Abstract

Based on the Snyder–Mitchell model in the Cartesian coordinate system, exact analytical Hermite–Gaussian (HG) solutions are obtained in strongly nonlocal nonlinear media. The comparisons of analytical solutions with numerical simulations of the nonlocal nonlinear Schrödinger equation show that the analytical HG solutions are in good agreement with the numerical simulations in the case of strong nonlocality. Furthermore, we demonstrate that HG functions can be expressed as a linear superposition of individual Gaussian functions with a π phase difference under the appropriate conditions.

© 2007 Optical Society of America

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References

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  1. A. W. Snyder and D. J. Mitchell, "Accessible solitons," Science 276, 1538-1541 (1997).
    [CrossRef]
  2. M. Peccianti, C. Conti, G. Assanto, A. D. Luca, and C. Umeton, "All-optical switching and logic gating with spatial solitons in liquid crystals," Appl. Phys. Lett. 81, 3335-3337 (2002).
    [CrossRef]
  3. G. Assanto, M. Peccianti, and C. Conti, "Nematicons: optical spatial solitons in nematic liquid crystals," Opt. Photonics News 14, 44-48 (2003).
    [CrossRef]
  4. D. J. Mitchell and A. W. Snyder, "Soliton dynamics in a nonlocal medium," J. Opt. Soc. Am. B 16, 236-239 (1999).
    [CrossRef]
  5. W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
    [CrossRef]
  6. C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003).
    [CrossRef] [PubMed]
  7. C. Conti, M. Peccianti, and G. Assanto, "Observation of optical spatial solitons in a highly nonlocal medium," Phys. Rev. Lett. 92, 113902 (2004).
    [CrossRef] [PubMed]
  8. Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, "Large phase shift of nonlocal optical spatial solitons," Phys. Rev. E 69, 016602 (2004).
    [CrossRef]
  9. Q. Guo, "Nonlocal spatial solitons and their interactions," in Optical Transmission, Switching, and Subsystems, C.F.Lam, C.Fan, N.Hanik, and K.Oguchi, eds., Proc. SPIE 5281, 581-594 (2004).
  10. Y. Xie and Q. Guo, "Phase modulations due to collisions of beam pairs in nonlocal nonlinear media," Opt. Quantum Electron. 36, 1335-1351 (2004).
    [CrossRef]
  11. A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, "Bound dipole solitary solutions in anisotropic nonlocal self-focusing media," Phys. Rev. A 56, R1110-R1113 (1997).
    [CrossRef]
  12. M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, "Nonlocal spatial soliton interactions in nematic liquid crystals," Opt. Lett. 27, 1460-1462 (2002).
    [CrossRef]
  13. X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, "Single-component higher-order mode solitons in liquid crystals," Opt. Commun. 233, 211-217 (2004).
    [CrossRef]
  14. Z. Xu, Y. V. Kartashov, and L. Torner, "Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media," Opt. Lett. 30, 3171-3173 (2005).
    [CrossRef] [PubMed]
  15. S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, "Stability of two-dimensional spatial solitons in nonlocal nonlinear media," Phys. Rev. E 73, 066603 (2006).
    [CrossRef]
  16. A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, "Dynamics of two-dimensional coherent structures in nonlocal nonlinear media," Phys. Rev. E 73, 066605 (2006).
    [CrossRef]
  17. N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, "Attraction of nonlocal dark optical solitons," Opt. Lett. 29, 286-288 (2004).
    [CrossRef] [PubMed]
  18. A. Dreischuh, D. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, "Observation of attraction between dark solitons," Phys. Rev. Lett. 96, 043901 (2006).
    [CrossRef] [PubMed]
  19. D. W. McLaughlin, D. J. Muraki, M. J. Shelley, "Self-focussed optical structures in a nematic liquid crystal," Physica D 97, 471-497 (1996).
    [CrossRef]
  20. C. Rotschild, M. Segev, Z. Y. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, "Two-dimensional multipole solitons in nonlocal nonlinear media," Opt. Lett. 31, 3312-3314 (2006).
    [CrossRef] [PubMed]
  21. D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, "Laguerre and Hermite soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
    [CrossRef] [PubMed]
  22. A. W. Snyder and Y. Kivshar, "Bright spatial solitons in non-Kerr media: stationary beams and dynamical evolution," J. Opt. Soc. Am. B 11, 3025-3031 (1997).
    [CrossRef]
  23. S. Abe and A. Ogura, "Solitary waves and their critical behavior in a nonlinear nonlocal medium with power-law response," Phys. Rev. E 57, 6066-6070 (1998).
    [CrossRef]
  24. B. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "Sub-wavelength non-local spatial solitons," Opt. Commun. 166, 121-126 (1999).
    [CrossRef]
  25. W. Krolikowski and O. Bang, "Solitons in nonlocal nonlinear media: exact solutions," Phys. Rev. E 63, 016610 (2000).
    [CrossRef]
  26. S. G. Ouyang, Q. Guo, and W. Hu, "Perturbative analysis of generally nonlocal spatial optical solitons," Phys. Rev. E 74, 036622 (2006).
    [CrossRef]
  27. P. D. Rasmusen, O. Bang, W. Królikowski, "Theory of nonlocal soliton interaction in nematic liquid crystals," Phys. Rev. E 72, 066611 (2005).
    [CrossRef]
  28. X. P. Zhang and Q. Guo, "Analytical solution in the Hermite-Gaussian form of the beam propagating in the strong nonlocal media," Acta Phys. Sin. 54, 3178-3182 (2005) (in Chinese).
  29. X. P. Zhang and Q. Guo, "Analytical solution to the spatial optical solitons propagating in the strong nonlocal media," Acta Phys. Sin. 54, 5189-5193 (2005) (in Chinese).
  30. W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, "Nonlocality-controlled interaction of spatial solitons in nematic liquid crystals," Appl. Phys. Lett. 89, 071111 (2006).
    [CrossRef]
  31. Y. R. Shen, "Optical physics--solitons made simple," Science 276, 1520-1520 (1997).
    [CrossRef]
  32. W. Krolikowski, M. Saffman, B. Luther-Davies, and C. Denz, "Anomalous interaction of spatial solitons in photorefractive media," Phys. Rev. Lett. 80, 3240-3243 (1998).
    [CrossRef]
  33. G. E. Andrews, R. Askey, and R. Roy, Special Functions (Cambridge U. Press, 2000), Sec. 6.1.

