But what about the intermediate circle in (b)? The ZC sequence on this circle does not correspond to any of the sequences in Fig. 2, so we are unable to determine by inspection what the twist index is on this circle. Returning to basics, we start at the positive (red) ZC, go around the circle in the positive counterclockwise direction, and obtain the zero crossing sequence ZZZρZρZρZZZZ. Carrying out the erasures of the phase ratchet rules yields ZZZρ, so the absolute value of the index is 1/2, and the 3D figure on the circle is a one-half-twist Möbius strip. All cyclic permutations of the original ZC sequence give the same result, as does traversing the path in the negative, clockwise direction. But what is the sign of the index? The τM sum rule is not applicable because there are unattached ZCs that cross the circle. Resorting to direct measurement, we find that the index here is +1/2, as shown.

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