Abstract

We have introduced a novel (to our knowledge) class of complex-variable-function (CVF)—Gaussian solitons that constitute the exact soliton solutions of the Snyder–Mitchell model. The CVF–Gaussian solitons whose transverse structure has an inherent rotation are the product of an arbitrary analytic CVF and a Gaussian function. A distribution factor that is the parameter for the description of the transverse distribution of the CVF–Gaussian solitons is discussed.

© 2008 Optical Society of America

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  1. A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).
  2. A. W. Snyder and Y. Kivshar, J. Opt. Soc. Am. B 14, 3025 (1997).
  3. D. J. Mitchell and A. W. Snyder, J. Opt. Soc. Am. B 16, 236 (1999).
  4. W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B 6, S288 (2004).
  5. D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 98, 053901 (2007).
    [PubMed]
  6. D. M. Deng, X. Zhao, Q. Guo, and S. Lan, J. Opt. Soc. Am. B 24, 2537 (2007).
  7. D. M. Deng and Q. Guo, J. Opt. A 10, 035101 (2008).
  8. Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, Phys. Rev. E 69, 016602 (2004).
  9. Q. Guo, Proc. SPIE 5281, 581 (2004).
  10. Y. Xie and Q. Guo, Opt. Quantum Electron. 36, 1335 (2004).
  11. D. M. Deng and Q. Guo, Opt. Lett. 32, 3206 (2007).
    [PubMed]
  12. D. M. Deng and Q. Guo, J. Phys. B 14, 145401 (2008).
  13. S. G. Ouyang, Q. Guo, and W. Hu, Phys. Rev. E 74, 036622 (2006).
  14. S. G. Ouyang, W. Hu, and Q. Guo, Phys. Rev. A 76, 053832 (2007).
  15. C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 92, 113902 (2004).
    [PubMed]
  16. M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Nat. Phys. 2, 737 (2006).
  17. C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
    [PubMed]
  18. C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, Nat. Phys. 2, 769 (2006).
  19. C. Rotschild, M. Segev, Z. Y. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, Opt. Lett. 31, 3312 (2006).
    [PubMed]
  20. X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, Opt. Commun. 233, 211 (2004).
  21. W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, Appl. Phys. Lett. 89, 071111 (2006).
  22. W. Hu, S. G. Ouyang, P. B. Yang, Q. Guo, and S. Lan, Phys. Rev. A 77, 033842 (2008).
  23. M. Peccianti, C. Conti, G. Assanto, A. D. Luca, and C. Umeton, Appl. Phys. Lett. 81, 3335 (2002).
  24. G. Assanto, M. Peccianti, and C. Conti, Opt. Photonics News 14, 44 (2003).
  25. S. Lopez-Aguayo, A. S. Desyatnikov, Y. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, Opt. Lett. 31, 1100 (2006).
    [PubMed]
  26. A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, Phys. Rev. E 71, 065603 (2005).
  27. S. Lopez-Aguayo, A. S. Desyatnikov, and Y. S. Kivshar, Opt. Express 14, 7903 (2006).
    [PubMed]
  28. S. Lopez-Aguayo and J. C. Gutiérrez-Vega, Opt. Express 15, 18326 (2007).
    [PubMed]
  29. D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Opt. Lett. 33, 198 (2008).
    [PubMed]
  30. A. Fratalocchi, A. Piccardi, M. Peccianti, and G. Assanto, Opt. Lett. 32, 1447 (2007).
    [PubMed]
  31. A. E. Siegman, Proc. SPIE 1224, 2 (1990).

2008 (4)

D. M. Deng and Q. Guo, J. Opt. A 10, 035101 (2008).

D. M. Deng and Q. Guo, J. Phys. B 14, 145401 (2008).

W. Hu, S. G. Ouyang, P. B. Yang, Q. Guo, and S. Lan, Phys. Rev. A 77, 033842 (2008).

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Opt. Lett. 33, 198 (2008).
[PubMed]

2007 (6)

2006 (7)

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, Appl. Phys. Lett. 89, 071111 (2006).

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, Nat. Phys. 2, 769 (2006).

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Nat. Phys. 2, 737 (2006).