2007 (1)

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, "Laguerre and Hermite soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

2006 (6)

S. G. Ouyang, Q. Guo, and W. Hu, "Perturbative analysis of generally nonlocal spatial optical solitons," Phys. Rev. E 74, 036622 (2006).
[CrossRef]

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, "Nonlocality-controlled interaction of spatial solitons in nematic liquid crystals," Appl. Phys. Lett. 89, 071111 (2006).
[CrossRef]

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, "Stability of two-dimensional spatial solitons in nonlocal nonlinear media," Phys. Rev. E 73, 066603 (2006).
[CrossRef]

A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, "Dynamics of two-dimensional coherent structures in nonlocal nonlinear media," Phys. Rev. E 73, 066605 (2006).
[CrossRef]

A. Dreischuh, D. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, "Observation of attraction between dark solitons," Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

C. Rotschild, M. Segev, Z. Y. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, "Two-dimensional multipole solitons in nonlocal nonlinear media," Opt. Lett. 31, 3312-3314 (2006).
[CrossRef] [PubMed]

2005 (4)

Z. Xu, Y. V. Kartashov, and L. Torner, "Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media," Opt. Lett. 30, 3171-3173 (2005).
[CrossRef] [PubMed]

P. D. Rasmusen, O. Bang, W. Królikowski, "Theory of nonlocal soliton interaction in nematic liquid crystals," Phys. Rev. E 72, 066611 (2005).
[CrossRef]

X. P. Zhang and Q. Guo, "Analytical solution in the Hermite-Gaussian form of the beam propagating in the strong nonlocal media," Acta Phys. Sin. 54, 3178-3182 (2005) (in Chinese).

X. P. Zhang and Q. Guo, "Analytical solution to the spatial optical solitons propagating in the strong nonlocal media," Acta Phys. Sin. 54, 5189-5193 (2005) (in Chinese).

