Abstract

Circularly polarized Gauss–Laguerre GL00 and GL01 laser beams that cross at their waists at a small angle are shown to generate a quasi-paraxial field that contains a line of circular polarization, a C line, surrounded by polarization ellipses whose major and minor axes generate multitwist Möbius strips with twist numbers that increase with the distance from the C point.

© 2010 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. London Ser. A 389, 279 (1983).
    [CrossRef]
  3. M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 457, 141 (2001).
    [CrossRef]
  4. M. R. Dennis, Opt. Commun. 213, 201 (2002).
    [CrossRef]
  5. M. V. Berry, J. Opt. A 6, 675 (2004).
    [CrossRef]
  6. I. Freund, Opt. Commun. 283, 1 (2010).
    [CrossRef]
  7. M. S. Soskin and M. V. Vasnetsov, in Optical Vortices, Vol. 228 of Horizons in World Physics, M.Vasnetsov and K.Staliunas, eds. (Nova Science, 1999), pp. 1-35.
  8. Such a field can be created, for example, as described by L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
    [CrossRef] [PubMed]
  9. N. B. Baranova, B. Ya Zel'dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).
  10. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 119, 604 (1995).
    [CrossRef]
  11. R. Dandliker, I. Marki, M. Salt, and A. Nesci, J. Opt. A, Pure Appl. Opt. 6, S189 (2004).
    [CrossRef]
  12. P. Tortora, R. Dandliker, W. Nakagawa, and L. Vaccaro, Opt. Commun. 259, 876 (2006).
    [CrossRef]
  13. C. Rockstuhl, I. Marki, T. Scharf, M. Salt, H. P. Herzig, and R. Dandliker, Curr. Nanosci. 2, 337 (2006).
    [CrossRef]
  14. P. Tortora, E. Descrovi, L. Aeschimann, L. Vaccaro, H. P. Herzig, and R. Dandliker, Ultramicroscopy 107, 158 (2007).
    [CrossRef]
  15. K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, Nat. Photonics 1, 53 (2007).
    [CrossRef]
  16. Z. H. Kim and S. R. Leone, Opt. Express 16, 1733 (2008).
    [CrossRef] [PubMed]
  17. R. J. Engelen, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 023902 (2009)
    [CrossRef] [PubMed]
  18. R. J. Engelen, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 049904(E) (2009).
    [CrossRef]
  19. 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]

2010

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

2009

R. J. Engelen, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 023902 (2009)
[CrossRef] [PubMed]

R. J. Engelen, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 049904(E) (2009).
[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] [PubMed]

2008

2007

P. Tortora, E. Descrovi, L. Aeschimann, L. Vaccaro, H. P. Herzig, and R. Dandliker, Ultramicroscopy 107, 158 (2007).
[CrossRef]

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, Nat. Photonics 1, 53 (2007).
[CrossRef]

2006

P. Tortora, R. Dandliker, W. Nakagawa, and L. Vaccaro, Opt. Commun. 259, 876 (2006).
[CrossRef]

C. Rockstuhl, I. Marki, T. Scharf, M. Salt, H. P. Herzig, and R. Dandliker, Curr. Nanosci. 2, 337 (2006).
[CrossRef]

Such a field can be created, for example, as described by L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

2004

R. Dandliker, I. Marki, M. Salt, and A. Nesci, J. Opt. A, Pure Appl. Opt. 6, S189 (2004).
[CrossRef]

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

2002

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

2001

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

1995

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 119, 604 (1995).
[CrossRef]

1983

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

1981

N. B. Baranova, B. Ya Zel'dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

Aeschimann, L.

P. Tortora, E. Descrovi, L. Aeschimann, L. Vaccaro, H. P. Herzig, and R. Dandliker, Ultramicroscopy 107, 158 (2007).
[CrossRef]

Baba, T.

R. J. Engelen, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 049904(E) (2009).
[CrossRef]

R. J. Engelen, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 023902 (2009)
[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]

Baranova, N. B.

N. B. Baranova, B. Ya Zel'dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

Basistiy, I. V.

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 119, 604 (1995).
[CrossRef]

Berry, M. V.

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

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

Choi, S. B.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, Nat. Photonics 1, 53 (2007).
[CrossRef]

Choi, W. J.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, Nat. Photonics 1, 53 (2007).
[CrossRef]

Dandliker, R.

P. Tortora, E. Descrovi, L. Aeschimann, L. Vaccaro, H. P. Herzig, and R. Dandliker, Ultramicroscopy 107, 158 (2007).
[CrossRef]

P. Tortora, R. Dandliker, W. Nakagawa, and L. Vaccaro, Opt. Commun. 259, 876 (2006).
[CrossRef]

C. Rockstuhl, I. Marki, T. Scharf, M. Salt, H. P. Herzig, and R. Dandliker, Curr. Nanosci. 2, 337 (2006).
[CrossRef]

R. Dandliker, I. Marki, M. Salt, and A. Nesci, J. Opt. A, Pure Appl. Opt. 6, S189 (2004).
[CrossRef]

Dennis, M. R.

