Abstract

We analyze self-trapping of optical beams in photorefractive media for low irradiances that are typical of cw lasers and high irradiances that are typical of short-pulse lasers and obtain bright and dark soliton solutions in one transverse dimension. Whereas in the low-intensity regime photorefractive screening solitons are trapped by the refractive-index change that is related to the electric field associated with ionized donors or acceptors, in the high-intensity regime the index is related to the field that is also associated with free carriers. Since the time necessary to form the soliton scales inversely with the intensity, one expects a very fast response time in the high-intensity regime, and this suggests attractive applications in optical switching and reconfigurable optical waveguides. We show that throughout the entire intensity range dark solitons require lower voltages and less energy per pulse for applications than do bright solitons.

© 1996 Optical Society of America

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  1. M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992).
    [CrossRef] [PubMed]
  2. B. Crosignani, M. Segev, D. Engin, P. DiPorto, A. Yariv, and G. Salamo, J. Opt. Soc. Am. B 10, 449 (1993).
    [CrossRef]
  3. D. N. Christodoulides and M. I. Carvalho, Opt. Lett. 19, 1714 (1994).
    [CrossRef] [PubMed]
  4. G. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
    [CrossRef] [PubMed]
  5. M. Segev, A. Yariv, G. Salamo, G. Duree, J. Shultz, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Photon. News, December, 1993, p. 8.
  6. M. Segev, A. Yariv, B. Crosignani, P. DiPorto, G. Duree, G. Salamo, and E. Sharp, Opt. Lett. 19, 1296 (1994).
    [CrossRef] [PubMed]
  7. G. Duree, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Lett. 19, 1195 (1994).
    [CrossRef] [PubMed]
  8. G. Duree, M. Morin, G. Salamo, M. Segev, A. Yariv, B. Crosignani, and P. DiPorto, Phys. Rev. Lett. 74, 1978 (1995).
    [CrossRef] [PubMed]
  9. M. Segev, G. Salamo, G. Duree, M. Morin, B. Crosignani, P. DiPorto, and A. Yariv, Opt. Photon. News, December1994, p. 9.
    [CrossRef]
  10. M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
    [CrossRef] [PubMed]
  11. D. N. Christodoulides and M. I. Carvalho, J. Opt. Soc. Am. B 12, 1628 (1995).
    [CrossRef]
  12. S. R. Singh and D. N. Christodoulides, Opt. Commun. 118, 569 (1995).
    [CrossRef]
  13. M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
    [CrossRef]
  14. M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995).
    [CrossRef]
  15. G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. Bashaw, Phys. Rev. A 50, R4457 (1994).
    [CrossRef]
  16. M. Taya, M. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, Phys. Rev. A 52, 3095 (1995). This observation was reported first by G. C. Valley, M. Segev, A. Yariv, M. Taya, M. Fejer, M. Bashaw, B. Crosignani, and P. DiPorto, “Spatial photovoltaic and screening solitons in photorefractive materials,” in Proceedings of the IEEE Conference on Nonlinear Optics (Institute of Electric and Electronics Engineers, New York, 1994), postdeadline paper PD7.
    [CrossRef] [PubMed]
  17. In this paper, as in all other papers on self-trapping of optical beams in photorefractive media, we use the term soliton in conjunction with nondiffracting self-trapped optical beams. That is, we use the broader definition of solitons that includes those in nonintegrable systems, as discussed in detail by V. G. Makhankov in his review article, Phys. Rep. 35, 1–128 (1978).
    [CrossRef]
  18. J. S. Aitchinson, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. Smith, Opt. Lett. 15, 471 (1990).
    [CrossRef]
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    [PubMed]
  20. M. Segev, G. C. Valley, S. R. Singh, M. I. Carvalho, and D. N. Christodoulides, Opt. Lett. 20, 1764 (1995).
    [CrossRef]
  21. S. R. Singh, M. I. Carvalho, and D. N. Christodoulides, Opt. Lett. 20, 2177 (1995).
    [CrossRef]
  22. A photorefractive spatial soliton can also guide a second beam, which may be of a different wavelength. This was first demonstrated for photovoltaic solitons in Ref. 16. Recently the photorefractive quasi-steady-state solitons have been shown to possess excellent guiding properties that may exceed those of the other types. See M. Morin, G. Duree, G. Salamo, and M. Segev, Opt. Lett. 20, 2066 (1995).
    [CrossRef] [PubMed]
  23. S. Gatz and J. Herrmann, J. Opt. Soc. Am. B 8, 2296 (1991).
    [CrossRef]
  24. G. C. Valley, J. Dubard, and A. L. Smirl, IEEE J. Quantum Electron. 26, 1058 (1990).
    [CrossRef]
  25. A. L. Smirl, K. Bohnert, G. C. Valley, R. A. Mullen, and T. F. Boggess, J. Opt. Soc. Am. B 6, 606 (1989).
    [CrossRef]
  26. G. C. Valley, J. Dubard, A. L. Smirl, and A. M. Glass, Opt. Lett. 14, 961 (1989).
    [CrossRef] [PubMed]
  27. T. F. Boggess, J. O. White, and G. C. Valley, J. Opt. Soc. Am. B 7, 2255 (1990).
    [CrossRef]
  28. S. M. Sze and G. Gibbons, Appl. Phys. Lett. 8, 111 (1966).
    [CrossRef]