2004 (5)

C. Conti, M. Peccianti, and G. Assanto, "Observation of optical spatial solitons in a highly nonlocal medium," Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, "Large phase shift of nonlocal optical spatial solitons," Phys. Rev. E 69, 016602 (2004).
[CrossRef]

Y. Xie and Q. Guo, "Phase modulations due to collisions of beam pairs in nonlocal nonlinear media," Opt. Quantum Electron. 36, 1335-1351 (2004).
[CrossRef]

N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, "Attraction of nonlocal dark optical solitons," Opt. Lett. 29, 286-288 (2004).
[CrossRef] [PubMed]

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, "Single-component higher-order mode solitons in liquid crystals," Opt. Commun. 233, 211-217 (2004).
[CrossRef]

2003 (2)

G. Assanto, M. Peccianti, and C. Conti, "Nematicons: optical spatial solitons in nematic liquid crystals," Opt. Photonics News 14, 44-48 (2003).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef] [PubMed]

2002 (2)

M. Peccianti, C. Conti, G. Assanto, A. D. Luca, and C. Umeton, "All-optical switching and logic gating with spatial solitons in liquid crystals," Appl. Phys. Lett. 81, 3335-3337 (2002).
[CrossRef]

M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, "Nonlocal spatial soliton interactions in nematic liquid crystals," Opt. Lett. 27, 1460-1462 (2002).
[CrossRef]

2001 (1)

W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[CrossRef]

2000 (1)

W. Krolikowski and O. Bang, "Solitons in nonlocal nonlinear media: exact solutions," Phys. Rev. E 63, 016610 (2000).
[CrossRef]

1999 (2)

B. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "Sub-wavelength non-local spatial solitons," Opt. Commun. 166, 121-126 (1999).
[CrossRef]

D. J. Mitchell and A. W. Snyder, "Soliton dynamics in a nonlocal medium," J. Opt. Soc. Am. B 16, 236-239 (1999).
[CrossRef]

1998 (2)

W. Krolikowski, M. Saffman, B. Luther-Davies, and C. Denz, "Anomalous interaction of spatial solitons in photorefractive media," Phys. Rev. Lett. 80, 3240-3243 (1998).
[CrossRef]

S. Abe and A. Ogura, "Solitary waves and their critical behavior in a nonlinear nonlocal medium with power-law response," Phys. Rev. E 57, 6066-6070 (1998).
[CrossRef]

1997 (4)

A. W. Snyder and Y. Kivshar, "Bright spatial solitons in non-Kerr media: stationary beams and dynamical evolution," J. Opt. Soc. Am. B 11, 3025-3031 (1997).
[CrossRef]

A. W. Snyder and D. J. Mitchell, "Accessible solitons," Science 276, 1538-1541 (1997).
[CrossRef]

A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, "Bound dipole solitary solutions in anisotropic nonlocal self-focusing media," Phys. Rev. A 56, R1110-R1113 (1997).
[CrossRef]

Y. R. Shen, "Optical physics--solitons made simple," Science 276, 1520-1520 (1997).
[CrossRef]

1996 (1)

D. W. McLaughlin, D. J. Muraki, M. J. Shelley, "Self-focussed optical structures in a nematic liquid crystal," Physica D 97, 471-497 (1996).
[CrossRef]

Acta Phys. Sin. (2)

X. P. Zhang and Q. Guo, "Analytical solution in the Hermite-Gaussian form of the beam propagating in the strong nonlocal media," Acta Phys. Sin. 54, 3178-3182 (2005) (in Chinese).

X. P. Zhang and Q. Guo, "Analytical solution to the spatial optical solitons propagating in the strong nonlocal media," Acta Phys. Sin. 54, 5189-5193 (2005) (in Chinese).

Appl. Phys. Lett. (2)

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, "Nonlocality-controlled interaction of spatial solitons in nematic liquid crystals," Appl. Phys. Lett. 89, 071111 (2006).
[CrossRef]

M. Peccianti, C. Conti, G. Assanto, A. D. Luca, and C. Umeton, "All-optical switching and logic gating with spatial solitons in liquid crystals," Appl. Phys. Lett. 81, 3335-3337 (2002).
[CrossRef]

J. Opt. Soc. Am. B (2)

A. W. Snyder and Y. Kivshar, "Bright spatial solitons in non-Kerr media: stationary beams and dynamical evolution," J. Opt. Soc. Am. B 11, 3025-3031 (1997).
[CrossRef]

D. J. Mitchell and A. W. Snyder, "Soliton dynamics in a nonlocal medium," J. Opt. Soc. Am. B 16, 236-239 (1999).
[CrossRef]