S. G. Ouyang, Q. Guo, and W. Hu, Phys. Rev. E 74, 036622 (2006).

S. Lopez-Aguayo, A. S. Desyatnikov, Y. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, Opt. Lett. 31, 1100 (2006).
[PubMed]

S. Lopez-Aguayo, A. S. Desyatnikov, and Y. S. Kivshar, Opt. Express 14, 7903 (2006).
[PubMed]

C. Rotschild, M. Segev, Z. Y. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, Opt. Lett. 31, 3312 (2006).
[PubMed]

2005 (2)

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, Phys. Rev. E 71, 065603 (2005).

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
[PubMed]

2004 (6)

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 92, 113902 (2004).
[PubMed]

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, Opt. Commun. 233, 211 (2004).

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B 6, S288 (2004).

Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, Phys. Rev. E 69, 016602 (2004).

Q. Guo, Proc. SPIE 5281, 581 (2004).

Y. Xie and Q. Guo, Opt. Quantum Electron. 36, 1335 (2004).

2003 (1)

G. Assanto, M. Peccianti, and C. Conti, Opt. Photonics News 14, 44 (2003).

2002 (1)

M. Peccianti, C. Conti, G. Assanto, A. D. Luca, and C. Umeton, Appl. Phys. Lett. 81, 3335 (2002).

1999 (1)

1997 (2)

A. W. Snyder and Y. Kivshar, J. Opt. Soc. Am. B 14, 3025 (1997).

A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).

1990 (1)

A. E. Siegman, Proc. SPIE 1224, 2 (1990).

Adamski, A.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, Opt. Commun. 233, 211 (2004).

Alfassi, B.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, Nat. Phys. 2, 769 (2006).

Assanto, G.

A. Fratalocchi, A. Piccardi, M. Peccianti, and G. Assanto, Opt. Lett. 32, 1447 (2007).
[PubMed]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Nat. Phys. 2, 737 (2006).

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 92, 113902 (2004).
[PubMed]

G. Assanto, M. Peccianti, and C. Conti, Opt. Photonics News 14, 44 (2003).

M. Peccianti, C. Conti, G. Assanto, A. D. Luca, and C. Umeton, Appl. Phys. Lett. 81, 3335 (2002).

Bang, O.

S. Lopez-Aguayo, A. S. Desyatnikov, Y. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, Opt. Lett. 31, 1100 (2006).
[PubMed]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B 6, S288 (2004).

Buccoliero, D.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Opt. Lett. 33, 198 (2008).
[PubMed]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 98, 053901 (2007).
[PubMed]

Cambournac, C.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, Opt. Commun. 233, 211 (2004).

Carmon, T.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
[PubMed]

Chi, S.

Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, Phys. Rev. E 69, 016602 (2004).

Cohen, O.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, Nat. Phys. 2, 769 (2006).

C. Rotschild, M. Segev, Z. Y. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, Opt. Lett. 31, 3312 (2006).
[PubMed]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
[PubMed]

Conti, C.

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 92, 113902 (2004).
[PubMed]

G. Assanto, M. Peccianti, and C. Conti, Opt. Photonics News 14, 44 (2003).

M. Peccianti, C. Conti, G. Assanto, A. D. Luca, and C. Umeton, Appl. Phys. Lett. 81, 3335 (2002).

Deng, D. M.

D. M. Deng and Q. Guo, J. Phys. B 14, 145401 (2008).

D. M. Deng and Q. Guo, J. Opt. A 10, 035101 (2008).

D. M. Deng, X. Zhao, Q. Guo, and S. Lan, J. Opt. Soc. Am. B 24, 2537 (2007).

D. M. Deng and Q. Guo, Opt. Lett. 32, 3206 (2007).
[PubMed]

Desyatnikov, A. S.

Dyadyusha, A.

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Nat. Phys. 2, 737 (2006).

Edmundson, D.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B 6, S288 (2004).

Fratalocchi, A.

Guo, Q.

D. M. Deng and Q. Guo, J. Opt. A 10, 035101 (2008).

D. M. Deng and Q. Guo, J. Phys. B 14, 145401 (2008).

W. Hu, S. G. Ouyang, P. B. Yang, Q. Guo, and S. Lan, Phys. Rev. A 77, 033842 (2008).

S. G. Ouyang, W. Hu, and Q. Guo, Phys. Rev. A 76, 053832 (2007).

D. M. Deng and Q. Guo, Opt. Lett. 32, 3206 (2007).
[PubMed]

D. M. Deng, X. Zhao, Q. Guo, and S. Lan, J. Opt. Soc. Am. B 24, 2537 (2007).

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, Appl. Phys. Lett. 89, 071111 (2006).