1995 (8)

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

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

S. R. Singh and D. N. Christodoulides, Opt. Commun. 118, 569 (1995).
[CrossRef]

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995).
[CrossRef]

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, Phys. Rev. A 52, 3095 (1995). This observation was reported first by G. C. Valley, M. Segev, A. Yariv, M. Taya, M. Fejer, M. Bashaw, B. Crosignani, and P. DiPorto, “Spatial photovoltaic and screening solitons in photorefractive materials,” in Proceedings of the IEEE Conference on Nonlinear Optics (Institute of Electric and Electronics Engineers, New York, 1994), postdeadline paper PD7.
[CrossRef] [PubMed]

M. Segev, G. C. Valley, S. R. Singh, M. I. Carvalho, and D. N. Christodoulides, Opt. Lett. 20, 1764 (1995).
[CrossRef]

S. R. Singh, M. I. Carvalho, and D. N. Christodoulides, Opt. Lett. 20, 2177 (1995).
[CrossRef]

A photorefractive spatial soliton can also guide a second beam, which may be of a different wavelength. This was first demonstrated for photovoltaic solitons in Ref. 16. Recently the photorefractive quasi-steady-state solitons have been shown to possess excellent guiding properties that may exceed those of the other types. See M. Morin, G. Duree, G. Salamo, and M. Segev, Opt. Lett. 20, 2066 (1995).
[CrossRef] [PubMed]

1994 (7)

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

M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

M. Segev, G. Salamo, G. Duree, M. Morin, B. Crosignani, P. DiPorto, and A. Yariv, Opt. Photon. News, December1994, p. 9.
[CrossRef]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

M. Segev, A. Yariv, B. Crosignani, P. DiPorto, G. Duree, G. Salamo, and E. Sharp, Opt. Lett. 19, 1296 (1994).
[CrossRef] [PubMed]

G. Duree, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Lett. 19, 1195 (1994).
[CrossRef] [PubMed]

D. N. Christodoulides and M. I. Carvalho, Opt. Lett. 19, 1714 (1994).
[CrossRef] [PubMed]

1993 (3)

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

M. Segev, A. Yariv, G. Salamo, G. Duree, J. Shultz, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Photon. News, December, 1993, p. 8.

B. Crosignani, M. Segev, D. Engin, P. DiPorto, A. Yariv, and G. Salamo, J. Opt. Soc. Am. B 10, 449 (1993).
[CrossRef]

1992 (1)

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

1991 (2)

1990 (3)

1989 (2)

1978 (1)

In this paper, as in all other papers on self-trapping of optical beams in photorefractive media, we use the term soliton in conjunction with nondiffracting self-trapped optical beams. That is, we use the broader definition of solitons that includes those in nonintegrable systems, as discussed in detail by V. G. Makhankov in his review article, Phys. Rep. 35, 1–128 (1978).
[CrossRef]

1966 (1)

S. M. Sze and G. Gibbons, Appl. Phys. Lett. 8, 111 (1966).
[CrossRef]

Aitchinson, J. S.

Allan, G. R.

Andersen, D. A.

Bashaw, M.

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, Phys. Rev. A 52, 3095 (1995). This observation was reported first by G. C. Valley, M. Segev, A. Yariv, M. Taya, M. Fejer, M. Bashaw, B. Crosignani, and P. DiPorto, “Spatial photovoltaic and screening solitons in photorefractive materials,” in Proceedings of the IEEE Conference on Nonlinear Optics (Institute of Electric and Electronics Engineers, New York, 1994), postdeadline paper PD7.
[CrossRef] [PubMed]

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

Boggess, T. F.

Bohnert, K.

Carvalho, M. I.

Christodoulides, D. N.

Crosignani, B.