Opt. Commun. (2)

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, "Single-component higher-order mode solitons in liquid crystals," Opt. Commun. 233, 211-217 (2004).
[CrossRef]

B. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "Sub-wavelength non-local spatial solitons," Opt. Commun. 166, 121-126 (1999).
[CrossRef]

Opt. Lett. (4)

Opt. Photonics News (1)

G. Assanto, M. Peccianti, and C. Conti, "Nematicons: optical spatial solitons in nematic liquid crystals," Opt. Photonics News 14, 44-48 (2003).
[CrossRef]

Opt. Quantum Electron. (1)

Y. Xie and Q. Guo, "Phase modulations due to collisions of beam pairs in nonlocal nonlinear media," Opt. Quantum Electron. 36, 1335-1351 (2004).
[CrossRef]

Phys. Rev. A (1)

A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, "Bound dipole solitary solutions in anisotropic nonlocal self-focusing media," Phys. Rev. A 56, R1110-R1113 (1997).
[CrossRef]

Phys. Rev. E (8)

W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[CrossRef]

Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, "Large phase shift of nonlocal optical spatial solitons," Phys. Rev. E 69, 016602 (2004).
[CrossRef]

W. Krolikowski and O. Bang, "Solitons in nonlocal nonlinear media: exact solutions," Phys. Rev. E 63, 016610 (2000).
[CrossRef]

S. G. Ouyang, Q. Guo, and W. Hu, "Perturbative analysis of generally nonlocal spatial optical solitons," Phys. Rev. E 74, 036622 (2006).
[CrossRef]

P. D. Rasmusen, O. Bang, W. Królikowski, "Theory of nonlocal soliton interaction in nematic liquid crystals," Phys. Rev. E 72, 066611 (2005).
[CrossRef]

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, "Stability of two-dimensional spatial solitons in nonlocal nonlinear media," Phys. Rev. E 73, 066603 (2006).
[CrossRef]

A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, "Dynamics of two-dimensional coherent structures in nonlocal nonlinear media," Phys. Rev. E 73, 066605 (2006).
[CrossRef]

S. Abe and A. Ogura, "Solitary waves and their critical behavior in a nonlinear nonlocal medium with power-law response," Phys. Rev. E 57, 6066-6070 (1998).
[CrossRef]

Phys. Rev. Lett. (5)

W. Krolikowski, M. Saffman, B. Luther-Davies, and C. Denz, "Anomalous interaction of spatial solitons in photorefractive media," Phys. Rev. Lett. 80, 3240-3243 (1998).
[CrossRef]

A. Dreischuh, D. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, "Observation of attraction between dark solitons," Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, "Laguerre and Hermite soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, "Observation of optical spatial solitons in a highly nonlocal medium," Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

Physica D (1)

D. W. McLaughlin, D. J. Muraki, M. J. Shelley, "Self-focussed optical structures in a nematic liquid crystal," Physica D 97, 471-497 (1996).
[CrossRef]

Science (2)

Y. R. Shen, "Optical physics--solitons made simple," Science 276, 1520-1520 (1997).
[CrossRef]

A. W. Snyder and D. J. Mitchell, "Accessible solitons," Science 276, 1538-1541 (1997).
[CrossRef]

Other (2)

Q. Guo, "Nonlocal spatial solitons and their interactions," in Optical Transmission, Switching, and Subsystems, C.F.Lam, C.Fan, N.Hanik, and K.Oguchi, eds., Proc. SPIE 5281, 581-594 (2004).

G. E. Andrews, R. Askey, and R. Roy, Special Functions (Cambridge U. Press, 2000), Sec. 6.1.

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Figures (4)

Fig. 1
Fig. 1

Evolution of the normalized intensity profiles for ( 1 + 1 ) -dimensional (a) first-, (b) second-, and (c) third-order-mode HG breathers in the Gaussian-shaped response material. Solid curves, numerical simulation; open circles, analytical solution. Solid curves and open circles in (d), (e), and (f) are the normalized intensity distributions at Z = 4 corresponding to (a), (b), and (c), respectively. The parameters are chosen as P 0 P c = 0.7 , α = 0.1 .

