Abstract

We propose multihump N-component two-dimensional vector solitons for which each constituent carries a different topological charge. These new structures exhibit a unique triple-point phase diagram that is completely absent in the two-component limit.

© 2000 Optical Society of America

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  1. S. V. Manakov, Sov. Phys. JETP 38, 248 (1974).
  2. J. U. Kang, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, Phys. Rev. Lett. 76, 3699 (1996).
    [CrossRef] [PubMed]
  3. D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 53 (1988); M. V. Tratnik and J. E. Sipe, Phys. Rev. A 38, 2001 (1988).
    [CrossRef] [PubMed]
  4. M. Mitchell, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 80, 4657 (1998).
    [CrossRef]
  5. A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, Phys. Rev. Lett. 72, 1012 (1994).
    [CrossRef] [PubMed]
  6. S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, Opt. Lett. 13, 871 (1988); M. Shalaby and A. J. Barthelemy, IEEE J. Quantum Electron. 28, 2736 (1992).
    [CrossRef] [PubMed]
  7. D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, Appl. Phys. Lett. 68, 1763 (1996); Z. Chen, M. Segev, T. Coskun, and D. N. Christodoulides, Opt. Lett. 21, 1436 (1996).
    [CrossRef] [PubMed]
  8. N. Akhmediev, W. Krolikowski, and A. W. Snyder, Phys. Rev. Lett. 81, 4632 (1998).
    [CrossRef]
  9. C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, Phys. Rev. Lett. 83, 2332 (1999).
    [CrossRef]
  10. Note the distinction from quadratic solitons that carry a topological charge proposed by J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, Opt. Commun. 149, 77 (1998); G. Molina-Terriza, J. P. Torres, L. Torner, and J. M. Soto-Crespo, Opt. Commun. 158, 170 (1998). There the components are phase related, and the nonlinearity results from terms such as UiUj*, whereas in our case all such terms average out to zero.
    [CrossRef]
  11. Similar dark charge-carrying two-component solitons in self-defocusing media were proposed by A. P. Sheppard and M. Haelterman, Opt. Lett. 19, 859 (1994).
    [CrossRef] [PubMed]
  12. A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985).
    [CrossRef] [PubMed]
  13. M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
    [CrossRef]
  14. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  15. Z. H. Musslimani, M. Segev, and D. N. Christodoulides, “Composite multi-hump vector solitons carrying topological change,” submitted to Phys. Rev. Lett.
  16. W. J. Firth and D. V. Skryabin, Phys. Rev. Lett. 79, 2450 (1997).
    [CrossRef]
  17. M. Soljacic, S. Sears, and M. Segev, Phys. Rev. Lett. 81, 4851 (1998).
    [CrossRef]
  18. E. L. Dawes and J. H. Marburger, Phys. Rev. 179, 179 (1969); J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974); M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995); V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
    [CrossRef] [PubMed]
  19. E. A. Ostrovskaya, Y. S. Kivshar, D. V. Skryabin, and W. J. Firth, Phys. Rev. Lett. 83, 296 (1999).
    [CrossRef]
  20. N. N. Akhmediev, V. I. Korneev, and R. F. Nabiev, Opt. Lett. 17, 393 (1992); B. Luther-Davies, J. Christou, V. Tikhonenko, and Y. S. Kivshar, J. Opt. Soc. Am. B 14, 3045 (1997).
    [CrossRef] [PubMed]
  21. Z. H. Musslimani, M. Segev, A. Nepomnyashchy, and Y. S. Kivshar, Phys. Rev. E 60, R1170 (1999).
    [CrossRef]

1999 (3)

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, Phys. Rev. Lett. 83, 2332 (1999).
[CrossRef]

E. A. Ostrovskaya, Y. S. Kivshar, D. V. Skryabin, and W. J. Firth, Phys. Rev. Lett. 83, 296 (1999).
[CrossRef]

Z. H. Musslimani, M. Segev, A. Nepomnyashchy, and Y. S. Kivshar, Phys. Rev. E 60, R1170 (1999).
[CrossRef]

1998 (4)

N. Akhmediev, W. Krolikowski, and A. W. Snyder, Phys. Rev. Lett. 81, 4632 (1998).
[CrossRef]

M. Soljacic, S. Sears, and M. Segev, Phys. Rev. Lett. 81, 4851 (1998).
[CrossRef]

Note the distinction from quadratic solitons that carry a topological charge proposed by J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, Opt. Commun. 149, 77 (1998); G. Molina-Terriza, J. P. Torres, L. Torner, and J. M. Soto-Crespo, Opt. Commun. 158, 170 (1998). There the components are phase related, and the nonlinearity results from terms such as UiUj*, whereas in our case all such terms average out to zero.
[CrossRef]