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

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995).
[CrossRef]

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

M. Segev, G. Salamo, G. Duree, M. Morin, B. Crosignani, P. DiPorto, and A. Yariv, Opt. Photon. News, December1994, p. 9.
[CrossRef]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

M. Segev, A. Yariv, B. Crosignani, P. DiPorto, G. Duree, G. Salamo, and E. Sharp, Opt. Lett. 19, 1296 (1994).
[CrossRef] [PubMed]

G. Duree, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Lett. 19, 1195 (1994).
[CrossRef] [PubMed]

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

M. Segev, A. Yariv, G. Salamo, G. Duree, J. Shultz, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Photon. News, December, 1993, p. 8.

B. Crosignani, M. Segev, D. Engin, P. DiPorto, A. Yariv, and G. Salamo, J. Opt. Soc. Am. B 10, 449 (1993).
[CrossRef]

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

DiPorto, P.

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995).
[CrossRef]

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

M. Segev, G. Salamo, G. Duree, M. Morin, B. Crosignani, P. DiPorto, and A. Yariv, Opt. Photon. News, December1994, p. 9.
[CrossRef]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

G. Duree, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Lett. 19, 1195 (1994).
[CrossRef] [PubMed]

M. Segev, A. Yariv, B. Crosignani, P. DiPorto, G. Duree, G. Salamo, and E. Sharp, Opt. Lett. 19, 1296 (1994).
[CrossRef] [PubMed]

M. Segev, A. Yariv, G. Salamo, G. Duree, J. Shultz, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Photon. News, December, 1993, p. 8.

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

B. Crosignani, M. Segev, D. Engin, P. DiPorto, A. Yariv, and G. Salamo, J. Opt. Soc. Am. B 10, 449 (1993).
[CrossRef]

Dubard, J.

G. C. Valley, J. Dubard, and A. L. Smirl, IEEE J. Quantum Electron. 26, 1058 (1990).
[CrossRef]

G. C. Valley, J. Dubard, A. L. Smirl, and A. M. Glass, Opt. Lett. 14, 961 (1989).
[CrossRef] [PubMed]

Duree, G.

A photorefractive spatial soliton can also guide a second beam, which may be of a different wavelength. This was first demonstrated for photovoltaic solitons in Ref. 16. Recently the photorefractive quasi-steady-state solitons have been shown to possess excellent guiding properties that may exceed those of the other types. See M. Morin, G. Duree, G. Salamo, and M. Segev, Opt. Lett. 20, 2066 (1995).
[CrossRef] [PubMed]

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

M. Segev, G. Salamo, G. Duree, M. Morin, B. Crosignani, P. DiPorto, and A. Yariv, Opt. Photon. News, December1994, p. 9.
[CrossRef]

M. Segev, A. Yariv, B. Crosignani, P. DiPorto, G. Duree, G. Salamo, and E. Sharp, Opt. Lett. 19, 1296 (1994).
[CrossRef] [PubMed]

G. Duree, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Lett. 19, 1195 (1994).
[CrossRef] [PubMed]

M. Segev, A. Yariv, G. Salamo, G. Duree, J. Shultz, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Photon. News, December, 1993, p. 8.

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

Engin, D.

B. Crosignani, M. Segev, D. Engin, P. DiPorto, A. Yariv, and G. Salamo, J. Opt. Soc. Am. B 10, 449 (1993).
[CrossRef]

Fejer, M. M.

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, Phys. Rev. A 52, 3095 (1995). This observation was reported first by G. C. Valley, M. Segev, A. Yariv, M. Taya, M. Fejer, M. Bashaw, B. Crosignani, and P. DiPorto, “Spatial photovoltaic and screening solitons in photorefractive materials,” in Proceedings of the IEEE Conference on Nonlinear Optics (Institute of Electric and Electronics Engineers, New York, 1994), postdeadline paper PD7.
[CrossRef] [PubMed]

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

Fischer, B.

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

Gatz, S.

Gibbons, G.

S. M. Sze and G. Gibbons, Appl. Phys. Lett. 8, 111 (1966).
[CrossRef]

Glass, A. M.

Herrmann, J.

Iturbe-Castillo, M. D.

M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Jackel, J. L.

Leaird, D. E.

Makhankov, V. G.

In this paper, as in all other papers on self-trapping of optical beams in photorefractive media, we use the term soliton in conjunction with nondiffracting self-trapped optical beams. That is, we use the broader definition of solitons that includes those in nonintegrable systems, as discussed in detail by V. G. Makhankov in his review article, Phys. Rep. 35, 1–128 (1978).
[CrossRef]

Marquez-Aguilar, P. A.

M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Morin, M.

Mullen, R. A.

Neurgaonkar, R.

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

Oliver, M. K.

Salamo, G.