S. G. Ouyang, Q. Guo, and W. Hu, Phys. Rev. E 74, 036622 (2006).

Y. Xie and Q. Guo, Opt. Quantum Electron. 36, 1335 (2004).

Q. Guo, Proc. SPIE 5281, 581 (2004).

Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, Phys. Rev. E 69, 016602 (2004).

Gutiérrez-Vega, J. C.

Haelterman, M.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, Opt. Commun. 233, 211 (2004).

Hu, W.

W. Hu, S. G. Ouyang, P. B. Yang, Q. Guo, and S. Lan, Phys. Rev. A 77, 033842 (2008).

S. G. Ouyang, W. Hu, and Q. Guo, Phys. Rev. A 76, 053832 (2007).

S. G. Ouyang, Q. Guo, and W. Hu, Phys. Rev. E 74, 036622 (2006).

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, Appl. Phys. Lett. 89, 071111 (2006).

Hutsebaut, X.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, Opt. Commun. 233, 211 (2004).

Kaczmarek, M.

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Nat. Phys. 2, 737 (2006).

Kartashov, Y. V.

Kivshar, Y.

Kivshar, Y. S.

Krolikowski, W.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Opt. Lett. 33, 198 (2008).
[PubMed]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 98, 053901 (2007).
[PubMed]

S. Lopez-Aguayo, A. S. Desyatnikov, Y. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, Opt. Lett. 31, 1100 (2006).
[PubMed]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B 6, S288 (2004).

Lan, S.

W. Hu, S. G. Ouyang, P. B. Yang, Q. Guo, and S. Lan, Phys. Rev. A 77, 033842 (2008).

D. M. Deng, X. Zhao, Q. Guo, and S. Lan, J. Opt. Soc. Am. B 24, 2537 (2007).

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, Appl. Phys. Lett. 89, 071111 (2006).

Lopez-Aguayo, S.

Luca, A. D.

M. Peccianti, C. Conti, G. Assanto, A. D. Luca, and C. Umeton, Appl. Phys. Lett. 81, 3335 (2002).

Luo, B.

Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, Phys. Rev. E 69, 016602 (2004).

Manela, O.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
[PubMed]

Mitchell, D. J.

D. J. Mitchell and A. W. Snyder, J. Opt. Soc. Am. B 16, 236 (1999).

A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).

Neshev, D.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B 6, S288 (2004).

Neyts, K.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, Opt. Commun. 233, 211 (2004).

Nikolov, N. I.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B 6, S288 (2004).

Ouyang, S. G.

W. Hu, S. G. Ouyang, P. B. Yang, Q. Guo, and S. Lan, Phys. Rev. A 77, 033842 (2008).

S. G. Ouyang, W. Hu, and Q. Guo, Phys. Rev. A 76, 053832 (2007).

S. G. Ouyang, Q. Guo, and W. Hu, Phys. Rev. E 74, 036622 (2006).

Peccianti, M.

A. Fratalocchi, A. Piccardi, M. Peccianti, and G. Assanto, Opt. Lett. 32, 1447 (2007).
[PubMed]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Nat. Phys. 2, 737 (2006).

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 92, 113902 (2004).
[PubMed]

G. Assanto, M. Peccianti, and C. Conti, Opt. Photonics News 14, 44 (2003).

M. Peccianti, C. Conti, G. Assanto, A. D. Luca, and C. Umeton, Appl. Phys. Lett. 81, 3335 (2002).

Piccardi, A.

Rasmussen, J. J.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B 6, S288 (2004).

Rotschild, C.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, Nat. Phys. 2, 769 (2006).

C. Rotschild, M. Segev, Z. Y. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, Opt. Lett. 31, 3312 (2006).
[PubMed]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
[PubMed]

Segev, M.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, Nat. Phys. 2, 769 (2006).

C. Rotschild, M. Segev, Z. Y. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, Opt. Lett. 31, 3312 (2006).
[PubMed]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
[PubMed]

Siegman, A. E.

A. E. Siegman, Proc. SPIE 1224, 2 (1990).

Skupin, S.

Snyder, A. W.

Torner, L.

Umeton, C.

M. Peccianti, C. Conti, G. Assanto, A. D. Luca, and C. Umeton, Appl. Phys. Lett. 81, 3335 (2002).

Wyller, J.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B 6, S288 (2004).

Xie, Y.

Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, Phys. Rev. E 69, 016602 (2004).

Y. Xie and Q. Guo, Opt. Quantum Electron. 36, 1335 (2004).

Xu, Z. Y.

