Abstract

We propose a new kind of an optical spatial soliton: the holographic soliton. This soliton consists of two mutually coherent field components that interfere, induce a periodic change in the refractive index, and simultaneously are Bragg diffracted from the grating. Holographic solitons are formed when the broadening tendency of diffraction is balanced by phase modulation that is due to Bragg diffraction from the induced grating. Holographic solitons are solely supported by cross-phase modulation arising from the induced grating, not involving self-phase modulation at all.

© 2002 Optical Society of America

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  1. For a review of optical spatial solitons see G. I. Stegman and M. Segev, Science 286, 1518 (1999).
    [CrossRef]
  2. A. W. Snyder, D. J. Mitchell, L. Polodian, and F. Ladouceur, Opt. Lett. 16, 21 (1991).
    [CrossRef] [PubMed]
  3. A. Barthelemy, S. Maneuf, and C. Froehly, Opt. Commun. 55, 201 (1985).
    [CrossRef]
  4. J. S. Aitchison, 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] [PubMed]
  5. J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
    [CrossRef]
  6. V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
    [CrossRef] [PubMed]
  7. G. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. Neurgaonkar, Phys. Rev. Lett. 71, 633 (1993).
    [CrossRef]
  8. M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. Diporto, Electron. Lett. 31, 826 (1995).
    [CrossRef]
  9. M. Taya, M. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, Phys. Rev. A 52, 3095 (1995).
    [CrossRef] [PubMed]
  10. G. A. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
    [CrossRef] [PubMed]
  11. M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, Appl. Phys. Lett. 77, 7 (2000).
    [CrossRef]
  12. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
    [CrossRef] [PubMed]
  13. R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996).
    [CrossRef]
  14. O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
    [CrossRef]
  15. M. Vaupel, C. Seror, and R. Dykstra, Opt. Lett. 22, 1470 (1997).
    [CrossRef]
  16. M. Vaupel, O. Mandel, and N. R. Heckenberg, J. Opt. B 1, 96 (1999).
    [CrossRef]
  17. G. Shvets and A. Pukhov, Phys. Rev. E 59, 1033 (1999).
    [CrossRef]
  18. O. Cohen, S. Lan, T. Carmon, J. A. Giordmaine, and M. Segev, Opt. Lett. 27, 2013 (2002).
    [CrossRef]
  19. P. Gunter and J. P. Huignard, eds., Photorefractive Materials and Their Applications (Springer-Verlag, Berlin, 1988), Vol. II.
    [CrossRef]
  20. One can prove this expression under the same approximation that applies to photorefractive two-wave mixing [P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993)].
  21. This requires that the applied field be sufficiently large; e.g., in SBN:60 for θ=0.1°, E≅1 kV/cm.

2002 (2)

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef]

O. Cohen, S. Lan, T. Carmon, J. A. Giordmaine, and M. Segev, Opt. Lett. 27, 2013 (2002).
[CrossRef]

2000 (1)

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, Appl. Phys. Lett. 77, 7 (2000).
[CrossRef]

1999 (3)

M. Vaupel, O. Mandel, and N. R. Heckenberg, J. Opt. B 1, 96 (1999).
[CrossRef]

G. Shvets and A. Pukhov, Phys. Rev. E 59, 1033 (1999).
[CrossRef]

For a review of optical spatial solitons see G. I. Stegman and M. Segev, Science 286, 1518 (1999).
[CrossRef]

1997 (1)

1996 (2)

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996).
[CrossRef]

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

1995 (3)

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).
[CrossRef] [PubMed]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

1993 (1)

G. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. Neurgaonkar, Phys. Rev. Lett. 71, 633 (1993).
[CrossRef]

1991 (2)

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

A. W. Snyder, D. J. Mitchell, L. Polodian, and F. Ladouceur, Opt. Lett. 16, 21 (1991).
[CrossRef] [PubMed]

1990 (1)

1985 (1)

A. Barthelemy, S. Maneuf, and C. Froehly, Opt. Commun. 55, 201 (1985).
[CrossRef]

1974 (1)

J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
[CrossRef]

Aitchison, J. S.

Andersen, D. R.

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

Ashkin, A.

J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
[CrossRef]

Assanto, G.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, Appl. Phys. Lett. 77, 7 (2000).
[CrossRef]

Baek, Y.

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996).
[CrossRef]

Barthelemy, A.

A. Barthelemy, S. Maneuf, and C. Froehly, Opt. Commun. 55, 201 (1985).
[CrossRef]

Bashaw, M.

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

Bjorkholm, J. E.

J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
[CrossRef]

Carmon, T.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef]

O. Cohen, S. Lan, T. Carmon, J. A. Giordmaine, and M. Segev, Opt. Lett. 27, 2013 (2002).
[CrossRef]

Christou, J.

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

Cohen, O.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef]

O. Cohen, S. Lan, T. Carmon, J. A. Giordmaine, and M. Segev, Opt. Lett. 27, 2013 (2002).
[CrossRef]

Crosignani, B.

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. Diporto, Electron. Lett. 31, 826 (1995).
[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, 633 (1993).
[CrossRef]

De Luca, A.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, Appl. Phys. Lett. 77, 7 (2000).
[CrossRef]

De Rossi, A.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, Appl. Phys. Lett. 77, 7 (2000).
[CrossRef]

Diporto, P.

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. Diporto, Electron. Lett. 31, 826 (1995).
[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, 633 (1993).
[CrossRef]

Duree, G.

G. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. Neurgaonkar, Phys. Rev. Lett. 71, 633 (1993).
[CrossRef]

Dykstra, R.

Fejer, M. M.

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

Fleischer, J. W.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef]

Froehly, C.