A photorefractive spatial soliton can also guide a second beam, which may be of a different wavelength. This was first demonstrated for photovoltaic solitons in Ref. 16. Recently the photorefractive quasi-steady-state solitons have been shown to possess excellent guiding properties that may exceed those of the other types. See M. Morin, G. Duree, G. Salamo, and M. Segev, Opt. Lett. 20, 2066 (1995).
[CrossRef] [PubMed]

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

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995).
[CrossRef]

M. Segev, G. Salamo, G. Duree, M. Morin, B. Crosignani, P. DiPorto, and A. Yariv, Opt. Photon. News, December1994, p. 9.
[CrossRef]

M. Segev, A. Yariv, B. Crosignani, P. DiPorto, G. Duree, G. Salamo, and E. Sharp, Opt. Lett. 19, 1296 (1994).
[CrossRef] [PubMed]

G. Duree, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Lett. 19, 1195 (1994).
[CrossRef] [PubMed]

M. Segev, A. Yariv, G. Salamo, G. Duree, J. Shultz, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Photon. News, December, 1993, p. 8.

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

B. Crosignani, M. Segev, D. Engin, P. DiPorto, A. Yariv, and G. Salamo, J. Opt. Soc. Am. B 10, 449 (1993).
[CrossRef]

Sanchez-Mondragon, J. J.

M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Segev, M.

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

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995).
[CrossRef]

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, Phys. Rev. A 52, 3095 (1995). This observation was reported first by G. C. Valley, M. Segev, A. Yariv, M. Taya, M. Fejer, M. Bashaw, B. Crosignani, and P. DiPorto, “Spatial photovoltaic and screening solitons in photorefractive materials,” in Proceedings of the IEEE Conference on Nonlinear Optics (Institute of Electric and Electronics Engineers, New York, 1994), postdeadline paper PD7.
[CrossRef] [PubMed]

M. Segev, G. C. Valley, S. R. Singh, M. I. Carvalho, and D. N. Christodoulides, Opt. Lett. 20, 1764 (1995).
[CrossRef]

A photorefractive spatial soliton can also guide a second beam, which may be of a different wavelength. This was first demonstrated for photovoltaic solitons in Ref. 16. Recently the photorefractive quasi-steady-state solitons have been shown to possess excellent guiding properties that may exceed those of the other types. See M. Morin, G. Duree, G. Salamo, and M. Segev, Opt. Lett. 20, 2066 (1995).
[CrossRef] [PubMed]

G. Duree, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Lett. 19, 1195 (1994).
[CrossRef] [PubMed]

M. Segev, A. Yariv, B. Crosignani, P. DiPorto, G. Duree, G. Salamo, and E. Sharp, Opt. Lett. 19, 1296 (1994).
[CrossRef] [PubMed]

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

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

M. Segev, G. Salamo, G. Duree, M. Morin, B. Crosignani, P. DiPorto, and A. Yariv, Opt. Photon. News, December1994, p. 9.
[CrossRef]

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

M. Segev, A. Yariv, G. Salamo, G. Duree, J. Shultz, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Photon. News, December, 1993, p. 8.

B. Crosignani, M. Segev, D. Engin, P. DiPorto, A. Yariv, and G. Salamo, J. Opt. Soc. Am. B 10, 449 (1993).
[CrossRef]

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

Sharp, E.

G. Duree, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Lett. 19, 1195 (1994).
[CrossRef] [PubMed]

M. Segev, A. Yariv, B. Crosignani, P. DiPorto, G. Duree, G. Salamo, and E. Sharp, Opt. Lett. 19, 1296 (1994).
[CrossRef] [PubMed]

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

M. Segev, A. Yariv, G. Salamo, G. Duree, J. Shultz, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Photon. News, December, 1993, p. 8.

Shih, M.

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995).
[CrossRef]

Shultz, J.

M. Segev, A. Yariv, G. Salamo, G. Duree, J. Shultz, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Photon. News, December, 1993, p. 8.

Shultz, J. L.

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

Silberberg, Y.

Singh, S. R.

Skinner, S. R.

Smirl, A. L.

Smith, P. W.

Stepanov, S.

M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Sze, S. M.

S. M. Sze and G. Gibbons, Appl. Phys. Lett. 8, 111 (1966).
[CrossRef]

Taya, M.

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, Phys. Rev. A 52, 3095 (1995). This observation was reported first by G. C. Valley, M. Segev, A. Yariv, M. Taya, M. Fejer, M. Bashaw, B. Crosignani, and P. DiPorto, “Spatial photovoltaic and screening solitons in photorefractive materials,” in Proceedings of the IEEE Conference on Nonlinear Optics (Institute of Electric and Electronics Engineers, New York, 1994), postdeadline paper PD7.
[CrossRef] [PubMed]

Valley, G. C.