Fig. 2
Fig. 2

Evolution of the normalized intensity profiles for ( 1 + 1 ) -dimensional (a) first-, (b) second-, and (c) third-order-mode HG breathers in the Gaussian-shaped response material. Solid curves, numerical simulation; open circles, analytical solution. Solid curves and open circles in (d), (e), and (f) are the normalized intensity distributions at Z = 4 corresponding to (a), (b), and (c), respectively. The parameters are chosen as P 0 P c = 1.3 , α = 0.1 .

Fig. 3
Fig. 3

Stationary propagation of ( 1 + 1 ) -dimensional (a) first-, (b) second-, and (c) third-order-mode HG solitons in the Gaussian-shaped response material up to a distance of Z = 10 . Solid curves, numerical simulation; open circles, analytical solution. Solid curves and open circles in (d), (e), and (f) are the normalized intensity distributions at Z = 4 corresponding to (a), (b), and (c), respectively. The parameters are chosen as P 0 P c = 1 , α = 0.1 .

Fig. 4
Fig. 4

Stationary propagation of the HG solitons (solid curves) and copropopagation of several Gaussian solitons (open circles) for (a) the first-order-mode HG soliton and two out-of-phase Gaussian solitons; (b) the second-order-mode HG soliton, two in-phase Gaussian solitons, and one out-of-phase Gaussian soliton; (c) the third-order-mode HG solitons and two in-phase and two out-of-phase Gaussian solitons in the Gaussian-shaped response material up to a distance of Z = 10 . Solid curves and open circles in (d), (e), and (f) are the normalized intensity distributions at Z = 4 corresponding to (a), (b), and (c), respectively. The parameters are chosen as P 0 P c = 1 , α = 0.1 , A 1 = 2.65 , a 1 = 0.41 ; A 2 = 77.64 , a 2 = 0.23 , B 2 = 153.23 ; A 3 = 4091.01 , a 3 = 0.1 , B 3 = 8120.5 .

Equations (39)