M. Mitchell, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 80, 4657 (1998).
[CrossRef]

1997 (2)

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

W. J. Firth and D. V. Skryabin, Phys. Rev. Lett. 79, 2450 (1997).
[CrossRef]

1996 (2)

D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, Appl. Phys. Lett. 68, 1763 (1996); Z. Chen, M. Segev, T. Coskun, and D. N. Christodoulides, Opt. Lett. 21, 1436 (1996).
[CrossRef] [PubMed]

J. U. Kang, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, Phys. Rev. Lett. 76, 3699 (1996).
[CrossRef] [PubMed]

1994 (2)

1992 (1)

1988 (2)

1985 (1)

A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985).
[CrossRef] [PubMed]

1974 (1)

S. V. Manakov, Sov. Phys. JETP 38, 248 (1974).

1969 (1)

E. L. Dawes and J. H. Marburger, Phys. Rev. 179, 179 (1969); J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974); M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995); V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

Aitchison, J. S.

J. U. Kang, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, Phys. Rev. Lett. 76, 3699 (1996).
[CrossRef] [PubMed]

Akhmediev, N.

N. Akhmediev, W. Krolikowski, and A. W. Snyder, Phys. Rev. Lett. 81, 4632 (1998).
[CrossRef]

J. U. Kang, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, Phys. Rev. Lett. 76, 3699 (1996).
[CrossRef] [PubMed]

Akhmediev, N. N.

Anastassiou, C.

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, Phys. Rev. Lett. 83, 2332 (1999).
[CrossRef]

Carvalho, M. I.

D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, Appl. Phys. Lett. 68, 1763 (1996); Z. Chen, M. Segev, T. Coskun, and D. N. Christodoulides, Opt. Lett. 21, 1436 (1996).
[CrossRef] [PubMed]

Christodoulides, D. N.

M. Mitchell, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 80, 4657 (1998).
[CrossRef]

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, Appl. Phys. Lett. 68, 1763 (1996); Z. Chen, M. Segev, T. Coskun, and D. N. Christodoulides, Opt. Lett. 21, 1436 (1996).
[CrossRef] [PubMed]

D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 53 (1988); M. V. Tratnik and J. E. Sipe, Phys. Rev. A 38, 2001 (1988).
[CrossRef] [PubMed]

Z. H. Musslimani, M. Segev, and D. N. Christodoulides, “Composite multi-hump vector solitons carrying topological change,” submitted to Phys. Rev. Lett.

Coskun, T.

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

Dawes, E. L.

E. L. Dawes and J. H. Marburger, Phys. Rev. 179, 179 (1969); J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974); M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995); V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

Firth, W. J.

E. A. Ostrovskaya, Y. S. Kivshar, D. V. Skryabin, and W. J. Firth, Phys. Rev. Lett. 83, 296 (1999).
[CrossRef]

W. J. Firth and D. V. Skryabin, Phys. Rev. Lett. 79, 2450 (1997).
[CrossRef]

Giordmaine, J. A.

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, Phys. Rev. Lett. 83, 2332 (1999).
[CrossRef]

Haelterman, M.

Hewlett, S. J.

A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, Phys. Rev. Lett. 72, 1012 (1994).
[CrossRef] [PubMed]

Joseph, R. I.

Kang, J. U.

J. U. Kang, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, Phys. Rev. Lett. 76, 3699 (1996).
[CrossRef] [PubMed]

Kaplan, A. E.

A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985).
[CrossRef] [PubMed]

Kivshar, Y. S.

E. A. Ostrovskaya, Y. S. Kivshar, D. V. Skryabin, and W. J. Firth, Phys. Rev. Lett. 83, 296 (1999).
[CrossRef]

Z. H. Musslimani, M. Segev, A. Nepomnyashchy, and Y. S. Kivshar, Phys. Rev. E 60, R1170 (1999).
[CrossRef]

Korneev, V. I.

Krolikowski, W.

N. Akhmediev, W. Krolikowski, and A. W. Snyder, Phys. Rev. Lett. 81, 4632 (1998).
[CrossRef]

Lan, S.

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, Phys. Rev. Lett. 83, 2332 (1999).
[CrossRef]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Manakov, S. V.

S. V. Manakov, Sov. Phys. JETP 38, 248 (1974).

Marburger, J. H.

E. L. Dawes and J. H. Marburger, Phys. Rev. 179, 179 (1969); J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974); M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995); V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

Martin, J.