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, Phys. Rev. A 52, 3095 (1995). This observation was reported first by G. C. Valley, M. Segev, A. Yariv, M. Taya, M. Fejer, M. Bashaw, B. Crosignani, and P. DiPorto, “Spatial photovoltaic and screening solitons in photorefractive materials,” in Proceedings of the IEEE Conference on Nonlinear Optics (Institute of Electric and Electronics Engineers, New York, 1994), postdeadline paper PD7.
[CrossRef] [PubMed]

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995).
[CrossRef]

M. Segev, G. C. Valley, S. R. Singh, M. I. Carvalho, and D. N. Christodoulides, Opt. Lett. 20, 1764 (1995).
[CrossRef]

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

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

G. C. Valley, J. Dubard, and A. L. Smirl, IEEE J. Quantum Electron. 26, 1058 (1990).
[CrossRef]

T. F. Boggess, J. O. White, and G. C. Valley, J. Opt. Soc. Am. B 7, 2255 (1990).
[CrossRef]

A. L. Smirl, K. Bohnert, G. C. Valley, R. A. Mullen, and T. F. Boggess, J. Opt. Soc. Am. B 6, 606 (1989).
[CrossRef]

G. C. Valley, J. Dubard, A. L. Smirl, and A. M. Glass, Opt. Lett. 14, 961 (1989).
[CrossRef] [PubMed]

Vogel, E. M.

Vysloukh, V.

M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Weiner, A. M.

White, J. O.

Yariv, A.

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

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

M. Segev, G. Salamo, G. Duree, M. Morin, B. Crosignani, P. DiPorto, and A. Yariv, Opt. Photon. News, December1994, p. 9.
[CrossRef]

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

M. Segev, A. Yariv, B. Crosignani, P. DiPorto, G. Duree, G. Salamo, and E. Sharp, Opt. Lett. 19, 1296 (1994).
[CrossRef] [PubMed]

G. Duree, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Lett. 19, 1195 (1994).
[CrossRef] [PubMed]

B. Crosignani, M. Segev, D. Engin, P. DiPorto, A. Yariv, and G. Salamo, J. Opt. Soc. Am. B 10, 449 (1993).
[CrossRef]

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

M. Segev, A. Yariv, G. Salamo, G. Duree, J. Shultz, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Photon. News, December, 1993, p. 8.

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

Appl. Phys. Lett. (2)

M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

S. M. Sze and G. Gibbons, Appl. Phys. Lett. 8, 111 (1966).
[CrossRef]

Electron. Lett. (1)

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. C. Valley, J. Dubard, and A. L. Smirl, IEEE J. Quantum Electron. 26, 1058 (1990).
[CrossRef]

J. Opt. Soc. Am. B (5)

Opt. Commun. (1)

S. R. Singh and D. N. Christodoulides, Opt. Commun. 118, 569 (1995).
[CrossRef]

Opt. Lett. (9)

Opt. Photon. News (2)

M. Segev, G. Salamo, G. Duree, M. Morin, B. Crosignani, P. DiPorto, and A. Yariv, Opt. Photon. News, December1994, p. 9.
[CrossRef]

M. Segev, A. Yariv, G. Salamo, G. Duree, J. Shultz, B. Crosignani, P. DiPorto, and E. Sharp, Opt. Photon. News, December, 1993, p. 8.

Phys. Rep. (1)

In this paper, as in all other papers on self-trapping of optical beams in photorefractive media, we use the term soliton in conjunction with nondiffracting self-trapped optical beams. That is, we use the broader definition of solitons that includes those in nonintegrable systems, as discussed in detail by V. G. Makhankov in his review article, Phys. Rep. 35, 1–128 (1978).
[CrossRef]

Phys. Rev. A (2)

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

M. Taya, M. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, Phys. Rev. A 52, 3095 (1995). This observation was reported first by G. C. Valley, M. Segev, A. Yariv, M. Taya, M. Fejer, M. Bashaw, B. Crosignani, and P. DiPorto, “Spatial photovoltaic and screening solitons in photorefractive materials,” in Proceedings of the IEEE Conference on Nonlinear Optics (Institute of Electric and Electronics Engineers, New York, 1994), postdeadline paper PD7.
[CrossRef] [PubMed]

Phys. Rev. Lett. (4)

M. Segev, B. Crosignani, A. Yariv, and 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, and R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

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

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Figures (11)

Fig. 1
Fig. 1

Formation of a photorefractive low-intensity bright soliton.

Fig. 2
Fig. 2

Amplitude of a bright low-intensity soliton divided by the amplitude at 0, as a function of dimensionless length ξ, for several u0 values.

Fig. 3
Fig. 3

Width (FWHM of the intensity, u2, profile) of a bright low-intensity soliton (upper curve) as a function of u0 and of a dark low-intensity soliton (lower curve) as a function of u.

Fig. 4
Fig. 4

Formation of a photorefractive low-intensity dark soliton.