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

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

Descrovi, E.

P. Tortora, E. Descrovi, L. Aeschimann, L. Vaccaro, H. P. Herzig, and R. Dandliker, Ultramicroscopy 107, 158 (2007).
[CrossRef]

Engelen, R. J.

R. J. Engelen, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 049904(E) (2009).
[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] [PubMed]

R. J. Engelen, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 023902 (2009)
[CrossRef] [PubMed]

Freund, I.

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

Herzig, H. P.

P. Tortora, E. Descrovi, L. Aeschimann, L. Vaccaro, H. P. Herzig, and R. Dandliker, Ultramicroscopy 107, 158 (2007).
[CrossRef]

C. Rockstuhl, I. Marki, T. Scharf, M. Salt, H. P. Herzig, and R. Dandliker, Curr. Nanosci. 2, 337 (2006).
[CrossRef]

Kihm, H. W.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, Nat. Photonics 1, 53 (2007).
[CrossRef]

Kihm, J. E.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, Nat. Photonics 1, 53 (2007).
[CrossRef]

Kim, D. S.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, Nat. Photonics 1, 53 (2007).
[CrossRef]

Kim, H.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, Nat. Photonics 1, 53 (2007).
[CrossRef]

Kim, J.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, Nat. Photonics 1, 53 (2007).
[CrossRef]

Kim, Z. H.

Kuipers, L.

R. J. Engelen, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 023902 (2009)
[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]

R. J. Engelen, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 049904(E) (2009).
[CrossRef]

Lee, B.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, Nat. Photonics 1, 53 (2007).
[CrossRef]

Lee, K. G.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, Nat. Photonics 1, 53 (2007).
[CrossRef]

Leone, S. R.

Lienau, C.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, Nat. Photonics 1, 53 (2007).
[CrossRef]

Mamaev, A. V.

N. B. Baranova, B. Ya Zel'dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

Manzo, C.

Such a field can be created, for example, as described by L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

Marki, I.

C. Rockstuhl, I. Marki, T. Scharf, M. Salt, H. P. Herzig, and R. Dandliker, Curr. Nanosci. 2, 337 (2006).
[CrossRef]

R. Dandliker, I. Marki, M. Salt, and A. Nesci, J. Opt. A, Pure Appl. Opt. 6, S189 (2004).
[CrossRef]

Marrucci, L.

Such a field can be created, for example, as described by L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

Mori, D.

R. J. Engelen, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 049904(E) (2009).
[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] [PubMed]

R. J. Engelen, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 023902 (2009)
[CrossRef] [PubMed]

Nakagawa, W.

P. Tortora, R. Dandliker, W. Nakagawa, and L. Vaccaro, Opt. Commun. 259, 876 (2006).
[CrossRef]

Nesci, A.

R. Dandliker, I. Marki, M. Salt, and A. Nesci, J. Opt. A, Pure Appl. Opt. 6, S189 (2004).
[CrossRef]

Nye, J. F.

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

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

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]

Paparo, D.

Such a field can be created, for example, as described by L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

Park, D. J.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, Nat. Photonics 1, 53 (2007).
[CrossRef]

Park, Q. H.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, Nat. Photonics 1, 53 (2007).
[CrossRef]

Pilipetskii, N.

N. B. Baranova, B. Ya Zel'dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

Rockstuhl, C.

C. Rockstuhl, I. Marki, T. Scharf, M. Salt, H. P. Herzig, and R. Dandliker, Curr. Nanosci. 2, 337 (2006).
[CrossRef]

Ropers, C.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, Nat. Photonics 1, 53 (2007).
[CrossRef]

Salt, M.

C. Rockstuhl, I. Marki, T. Scharf, M. Salt, H. P. Herzig, and R. Dandliker, Curr. Nanosci. 2, 337 (2006).
[CrossRef]

R. Dandliker, I. Marki, M. Salt, and A. Nesci, J. Opt. A, Pure Appl. Opt. 6, S189 (2004).
[CrossRef]

Scharf, T.

C. Rockstuhl, I. Marki, T. Scharf, M. Salt, H. P. Herzig, and R. Dandliker, Curr. Nanosci. 2, 337 (2006).
[CrossRef]

Shkukov, V. V.

N. B. Baranova, B. Ya Zel'dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

Soskin, M. S.

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 119, 604 (1995).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, in Optical Vortices, Vol. 228 of Horizons in World Physics, M.Vasnetsov and K.Staliunas, eds. (Nova Science, 1999), pp. 1-35.

Tortora, P.