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, Phys. Rev. Lett. 83, 2332 (1999).
[CrossRef]

Mitchell, D. J.

A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, Phys. Rev. Lett. 72, 1012 (1994).
[CrossRef] [PubMed]

Mitchell, M.

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, Phys. Rev. Lett. 83, 2332 (1999).
[CrossRef]

M. Mitchell, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 80, 4657 (1998).
[CrossRef]

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

Musslimani, Z. H.

Z. H. Musslimani, M. Segev, A. Nepomnyashchy, and Y. S. Kivshar, Phys. Rev. E 60, R1170 (1999).
[CrossRef]

Z. H. Musslimani, M. Segev, and D. N. Christodoulides, “Composite multi-hump vector solitons carrying topological change,” submitted to Phys. Rev. Lett.

Nabiev, R. F.

Nepomnyashchy, A.

Z. H. Musslimani, M. Segev, A. Nepomnyashchy, and Y. S. Kivshar, Phys. Rev. E 60, R1170 (1999).
[CrossRef]

Ostrovskaya, E. A.

E. A. Ostrovskaya, Y. S. Kivshar, D. V. Skryabin, and W. J. Firth, Phys. Rev. Lett. 83, 296 (1999).
[CrossRef]

Petrov, D. V.

Note the distinction from quadratic solitons that carry a topological charge proposed by J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, Opt. Commun. 149, 77 (1998); G. Molina-Terriza, J. P. Torres, L. Torner, and J. M. Soto-Crespo, Opt. Commun. 158, 170 (1998). There the components are phase related, and the nonlinearity results from terms such as UiUj*, whereas in our case all such terms average out to zero.
[CrossRef]

Sears, S.

M. Soljacic, S. Sears, and M. Segev, Phys. Rev. Lett. 81, 4851 (1998).
[CrossRef]

Segev, M.

Z. H. Musslimani, M. Segev, A. Nepomnyashchy, and Y. S. Kivshar, Phys. Rev. E 60, R1170 (1999).
[CrossRef]

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, Phys. Rev. Lett. 83, 2332 (1999).
[CrossRef]

M. Mitchell, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 80, 4657 (1998).
[CrossRef]

M. Soljacic, S. Sears, and M. Segev, Phys. Rev. Lett. 81, 4851 (1998).
[CrossRef]

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, Appl. Phys. Lett. 68, 1763 (1996); Z. Chen, M. Segev, T. Coskun, and D. N. Christodoulides, Opt. Lett. 21, 1436 (1996).
[CrossRef] [PubMed]

Z. H. Musslimani, M. Segev, and D. N. Christodoulides, “Composite multi-hump vector solitons carrying topological change,” submitted to Phys. Rev. Lett.

Sheppard, A. P.

Shih, M.

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, Phys. Rev. Lett. 83, 2332 (1999).
[CrossRef]

Singh, S. R.

D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, Appl. Phys. Lett. 68, 1763 (1996); Z. Chen, M. Segev, T. Coskun, and D. N. Christodoulides, Opt. Lett. 21, 1436 (1996).
[CrossRef] [PubMed]

Skryabin, D. V.

E. A. Ostrovskaya, Y. S. Kivshar, D. V. Skryabin, and W. J. Firth, Phys. Rev. Lett. 83, 296 (1999).
[CrossRef]

W. J. Firth and D. V. Skryabin, Phys. Rev. Lett. 79, 2450 (1997).
[CrossRef]

Snyder, A. W.

N. Akhmediev, W. Krolikowski, and A. W. Snyder, Phys. Rev. Lett. 81, 4632 (1998).
[CrossRef]

A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, Phys. Rev. Lett. 72, 1012 (1994).
[CrossRef] [PubMed]

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Soljacic, M.

M. Soljacic, S. Sears, and M. Segev, Phys. Rev. Lett. 81, 4851 (1998).
[CrossRef]

Soto-Crespo, J. M.

Note the distinction from quadratic solitons that carry a topological charge proposed by J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, Opt. Commun. 149, 77 (1998); G. Molina-Terriza, J. P. Torres, L. Torner, and J. M. Soto-Crespo, Opt. Commun. 158, 170 (1998). There the components are phase related, and the nonlinearity results from terms such as UiUj*, whereas in our case all such terms average out to zero.
[CrossRef]

Stegeman, G. I.

Steiglitz, K.

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, Phys. Rev. Lett. 83, 2332 (1999).
[CrossRef]

Torner, L.