Fig. 5
Fig. 5

Amplitude of a dark low-intensity soliton divided by the amplitude at infinity, as a function of dimensionless length ξ, for several u values.

Fig. 6
Fig. 6

Amplitude of a bright high-intensity soliton divided by the amplitude at 0, as a function of dimensionless length ξ, for several u0 values.

Fig. 7
Fig. 7

Width (FWHM of the intensity) of a bright high-intensity soliton (upper curve) as a function of u0 and of a dark high-intensity soliton (lower curve) as a function of u.

Fig. 8
Fig. 8

Amplitude of a dark high-intensity soliton divided by the amplitude at 0, as a function of dimensionless length ξ, for several u values.

Fig. 9
Fig. 9

Width (FWHM of the intensity) of a bright saturation soliton as a function of u0 and of a dark saturation soliton as a function of u, for a = 10. Both curves coincide everywhere, and so appear as one curve.

Fig. 10
Fig. 10

Dimensionless functions I(u0)/u0 (lower curve, bright low-intensity soliton) and (1 + u2)I(u)/u (upper curve, dark low-intensity soliton) as functions of u0 and u, respectively.

Fig. 11
Fig. 11

Dimensionless functions I(u0)/u0 (upper curve, bright high-intensity soliton) and (1 + u2)1/4I(u) (lower curve, dark high-intensity soliton) as functions of u0 and u, respectively.

Equations (89)

