Abstract

We present an experimental study of steady-state dark photorefractive screening solitons trapped in a bulk strontium barium niobate crystal. We compare experimental measurements with theoretical calculations of the soliton properties and find good agreement between theory and experiments. We confirm the shapepreserving behavior of the dark soliton by measuring its beam profile as it propagates throughout a specially cut crystal and by guiding a beam of a different wavelength.

© 1996 Optical Society of America

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  1. M. Segev, B. Crosignani, A. Yariv, B. Fischer, Phys. Rev. Lett. 68, 923 (1992).
    [CrossRef] [PubMed]
  2. G. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
    [CrossRef] [PubMed]
  3. Observations of steady-state self focusing effects were presented by M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
    [CrossRef]
  4. M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, A. Yariv, Phys. Rev. Lett. 73, 3211 (1994); M. Segev, M. Shih, G. C. Valley, J. Opt. Soc. B 13, 706 (1996).
    [CrossRef] [PubMed]
  5. D. N. Christodoulides, M. I. Carvalho, J. Opt. Soc. Am. B 12, 1628 (1995).
    [CrossRef]
  6. M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, P. DiPorto, Electron. Lett. 31, 826 (1995); Opt. Lett. 21, 324 (1996).
    [CrossRef] [PubMed]
  7. G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, M. Bashaw, Phys. Rev. A 50, R4457 (1994).
    [CrossRef] [PubMed]
  8. M. Taya, M. Bashaw, M. M. Fejer, M. Segev, G. C. Valley, Phys. Rev. A 52, 3095 (1995).
    [CrossRef] [PubMed]
  9. G. Duree, M. Morin, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, Phys. Rev. Lett. 74, 1978 (1995).
    [CrossRef] [PubMed]
  10. M. D. Iturbe-Castillo, J. J. Sanchez-Mondragon, S. I. Stepanov, M. B. Klein, B. A. Wechsler, Opt. Commun. 118, 515 (1995).
    [CrossRef]
  11. M. Morin, G. Duree, G. Salamo, M. Segev, Opt. Lett. 20, 2066 (1995).
    [CrossRef] [PubMed]
  12. G. A. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
    [CrossRef] [PubMed]
  13. G. R. Allan, S. R. Skinner, D. R. Andersen, A. L. Smirl, Opt. Lett. 16, 156 (1991).
    [PubMed]
  14. Y. S. Kivshar, X. Yang, Opt. Commun. 107, 93 (1994).
    [CrossRef]
  15. B. Luther-Davies, Y. Xiaoping, Opt. Lett. 17, 496 (1992).
    [CrossRef] [PubMed]

1995 (6)

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, G. C. Valley, Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

G. Duree, M. Morin, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

M. D. Iturbe-Castillo, J. J. Sanchez-Mondragon, S. I. Stepanov, M. B. Klein, B. A. Wechsler, Opt. Commun. 118, 515 (1995).
[CrossRef]

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, P. DiPorto, Electron. Lett. 31, 826 (1995); Opt. Lett. 21, 324 (1996).
[CrossRef] [PubMed]

D. N. Christodoulides, M. I. Carvalho, J. Opt. Soc. Am. B 12, 1628 (1995).
[CrossRef]

M. Morin, G. Duree, G. Salamo, M. Segev, Opt. Lett. 20, 2066 (1995).
[CrossRef] [PubMed]

1994 (4)

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, M. Bashaw, Phys. Rev. A 50, R4457 (1994).
[CrossRef] [PubMed]

Y. S. Kivshar, X. Yang, Opt. Commun. 107, 93 (1994).
[CrossRef]

Observations of steady-state self focusing effects were presented by M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, A. Yariv, Phys. Rev. Lett. 73, 3211 (1994); M. Segev, M. Shih, G. C. Valley, J. Opt. Soc. B 13, 706 (1996).
[CrossRef] [PubMed]

1993 (1)

G. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

1992 (2)

M. Segev, B. Crosignani, A. Yariv, B. Fischer, Phys. Rev. Lett. 68, 923 (1992).
[CrossRef] [PubMed]

B. Luther-Davies, Y. Xiaoping, Opt. Lett. 17, 496 (1992).
[CrossRef] [PubMed]

1991 (2)

G. R. Allan, S. R. Skinner, D. R. Andersen, A. L. Smirl, Opt. Lett. 16, 156 (1991).
[PubMed]

G. A. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Allan, G. R.

Andersen, D. R.