Xuan, L.

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, Appl. Phys. Lett. 89, 071111 (2006).

Yakimenko, A. I.

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, Phys. Rev. E 71, 065603 (2005).

Yang, P. B.

W. Hu, S. G. Ouyang, P. B. Yang, Q. Guo, and S. Lan, Phys. Rev. A 77, 033842 (2008).

Yi, F.

Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, Phys. Rev. E 69, 016602 (2004).

Zaliznyak, Y. A.

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, Phys. Rev. E 71, 065603 (2005).

Zhang, T.

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, Appl. Phys. Lett. 89, 071111 (2006).

Zhao, X.

Appl. Phys. Lett. (2)

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, Appl. Phys. Lett. 89, 071111 (2006).

M. Peccianti, C. Conti, G. Assanto, A. D. Luca, and C. Umeton, Appl. Phys. Lett. 81, 3335 (2002).

J. Opt. A (1)

D. M. Deng and Q. Guo, J. Opt. A 10, 035101 (2008).

J. Opt. B (1)

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B 6, S288 (2004).

J. Opt. Soc. Am. B (3)

J. Phys. B (1)

D. M. Deng and Q. Guo, J. Phys. B 14, 145401 (2008).

Nat. Phys. (2)

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, Nat. Phys. 2, 769 (2006).

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Nat. Phys. 2, 737 (2006).

Opt. Commun. (1)

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, Opt. Commun. 233, 211 (2004).

Opt. Express (2)

Opt. Lett. (5)

Opt. Photonics News (1)

G. Assanto, M. Peccianti, and C. Conti, Opt. Photonics News 14, 44 (2003).

Opt. Quantum Electron. (1)

Y. Xie and Q. Guo, Opt. Quantum Electron. 36, 1335 (2004).

Phys. Rev. A (2)

W. Hu, S. G. Ouyang, P. B. Yang, Q. Guo, and S. Lan, Phys. Rev. A 77, 033842 (2008).

S. G. Ouyang, W. Hu, and Q. Guo, Phys. Rev. A 76, 053832 (2007).

Phys. Rev. E (3)

S. G. Ouyang, Q. Guo, and W. Hu, Phys. Rev. E 74, 036622 (2006).

Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, Phys. Rev. E 69, 016602 (2004).

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, Phys. Rev. E 71, 065603 (2005).

Phys. Rev. Lett. (3)

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 98, 053901 (2007).
[PubMed]

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 92, 113902 (2004).
[PubMed]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
[PubMed]

Proc. SPIE (2)

Q. Guo, Proc. SPIE 5281, 581 (2004).

A. E. Siegman, Proc. SPIE 1224, 2 (1990).

Science (1)

A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).

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Figures (2)

Fig. 1
Fig. 1

Propagation dynamics of the sine CVF-Gaussian beam in the Gaussian-shaped response material. (a)–(d) Transverse normalized intensity distributions from the analytical solution, (e)–(h) distributions from the numerical simulation ( α = 0.06 ) , (i)–(l) distributions from the numerical simulation ( α = 0.2 ) , (m)–(p) distributions from the numerical simulation ( α = 0.4 ) . The different columns represent the different propagation distances given in the top of the figure. The parameters are chosen as P 0 P c = 1 and b = 1 , and z R = k w 0 2 is the Rayleigh distance.

Fig. 2
Fig. 2

Same as Fig. 1 but with plots corresponding to Gaussian CVF-Gaussian beam and b = 2 .

Equations (8)

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i Φ z + μ Δ Φ + k Δ n n 0 Φ = 0 ,
i Φ z + μ Δ Φ n 2 2 n 0 k γ P 0 r 2 Φ = 0 ,
Φ = Φ F ( r , z ) Φ G ( r , z ) ,
Φ G = P 0 π w 0 exp ( r 2 2 w 0 2 i β 0 z ) ,
i Φ F z + μ Δ Φ F + 2 μ w 0 2 ( x Φ F x + y Φ F y ) = 0 .
( i ϑ y 2 μ w 0 2 x ) Φ F x ( i ϑ x + 2 μ w 0 2 y ) Φ F y + μ ( 2 x 2 + 2 y 2 ) Φ F = 0 .
Φ F ( r ) = f ( x ± i y b w 0 ) ,
Ψ ( r , z ) = C 0 f ( ζ ± ) w 0 exp ( r 2 2 w 0 2 i β 0 z ) ,

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