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( s | A | 2 + s I b + β ) ( N d N d i ) γ n ˆ N d i = 0 ,
· J ˆ = · ( q μ n ˆ E ˆ + k B T μ n ˆ ) = 0 ,
· E ˆ + ( q / s ) ( n ˆ + N A N d i ) = 0 ,
( z i 2 k 2 x 2 ) A ( x , z ) = i k n b Δ n ( E ˆ ) A ( x , z ) ,
V = l / 2 l / 2 d x · E ˆ ,
A ( x , z ) = u ( x ) exp ( i Γ z ) ( I dark + I b ) 1 / 2 ,
n a ( 1 + | u | 2 ) ( 1 N ) / N = 0 ,
J = n E + 2 n = const . ,
( N 1 / r n ) 1 E = 0 ,
l / d + / 2 d / 2 d d ξ E = 0 ,
u = ± ( Γ / b + E ) u .
n E + 2 n + [ d / / 2 d / 2 d d ξ / n ] 1 = 0 ,
u = u [ 0 ] + ( 1 + 2 ) u [ 1 ] + ( 1 + 2 ) 2 u [ 2 ] + O [ ( 1 + 2 ) 3 ] + ,
n = n [ 0 ] + ( 1 + 2 ) n [ 1 ] + ( 1 + 2 ) 2 n [ 2 ] + O [ ( 1 + 2 ) 3 ] + ,
N = N [ 0 ] + ( 1 + 2 ) N [ 1 ] + ( 1 + 2 ) 2 N [ 2 ] + O [ ( 1 + 2 ) 3 ] + ,
E = E [ 0 ] + ( 1 + 2 ) E [ 1 ] + ( 1 + 2 ) 2 E [ 2 ] + O [ ( 1 + 2 ) 3 ] + .
n [ 0 ] 2 + n [ 0 ] [ a ( u [ 0 ] 2 + 1 ) + 1 / r ] a ( u [ 0 ] 2 + 1 ) ( 1 1 / r ) = 0 ,
n [ 0 ] = 1 / 2 ( [ a ( u [ 0 ] 2 + 1 ) + 1 / r ] + { [ a ( u [ 0 ] 2 + 1 ) + 1 / r ] 2 + 4 a ( 1 1 / r ) ( u [ 0 ] 2 + 1 ) } 1 / 2 ) .
E [ 0 ] = [ n [ 0 ] ( d / ) / 2 d / 2 d d ξ / n [ 0 ] ] 1 .
4 a r ( u 2 + 1 ) 1 / r 1.
n = a r ( u 2 + 1 ) ,
E = η / ( u 2 + 1 ) ,
u ± ( η u 2 + 1 Γ b ) u = 0 ,
1 / r a ( u 2 + 1 ) 1.
n = [ a ( u 2 + 1 ) ] 1 / 2 ,
E = η / ( u 2 + 1 ) 1 / 2 ,
u ± ( η ( u 2 + 1 ) 1 / 2 Γ b ) u = 0 ,
1 / r 1 a ( u 2 + 1 ) / 4.
n = 1 1 a ( u 2 + 1 ) ,
E = [ 1 + 1 a ( u 2 + 1 ) ] ,
( d / ) / 2 d / 2 d d ξ ( 1 + 1 a ( u 2 + 1 ) ) 1
u ± [ 1 a ( u 2 + 1 ) ( Γ b 1 ) ] u = 0 ,
p 2 p 0 2 = ± [ Γ b ( u 2 u 0 2 ) η ln ( u 2 + 1 u 0 2 + 1 ) ] ,
u = ( δ 1 u 2 + 1 ) u ,
u = ( u 0 2 2 u 2 ) u ,
p 0 2 = ± [ ( u + 1 ) ln ( u + 1 ) u 2 ] .
u = ( 1 u 2 + 1 u 2 + 1 ) u ,
u = ( u 2 u 2 ) u ,
p 2 p 0 2 = ± { Γ b ( u 2 u 0 2 ) 2 η [ ( u 2 + 1 ) 1 / 2 ( u 0 2 + 1 ) 1 / 2 ] } ,
u = ( δ 1 ( u 2 + 1 ) 1 / 2 ) u ,
u = ( u 0 2 4 u 2 2 ) u ,
p 0 2 = ± { [ ( u 2 + 1 ) 1 / 2 1 ] 2 } .
u = [ 1 ( u 2 + 1 u 2 + 1 ) 1 / 2 ] u ,
p 2 p 0 2 = ± [ ( Γ b 1 ) ( u 2 u 0 2 ) 1 a ln ( u 2 + 1 u 0 2 + 1 ) ] ,
δ ( Γ b 1 ) = ln ( u 0 2 + 1 ) a u 0 2 .
u = ( δ 1 a ( u 2 + 1 ) ) u ,
δ ( Γ b 1 ) = 1 a ( u 2 + 1 ) ,
p 0 2 = ± 1 a [ ln ( u 2 + 1 ) u 2 u 2 + 1 ] .
u = ( δ 1 a ( u 2 + 1 ) ) u ,
u ( 0 ) = p 0 = 1 a [ ln ( u 2 + 1 ) u 2 u 2 + 1 ] 1 / 2 .
1 u 2 + 1 = 1 u 2 u 2 + 1 .
A = η 1 = 2 d 0 / 2 d d ξ / ( u 2 + 1 ) = 2 d u 0 0 d u ( d u / d ξ ) ( u 2 + 1 ) ,
A = A 1 + A 2 = 2 d × [ u 0 u 0 d u ( d u / d ξ ) ( u 2 + 1 ) + u 0 0 d u ( d u / d ξ ) ( u 2 + 1 ) ] ,
A 2 2 d lim 0 u 0 0 d u ( d u / d ξ ) = 2 d lim 0 ξ [ u 0 ] / 2 d d ξ = 1 2 d lim 0 ξ [ u 0 ] .
ξ [ u 0 ] = u 0 u 0 d u ( d u / d ξ ) ,
A = 1 + 2 d lim 0 u 0 u 0 d u ( d u / d ξ ) ( 1 1 + u 2 1 ) .
δ = Γ b = η ln ( u 0 2 + 1 ) u 0 2 .
d u d ξ = η [ ln ( u 2 + 1 ) u 2 u 0 2 ln ( u 0 2 + 1 ) ] 1 / 2