G. R. Allan, S. R. Skinner, D. R. Andersen, A. L. Smirl, Opt. Lett. 16, 156 (1991).
[PubMed]

G. A. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Bashaw, M.

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, G. C. Valley, Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, M. Bashaw, Phys. Rev. A 50, R4457 (1994).
[CrossRef] [PubMed]

Carvalho, M. I.

Christodoulides, D. N.

Crosignani, B.

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, P. DiPorto, Electron. Lett. 31, 826 (1995); Opt. Lett. 21, 324 (1996).
[CrossRef] [PubMed]

G. Duree, M. Morin, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, M. Bashaw, Phys. Rev. A 50, R4457 (1994).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, A. Yariv, Phys. Rev. Lett. 73, 3211 (1994); M. Segev, M. Shih, G. C. Valley, J. Opt. Soc. B 13, 706 (1996).
[CrossRef] [PubMed]

G. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

M. Segev, B. Crosignani, A. Yariv, B. Fischer, Phys. Rev. Lett. 68, 923 (1992).
[CrossRef] [PubMed]

DiPorto, P.

G. Duree, M. Morin, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, P. DiPorto, Electron. Lett. 31, 826 (1995); Opt. Lett. 21, 324 (1996).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, A. Yariv, Phys. Rev. Lett. 73, 3211 (1994); M. Segev, M. Shih, G. C. Valley, J. Opt. Soc. B 13, 706 (1996).
[CrossRef] [PubMed]

G. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Duree, G.

M. Morin, G. Duree, G. Salamo, M. Segev, Opt. Lett. 20, 2066 (1995).
[CrossRef] [PubMed]

G. Duree, M. Morin, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

G. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Fejer, M. M.

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, G. C. Valley, Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, M. Bashaw, Phys. Rev. A 50, R4457 (1994).
[CrossRef] [PubMed]

Fischer, B.

M. Segev, B. Crosignani, A. Yariv, B. Fischer, Phys. Rev. Lett. 68, 923 (1992).
[CrossRef] [PubMed]

Iturbe-Castillo, M. D.

M. D. Iturbe-Castillo, J. J. Sanchez-Mondragon, S. I. Stepanov, M. B. Klein, B. A. Wechsler, Opt. Commun. 118, 515 (1995).
[CrossRef]

Observations of steady-state self focusing effects were presented by M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Kaplan, A. E.

G. A. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Kivshar, Y. S.

Y. S. Kivshar, X. Yang, Opt. Commun. 107, 93 (1994).
[CrossRef]

Klein, M. B.

M. D. Iturbe-Castillo, J. J. Sanchez-Mondragon, S. I. Stepanov, M. B. Klein, B. A. Wechsler, Opt. Commun. 118, 515 (1995).
[CrossRef]

Luther-Davies, B.

Marquez-Aguilar, P. A.

Observations of steady-state self focusing effects were presented by M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Morin, M.

M. Morin, G. Duree, G. Salamo, M. Segev, Opt. Lett. 20, 2066 (1995).
[CrossRef] [PubMed]

G. Duree, M. Morin, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

Neurgaonkar, R. R.

G. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Regan, J. J.

G. A. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Salamo, G.

M. Morin, G. Duree, G. Salamo, M. Segev, Opt. Lett. 20, 2066 (1995).
[CrossRef] [PubMed]

G. Duree, M. Morin, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, P. DiPorto, Electron. Lett. 31, 826 (1995); Opt. Lett. 21, 324 (1996).
[CrossRef] [PubMed]

G. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Sanchez-Mondragon, J. J.

M. D. Iturbe-Castillo, J. J. Sanchez-Mondragon, S. I. Stepanov, M. B. Klein, B. A. Wechsler, Opt. Commun. 118, 515 (1995).
[CrossRef]

Observations of steady-state self focusing effects were presented by M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Segev, M.

M. Morin, G. Duree, G. Salamo, M. Segev, Opt. Lett. 20, 2066 (1995).
[CrossRef] [PubMed]

G. Duree, M. Morin, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, P. DiPorto, Electron. Lett. 31, 826 (1995); Opt. Lett. 21, 324 (1996).
[CrossRef] [PubMed]

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, G. C. Valley, Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, M. Bashaw, Phys. Rev. A 50, R4457 (1994).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, A. Yariv, Phys. Rev. Lett. 73, 3211 (1994); M. Segev, M. Shih, G. C. Valley, J. Opt. Soc. B 13, 706 (1996).
[CrossRef] [PubMed]

G. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

M. Segev, B. Crosignani, A. Yariv, B. Fischer, Phys. Rev. Lett. 68, 923 (1992).
[CrossRef] [PubMed]

Sharp, E.