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

i φ z + μ Δ φ + k η φ R ( x x ) φ ( x , z ) 2 d D x = 0 ,
i φ z + μ Δ φ 1 2 k η γ P 0 r 2 φ = 0 ,
i φ z + μ 2 φ x 2 1 2 k η γ P 0 x 2 φ = 0 .
φ = F ( x , z ) φ G ( x , z ) .
i F z φ G + μ 2 F x 2 φ G + 2 μ F x φ G x + F ( i φ G z + μ 2 φ G x 2 1 2 k η γ P 0 x 2 φ G ) = 0 .
i φ G z + μ 2 φ G x 2 1 2 k η γ P 0 x 2 φ G = 0 ,
i F z φ G + μ 2 F x 2 φ G + 2 μ F x φ G x = 0 .
φ G = P 0 exp [ i θ ( z ) ] [ π w ( z ) ] 1 2 exp [ x 2 2 w ( z ) 2 + i c ( z ) x 2 ] ,
w ( z ) 2 = w 0 2 ( P c P 0 sin 2 β 0 z + cos 2 β 0 z ) ,
c ( z ) = k β 0 ( P c P 0 1 ) sin 2 β 0 z 4 ( cos 2 β 0 z + P c P 0 sin 2 β 0 z ) ,
θ ( z ) = 1 2 arctan ( P c P 0 tan β 0 z ) ,
P c = 1 k 2 γ η w 0 4 ,
i F z + μ 2 F x 2 + ( 4 i μ c 2 μ w ( z ) 2 ) x F x = 0 .
ξ = x w ( z ) , ζ = z
2 F ξ 2 2 ξ F ξ + 2 i k w ( ζ ) 2 F ζ = 0 .
d 2 X d ξ 2 2 ξ d X d ξ + 2 n X = 0 ,
d Θ d ζ + i n k w ( ζ ) 2 Θ = 0 ,
X = H n ( ξ ) = H n ( x w ( z ) ) ,
Θ = exp [ i n arctan ( P c P 0 tan β 0 z ) ] .
φ n = C n w ( z ) H n ( x w ( z ) ) exp [ x 2 2 w ( z ) 2 ] exp { i [ c ( z ) x 2 + ( 2 n + 1 ) θ ( z ) ] } ,
φ 0 = P 0 [ π w ( z ) ] 1 2 exp [ x 2 2 w ( z ) 2 ] exp { i [ c ( z ) x 2 + θ ( z ) ] } .
φ 1 = P 0 [ 2 π w ( z ) ] 1 2 2 x w ( z ) exp [ x 2 2 w ( z ) 2 ] exp { i [ c ( z ) x 2 + 3 θ ( z ) ] } ,
φ 2 = P 0 [ 8 π w ( z ) ] 1 2 { 4 [ x w ( z ) ] 2 2 } exp [ x 2 2 w ( z ) 2 ] exp { i [ c ( z ) x 2 + 5 θ ( z ) ] } ,
φ 3 = P 0 [ 48 π w ( z ) ] 1 2 { 8 [ x w ( z ) ] 3 12 [ x w ( z ) ] } exp [ x 2 2 w ( z ) 2 ] exp { i [ c ( z ) x 2 + 7 θ ( z ) ] } .
φ n = C n 1 w 0 H n ( x w 0 ) exp ( x 2 2 w 0 2 ) exp ( i β n z ) ,
β n = ( n + 1 2 ) β 0 = ( n + 1 2 ) η γ P 0 .
φ 0 = P 0 ( π w 0 ) 1 2 exp ( x 2 2 w 0 2 ) exp ( i β 0 z ) .
φ 01 ( x ) = A 1 { exp [ ( x a 1 ) 2 2 ] exp [ ( x + a 1 ) 2 2 ] } = A 1 exp ( x 2 + a 1 2 2 ) [ exp ( a 1 x ) exp ( a 1 x ) ] .
φ 1 ( x ) = 2 x exp ( x 2 2 ) .
φ 01 ( x ) = A 1 exp ( x 2 + a 1 2 2 ) [ 2 a 1 x + 1 3 a 1 3 x 3 + ] .
φ 02 ( x ) = A 2 { exp [ ( x a 2 ) 2 2 ] + exp [ ( x + a 2 ) 2 2 ] } B 2 exp ( x 2 2 ) = A 2 exp ( x 2 + a 2 2 2 ) [ exp ( a 2 x ) + exp ( a 2 x ) ] B 2 exp ( x 2 2 ) .
φ 2 ( x ) = ( 4 x 2 2 ) exp ( x 2 2 ) .
φ 02 ( x ) = exp ( x 2 2 ) [ 2 A 2 exp ( a 2 2 2 ) B 2 + A 2 a 2 2 x 2 exp ( a 2 2 2 ) + 1 12 A 2 a 2 4 x 4 exp ( a 2 2 2 ) + ] .
φ 03 ( x ) = A 3 { exp [ ( x 2 a 3 ) 2 2 ] exp [ ( x + 2 a 3 ) 2 2 ] } + B 3 { exp [ ( x + a 3 ) 2 2 ] exp [ ( x a 3 ) 2 2 ] } = A 3 exp ( x 2 + 4 a 3 2 2 ) [ exp ( 2 a 3 x ) exp ( 2 a 3 x ) ] B 3 exp ( x 2 + a 3 2 2 ) [ exp ( a 3 x ) exp ( a 3 x ) ] .
φ 3 ( x ) = ( 8 x 3 12 x ) exp ( x 2 2 ) .
φ 03 ( x ) = exp ( x 2 2 ) { [ 4 a 3 A 3 exp ( 2 a 3 2 ) 2 a 3 B 3 exp ( a 3 2 2 ) ] x + [ 8 3 a 3 3 A 3 exp ( 2 a 3 2 ) B 3 3 a 3 3 exp ( a 3 2 2 ) ] x 3 + } .
i φ ( 2 ) z + μ ( 2 φ ( 2 ) x 2 + 2 φ ( 2 ) y 2 ) 1 2 k η γ P 0 ( x 2 + y 2 ) φ ( 2 ) = 0 .
φ ( 2 ) ( x , y , z ) = C m n 1 w ( z ) H m ( x w ( z ) ) H n ( y w ( z ) ) exp [ ( x 2 + y 2 ) 2 w ( z ) 2 ] exp [ i c ( z ) ( x 2 + y 2 ) ] × exp [ i 2 ( m + n + 1 ) θ ( z ) ] ,
φ 00 = P 0 π w ( z ) exp [ ( x 2 + y 2 ) 2 w ( z ) 2 ] exp [ i c ( z ) ( x 2 + y 2 ) ] exp [ i 2 θ ( z ) ] .

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