Note the distinction from quadratic solitons that carry a topological charge proposed by J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, Opt. Commun. 149, 77 (1998); G. Molina-Terriza, J. P. Torres, L. Torner, and J. M. Soto-Crespo, Opt. Commun. 158, 170 (1998). There the components are phase related, and the nonlinearity results from terms such as UiUj*, whereas in our case all such terms average out to zero.
[CrossRef]

Torres, J. P.

Note the distinction from quadratic solitons that carry a topological charge proposed by J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, Opt. Commun. 149, 77 (1998); G. Molina-Terriza, J. P. Torres, L. Torner, and J. M. Soto-Crespo, Opt. Commun. 158, 170 (1998). There the components are phase related, and the nonlinearity results from terms such as UiUj*, whereas in our case all such terms average out to zero.
[CrossRef]

Trillo, S.

Wabnitz, S.

Wright, E. M.

Appl. Phys. Lett. (1)

D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, Appl. Phys. Lett. 68, 1763 (1996); Z. Chen, M. Segev, T. Coskun, and D. N. Christodoulides, Opt. Lett. 21, 1436 (1996).
[CrossRef] [PubMed]

Opt. Commun. (1)

Note the distinction from quadratic solitons that carry a topological charge proposed by J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, Opt. Commun. 149, 77 (1998); G. Molina-Terriza, J. P. Torres, L. Torner, and J. M. Soto-Crespo, Opt. Commun. 158, 170 (1998). There the components are phase related, and the nonlinearity results from terms such as UiUj*, whereas in our case all such terms average out to zero.
[CrossRef]

Opt. Lett. (4)

Phys. Rev. (1)

E. L. Dawes and J. H. Marburger, Phys. Rev. 179, 179 (1969); J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974); M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995); V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

Phys. Rev. E (1)

Z. H. Musslimani, M. Segev, A. Nepomnyashchy, and Y. S. Kivshar, Phys. Rev. E 60, R1170 (1999).
[CrossRef]

Phys. Rev. Lett. (10)

E. A. Ostrovskaya, Y. S. Kivshar, D. V. Skryabin, and W. J. Firth, Phys. Rev. Lett. 83, 296 (1999).
[CrossRef]

N. Akhmediev, W. Krolikowski, and A. W. Snyder, Phys. Rev. Lett. 81, 4632 (1998).
[CrossRef]

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, Phys. Rev. Lett. 83, 2332 (1999).
[CrossRef]

W. J. Firth and D. V. Skryabin, Phys. Rev. Lett. 79, 2450 (1997).
[CrossRef]

M. Soljacic, S. Sears, and M. Segev, Phys. Rev. Lett. 81, 4851 (1998).
[CrossRef]

M. Mitchell, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 80, 4657 (1998).
[CrossRef]

A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, Phys. Rev. Lett. 72, 1012 (1994).
[CrossRef] [PubMed]

J. U. Kang, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, Phys. Rev. Lett. 76, 3699 (1996).
[CrossRef] [PubMed]

A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985).
[CrossRef] [PubMed]

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

Sov. Phys. JETP (1)

S. V. Manakov, Sov. Phys. JETP 38, 248 (1974).

Other (2)

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Z. H. Musslimani, M. Segev, and D. N. Christodoulides, “Composite multi-hump vector solitons carrying topological change,” submitted to Phys. Rev. Lett.

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Figures (3)

Fig. 1
Fig. 1

(a) Propagation constants μ0 (solid curves), μ1 (dotted curves), and μ2 (dashed curves) as a function of the normalized radius a of the induced waveguide (the V number). (b) Triple-point phase diagram for the (0, 1, 2) composite solitons at a=5.

Fig. 2
Fig. 2

u02 (dotted curves), u12 (dashed curves), u22 (dotted–dashed curves), and total intensity I (solid curves) for the (0, 1, 2) case. The parameters η1,η2 are (a) (1.5, 2), (b) (1, 2.7), (c) (1.8, 2.5), and (d) (1.8, 2.7).

Fig. 3
Fig. 3

u02 (dotted curves), u12 (short-dashed curves), u22 (dotted–dashed curves), u32 (long-dashed curves), and total intensity I (solid curves) for the (0, 1, 2, 3) case. The parameters η1,η2,η3 are (a) (1.2, 2, 2), (b) (1.8, 2.1, 2.1), (c) (1, 2, 2.5), and (d) (2.1, 2.1, 2.2).

Equations (5)

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

iUjz+2Uj+FIUj=0,
Ujr,θ,z=ujrexpimjθexpiμjz,
d2ujdr2+1rdujdr+κj2-mj2/r2uj=0,
ujr=ηjJmjκjr0raηjJmjκjaKmjμjaKmjμjrra,
μjKmj-1μjaKmjμja+κjJmj-1κjaJmjκja=0.

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