A. Barthelemy, S. Maneuf, and C. Froehly, Opt. Commun. 55, 201 (1985).
[CrossRef]

Giordmaine, J. A.

Hagan, D. J.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

M. Vaupel, O. Mandel, and N. R. Heckenberg, J. Opt. B 1, 96 (1999).
[CrossRef]

Jackel, J. L.

Kaplan, A. E.

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

Khoo, I. C.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, Appl. Phys. Lett. 77, 7 (2000).
[CrossRef]

Ladouceur, F.

Lan, S.

Leaird, D. E.

Luther-Davies, B.

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

Mandel, O.

M. Vaupel, O. Mandel, and N. R. Heckenberg, J. Opt. B 1, 96 (1999).
[CrossRef]

Maneuf, S.

A. Barthelemy, S. Maneuf, and C. Froehly, Opt. Commun. 55, 201 (1985).
[CrossRef]

Menyuk, C. R.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Mitchell, D. J.

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, 633 (1993).
[CrossRef]

Odoulov, S.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef]

Oliver, M. K.

Peccianti, M.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, Appl. Phys. Lett. 77, 7 (2000).
[CrossRef]

Polodian, L.

Pukhov, A.

G. Shvets and A. Pukhov, Phys. Rev. E 59, 1033 (1999).
[CrossRef]

Regan, J. J.

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

Salamo, G.

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. Diporto, Electron. Lett. 31, 826 (1995).
[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, 633 (1993).
[CrossRef]

Schiek, R.

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996).
[CrossRef]

Segev, M.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef]

O. Cohen, S. Lan, T. Carmon, J. A. Giordmaine, and M. Segev, Opt. Lett. 27, 2013 (2002).
[CrossRef]

For a review of optical spatial solitons see G. I. Stegman and M. Segev, Science 286, 1518 (1999).
[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).
[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, 633 (1993).
[CrossRef]

Seror, C.

Sharp, E.

G. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. Neurgaonkar, Phys. Rev. Lett. 71, 633 (1993).
[CrossRef]

Shih, M.

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

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, 633 (1993).
[CrossRef]

Shvets, G.

G. Shvets and A. Pukhov, Phys. Rev. E 59, 1033 (1999).
[CrossRef]

Silberberg, Y.

Smith, P. W.

Snyder, A. W.

Stegeman, G. I.

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Stegman, G. I.

For a review of optical spatial solitons see G. I. Stegman and M. Segev, Science 286, 1518 (1999).
[CrossRef]

Swartzlander, G. A.

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

Taya, M.

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

Tikhonenko, V.

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

Torner, L.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Torruellas, W. E.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Umeton, C.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, Appl. Phys. Lett. 77, 7 (2000).
[CrossRef]

Uzdin, R.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef]

Valley, G. C.

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

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

Van Stryland, E. W.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Vaupel, M.

M. Vaupel, O. Mandel, and N. R. Heckenberg, J. Opt. B 1, 96 (1999).
[CrossRef]

M. Vaupel, C. Seror, and R. Dykstra, Opt. Lett. 22, 1470 (1997).
[CrossRef]

Vogel, E. M.

Wang, Z.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Weiner, A. M.

Yariv, A.

G. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. Neurgaonkar, Phys. Rev. Lett. 71, 633 (1993).
[CrossRef]

Yeh, P.

One can prove this expression under the same approximation that applies to photorefractive two-wave mixing [P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993)].

Yin, H.

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

Appl. Phys. Lett. (1)

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, Appl. Phys. Lett. 77, 7 (2000).
[CrossRef]

Electron. Lett. (1)

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

J. Opt. B (1)

M. Vaupel, O. Mandel, and N. R. Heckenberg, J. Opt. B 1, 96 (1999).
[CrossRef]

Opt. Commun. (1)

A. Barthelemy, S. Maneuf, and C. Froehly, Opt. Commun. 55, 201 (1985).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (1)

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

Phys. Rev. E (2)

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996).
[CrossRef]

G. Shvets and A. Pukhov, Phys. Rev. E 59, 1033 (1999).
[CrossRef]

Phys. Rev. Lett. (6)

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

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

J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
[CrossRef]

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[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, 633 (1993).
[CrossRef]

Science (1)

For a review of optical spatial solitons see G. I. Stegman and M. Segev, Science 286, 1518 (1999).
[CrossRef]

Other (3)

P. Gunter and J. P. Huignard, eds., Photorefractive Materials and Their Applications (Springer-Verlag, Berlin, 1988), Vol. II.
[CrossRef]

One can prove this expression under the same approximation that applies to photorefractive two-wave mixing [P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993)].

This requires that the applied field be sufficiently large; e.g., in SBN:60 for θ=0.1°, E≅1 kV/cm.

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Figures (2)

Fig. 1
Fig. 1

Schematic of a holographic soliton in a photorefractive medium.

Fig. 2
Fig. 2

a, Normalized wave functions of the bright holographic soliton versus transverse coordinate ξ for several values of the peak amplitude u0. b, Normalized wave functions of the dark holographic soliton versus ξ for several values of the amplitude at infinity u. c, FWHM of the intensity of bright and dark holographic soliton versus u0 and u, respectively.

Equations (5)

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

Δn=Δn0AB* exp2ik sinθy+c.c.A2+B2+IB cos2θ,
2Ax2+2ik cosθAz+2k2Δn0 cos2θn0×B2A2+B2+IBA=0,  2Bx2+2ik cosθBz+2k2Δn0 cos2θn0×A2A2+B2+IBB=0.
u-βu±u2u2u2+1=0,
uξ2-u02-βuξ2-u02±uξ2-u02/2ln1+2uξ2/1+2u02/4=0.
u02=u2u2/2u2+1-1/2+ln2u2+1/4.

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