P. Tortora, E. Descrovi, L. Aeschimann, L. Vaccaro, H. P. Herzig, and R. Dandliker, Ultramicroscopy 107, 158 (2007).
[CrossRef]

P. Tortora, R. Dandliker, W. Nakagawa, and L. Vaccaro, Opt. Commun. 259, 876 (2006).
[CrossRef]

Vaccaro, L.

P. Tortora, E. Descrovi, L. Aeschimann, L. Vaccaro, H. P. Herzig, and R. Dandliker, Ultramicroscopy 107, 158 (2007).
[CrossRef]

P. Tortora, R. Dandliker, W. Nakagawa, and L. Vaccaro, Opt. Commun. 259, 876 (2006).
[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] [PubMed]

Vasnetsov, M. V.

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 119, 604 (1995).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, in Optical Vortices, Vol. 228 of Horizons in World Physics, M.Vasnetsov and K.Staliunas, eds. (Nova Science, 1999), pp. 1-35.

Woo, H.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, Nat. Photonics 1, 53 (2007).
[CrossRef]

Ya Zel'dovich, B.

N. B. Baranova, B. Ya Zel'dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

Yoon, Y. C.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, Nat. Photonics 1, 53 (2007).
[CrossRef]

Curr. Nanosci.

C. Rockstuhl, I. Marki, T. Scharf, M. Salt, H. P. Herzig, and R. Dandliker, Curr. Nanosci. 2, 337 (2006).
[CrossRef]

J. Opt. A

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

J. Opt. A, Pure Appl. Opt.

R. Dandliker, I. Marki, M. Salt, and A. Nesci, J. Opt. A, Pure Appl. Opt. 6, S189 (2004).
[CrossRef]

JETP Lett.

N. B. Baranova, B. Ya Zel'dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

Nat. Photonics

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, Nat. Photonics 1, 53 (2007).
[CrossRef]

Opt. Commun.

P. Tortora, R. Dandliker, W. Nakagawa, and L. Vaccaro, Opt. Commun. 259, 876 (2006).
[CrossRef]

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 119, 604 (1995).
[CrossRef]

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

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

Opt. Express

Phys. Rev. Lett.

Such a field can be created, for example, as described by L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

R. J. Engelen, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 023902 (2009)
[CrossRef] [PubMed]

R. J. Engelen, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett. 102, 049904(E) (2009).
[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] [PubMed]

Proc. R. Soc. London Ser. A

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

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

Ultramicroscopy

P. Tortora, E. Descrovi, L. Aeschimann, L. Vaccaro, H. P. Herzig, and R. Dandliker, Ultramicroscopy 107, 158 (2007).
[CrossRef]

Other

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

M. S. Soskin and M. V. Vasnetsov, in Optical Vortices, Vol. 228 of Horizons in World Physics, M.Vasnetsov and K.Staliunas, eds. (Nova Science, 1999), pp. 1-35.

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Figures (3)

Fig. 1
Fig. 1

Computed multitwist optical Möbius strip generated by the major axes of the polarization ellipses on a circle surrounding a C point. An observer walking (gray arrow) along this circle sees the ellipse axes rotate in the clockwise direction (black arrow), generating a Möbius strip containing 13 half-twists that form a circular RH screw (helical worm gear) containing 6.5 turns (teeth). Here and in Fig. 3, for clarity only half-axes (thin lines) are shown, the circle of ellipse centers is colored gray, and axis end points that lie above (below) the plane of this circle are colored white (black).

Fig. 2
Fig. 2

Rotation angle α x y , plotted 0 π black to white, of the projection of the major axis α onto the x y plane.

Fig. 3
Fig. 3

Möbius strip containing 29 half-twists generated by minor axis β . Here the radius of r = 42 λ is a little more than twice that of Fig. 1.

Equations (8)

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

G 0 l = ( w 0 / W 0 l ) B 0 l   exp ( ( ρ 0 l ) 2 / ( W 0 l ) 2 ) exp ( i k ( ρ 0 l ) 2 / ( 2 R 0 l ) ) exp ( i k Z 0 l ) ,
X 0 l = x   cos   θ 0 l z   sin   θ 0 l ,     Z 0 l = + x   sin   θ 0 l + z   cos   θ 0 l .
E x = ( E 0 0 ) x cos   θ 0 0 + ( E 0 1 ) x cos   θ 0 1 ,     E y = ( E 0 0 ) y + ( E 0 1 ) y ,
E z = ( E 0 0 ) x sin   θ 0 0 ( E 0 1 ) x sin   θ 0 1 ,
( E 0 0 ) x = G 0 0 ,     ( E 0 0 ) y = i G 0 0 ,
( E 0 1 ) x = G 0 1 B 0 1 ( X 0 1 + i σ y ) / W 0 1 ,
( E 0 1 ) y = i G 0 1 B 0 1 ( X 0 1 + i σ y ) / W 0 1 .
τ calc int ( 4 r   sin ( θ ) / λ ) + 1 / 2.

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