η + 2 η ( d / ) I ( u 0 ) 1 = 0 ,
I ( u 0 ) = lim 0 u 0 u 0 d u ( 1 1 + u 2 1 ) × [ ln ( u 2 + 1 ) u 2 u 0 2 ln ( u 0 2 + 1 ) ] 1 / 2 .
R = 2 ( d / ) I ( u 0 ) + 2 ( d / ) 2 I 2 ( u 0 ) + ,
A = η 1 = 2 d 0 / 2 d d ξ / ( u 2 + 1 ) = ( 2 d / ) 0 u d u ( d u / d ξ ) ( u 2 + 1 ) ,
A = A 1 + A 2 = 2 d [ 0 u ( 1 ) d u ( d u / d ξ ) ( u 2 + 1 ) + u ( 1 ) u d u ( d u / d ξ ) ( u 2 + 1 ) ]
A 2 2 d ( 1 + u 2 ) lim 0 u ( 1 ) u d u ( d u / d ξ ) = 2 d ( 1 + u 2 ) lim 0 ξ [ u ( 1 ) ] l / 2 d d ξ = 1 ( 1 + u 2 ) { 1 2 d lim 0 ξ [ u ( 1 ) ] } .
ξ [ u ( 1 ) ] = 0 u ( 1 ) d u ( d u / d ξ ) ,
A = 1 1 + u 2 + 2 d lim 0 0 u ( 1 ) d u ( d u / d ξ ) × ( 1 1 + u 2 1 1 + u 2 ) .
d u d ξ = η [ ln ( 1 + u 2 1 + u 2 ) u 2 u 2 1 + u 2 ] 1 / 2 .
η + 2 η ( d / ) ( 1 + u 2 ) 3 / 2 I ( u ) ( 1 + u 2 ) = 0 ,
I ( u ) = lim 0 0 u ( 1 ) d u ( 1 1 + u 2 1 1 + u 2 ) × [ ( 1 + u 2 ) ln ( 1 + u 2 1 + u 2 ) ( u 2 u 2 ) ] 1 / 2 .
R = 2 ( d / ) ( 1 + u 2 ) I ( u ) + 2 ( d / ) 2 ( 1 + u 2 ) 2 I 2 ( u ) + .
A = η 1 = 2 d 0 / 2 d d ξ / ( u 2 + 1 ) 1/2 = ( 2 d / ) u 0 0 d u ( d u / d ξ ) ( u 2 + 1 ) 1 / 2 ,
A = A 1 + A 2 = 2 d [ u 0 u 0 d u ( d u / d ξ ) ( u 2 + 1 ) 1 / 2 + u 0 0 d u ( d u / d ξ ) ( u 2 + 1 ) 1 / 2 ] ,
A 2 2 d lim 0 u 0 0 d u d u / d ξ = 2 d lim 0 ξ [ u 0 ] / 2 d d ξ = 1 2 d lim 0 ξ [ u 0 ] .
ξ [ u 0 ] = u 0 u 0 d u ( d u / d ξ ) ,
A = 1 + 2 d lim 0 u 0 u 0 d u ( d u / d ξ ) ( 1 ( 1 + u 2 ) 1 / 2 1 ) .
δ = Γ / b = 2 η [ ( u 0 2 + 1 ) 1 / 2 1 ] / u 0 2 .
d u d ξ = ( 2 η ) 1 / 2 { ( u 2 + 1 ) 1 / 2 1 u 2 u 0 2 [ ( u 0 2 + 1 ) 1 / 2 1 ] } 1 / 2
η + 2 η ( d / ) I ( u 0 ) 1 = 0 ,
I ( u 0 ) = 1 2 lim 0 u 0 u 0 d u [ 1 ( 1 + u 2 ) 1 / 2 1 ] × { ( u 2 + 1 ) 1 / 2 1 u 2 u 0 2 [ ( u 0 2 + 1 ) 1 / 2 1 ] } 1 / 2 .
R = 2 ( d / ) I ( u 0 ) + 2 ( d / ) 2 I 2 ( u 0 ) +
A = η 1 = 2 d 0 / 2 d d ξ / ( u 2 + 1 ) 1/2 = 2 d 0 u d u ( d u / d ξ ) ( u 2 + 1 ) 1 / 2 ,
A = A 1 + A 2 = 2 d [ 0 u ( 1 ) d u ( d u / d ξ ) ( u 2 + 1 ) 1 / 2 + u ( 1 ) u d u ( d u / d ξ ) ( u 2 + 1 ) 1 / 2 ] ,
A 2 2 d ( 1 + u 2 ) 1 / 2 lim 0 u ( 1 ) u d u ( d u / d ξ ) = 2 d ( 1 + u 2 ) 1 / 2 lim 0 lim 0 ξ [ u ( 1 ) ] / 2 d d ξ = 1 ( 1 + u 2 ) 1 / 2 { 1 2 d lim 0 ξ [ u ( 1 ) ] } .
ξ [ u ( 1 ) ] = 0 u ( 1 ) d u ( d u / d ξ ) ,
A = 1 ( 1 + u 2 ) 1 / 2 + 2 d lim 0 0 u ( 1 ) d u ( d u / d ξ ) × [ 1 ( 1 + u 2 ) 1 / 2 1 ( 1 + u 2 ) 1 / 2 ] .
d u / d ξ = η { 2 [ ( 1 + u 2 ) 1 / 2 ( 1 + u 2 ) 1 / 2 ] ( u 2 u 2 ) / ( 1 + u 2 ) 1 / 2 } 1 / 2 .
η + 2 η ( d / ) ( 1 + u 2 ) 1 / 2 I ( u ) ( 1 + u 2 ) 1 / 2 = 0 ,
I ( u ) = lim 0 0 u ( 1 ) d u [ 1 ( 1 + u 2 ) 1 / 2 1 ( 1 + u 2 ) 1 / 2 ] × { 2 [ ( 1 + u 2 ) 1 / 2 - ( 1 + u 2 ) 1/2 ] u 2 u 2 ( 1 + u 2 ) 1 / 2 } 1 / 2 .
R = 2 ( d / ) ( 1 + u 2 ) 1 / 4 I ( u ) + 2 ( d / ) 2 ( 1 + u 2 ) 1 / 2 I 2 ( u ) + .

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