G. Duree, M. Morin, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

G. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Shih, M.

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, P. DiPorto, Electron. Lett. 31, 826 (1995); Opt. Lett. 21, 324 (1996).
[CrossRef] [PubMed]

Shultz, J. L.

G. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Skinner, S. R.

Smirl, A. L.

Stepanov, S.

Observations of steady-state self focusing effects were presented by M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Stepanov, S. I.

M. D. Iturbe-Castillo, J. J. Sanchez-Mondragon, S. I. Stepanov, M. B. Klein, B. A. Wechsler, Opt. Commun. 118, 515 (1995).
[CrossRef]

Swartzlander, G. A.

G. A. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Taya, M.

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, G. C. Valley, Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

Valley, G. C.

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, G. C. Valley, Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, P. DiPorto, Electron. Lett. 31, 826 (1995); Opt. Lett. 21, 324 (1996).
[CrossRef] [PubMed]

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, M. Bashaw, Phys. Rev. A 50, R4457 (1994).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, A. Yariv, Phys. Rev. Lett. 73, 3211 (1994); M. Segev, M. Shih, G. C. Valley, J. Opt. Soc. B 13, 706 (1996).
[CrossRef] [PubMed]

Vysloukh, V.

Observations of steady-state self focusing effects were presented by M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Wechsler, B. A.

M. D. Iturbe-Castillo, J. J. Sanchez-Mondragon, S. I. Stepanov, M. B. Klein, B. A. Wechsler, Opt. Commun. 118, 515 (1995).
[CrossRef]

Xiaoping, Y.

Yang, X.

Y. S. Kivshar, X. Yang, Opt. Commun. 107, 93 (1994).
[CrossRef]

Yariv, A.

G. Duree, M. Morin, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, M. Bashaw, Phys. Rev. A 50, R4457 (1994).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, A. Yariv, Phys. Rev. Lett. 73, 3211 (1994); M. Segev, M. Shih, G. C. Valley, J. Opt. Soc. B 13, 706 (1996).
[CrossRef] [PubMed]

G. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

M. Segev, B. Crosignani, A. Yariv, B. Fischer, Phys. Rev. Lett. 68, 923 (1992).
[CrossRef] [PubMed]

Yin, H.

G. A. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

Observations of steady-state self focusing effects were presented by M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Electron. Lett. (1)

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, P. DiPorto, Electron. Lett. 31, 826 (1995); Opt. Lett. 21, 324 (1996).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

M. D. Iturbe-Castillo, J. J. Sanchez-Mondragon, S. I. Stepanov, M. B. Klein, B. A. Wechsler, Opt. Commun. 118, 515 (1995).
[CrossRef]

Y. S. Kivshar, X. Yang, Opt. Commun. 107, 93 (1994).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (2)

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, M. Bashaw, Phys. Rev. A 50, R4457 (1994).
[CrossRef] [PubMed]

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, G. C. Valley, Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett. (5)

G. Duree, M. Morin, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, A. Yariv, Phys. Rev. Lett. 73, 3211 (1994); M. Segev, M. Shih, G. C. Valley, J. Opt. Soc. B 13, 706 (1996).
[CrossRef] [PubMed]

M. Segev, B. Crosignani, A. Yariv, B. Fischer, Phys. Rev. Lett. 68, 923 (1992).
[CrossRef] [PubMed]

G. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

G. A. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Experimental setup.

Fig. 2
Fig. 2

Beam profiles (above) and photographs (below) of the input beam (left), the normally diffracting output (middle), and the soliton output beams (right) after propagation along a 5-mm SBN crystal.

Fig. 3
Fig. 3

Profiles of the normally diffracting notch (left) and of the dark soliton (middle) at various output planes of the specially cut SBN crystal. Right: The specially cut crystal and its crystalline orientation.

Fig. 4
Fig. 4

Theoretical (solid curve) and experimental (error bars) plots of the soliton width in dimensionless units of ξ as a function of the square root of the ratio between the soliton peak intensity and the background plus the dark irradiances (u). The measured soliton width in units of ξ does not change in the range 4.5 ≤ u ≤ 22.

Fig. 5
Fig. 5

Top-view photographs of a normally diffracting red beam in a 5-mm SBN crystal (above) and of the red beam guided in the waveguide induced by a dark screening soliton (below).

Equations (1)

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d 2 u / d ξ 2 - [ 1 - ( 1 + u 2 ) / ( 1 + u 2 ) ] u = 0 ,

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