Abstract

The propagation of solitary waves, so-called nematicons, in a nonlinear nematic liquid crystal is considered in the nonlocal regime. Approximate modulation equations governing the evolution of input beams into steady nematicons are derived by using suitable trial functions in a Lagrangian formulation of the equations for a nematic liquid crystal. The variational equations are then extended to include the effect of diffractive loss as the beam evolves. It is found that the nonlocal nature of the interaction between the light and the nematic has a significant effect on the form of this diffractive radiation. Furthermore, it is this shed radiation that allows the input beam to evolve to a steady nematicon. Finally, excellent agreement is found between solutions of the modulation equations and numerical solutions of the nematic liquid-crystal equations.

© 2007 Optical Society of America

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  1. G. Assanto, M. Peccianti, and C. Conti, "Nemations: optical spatial solitons in nematic liquid crystals," Opt. Photonics News Feb. 2003, pp. 44-48.
    [CrossRef]
  2. C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003).
    [CrossRef] [PubMed]
  3. 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]
  4. M. Peccianti, C. Conti, G. Assanto, A. de Luca, and C. Umeton, "Routing of anisotropic spatial solitons and modulational instability in liquid crystals," Nature 432, 733-737 (2004).
    [CrossRef] [PubMed]
  5. E. A. Kuznetsov and A. M. Rubenchik, "Soliton stabilization in plasmas and hydrodynamics," Phys. Rep. 142, 103-165 (1986).
    [CrossRef]
  6. W. L. Kath and N. F. Smyth, "Soliton evolution and radiation loss for the nonlinear Schrödinger equation," Phys. Rev. E 51, 1484-1492 (1995).
    [CrossRef]
  7. C. García Reimbert, A. A. Minzoni, and N. F. Smyth, "Spatial soliton evolution in nematic liquid crystals in the nonlinear local regime," J. Opt. Soc. Am. B 23, 294-301 (2006).
    [CrossRef]
  8. C. García Reimbert, A. A. Minzoni, N. F. Smyth, and A. L. Worthy, "Large-amplitude nematicon propagation in a liquid crystal with local response," J. Opt. Soc. Am. B 23, 2551-2558 (2006).
    [CrossRef]
  9. P. D. Rasmussen, O. Bang, and W. Królikowski, "Theory of nonlocal soliton interaction in nematic liquid crystals," Phys. Rev. E 72, 066611 (2005).
    [CrossRef]
  10. 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]
  11. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).
  12. J. Yang, "Vector solitons and their internal oscillations in birefringent nonlinear optical fibers," Stud. Appl. Math. 98, 61-97 (1997).
    [CrossRef]
  13. B. Fornberg and G. B. Whitham, "A numerical and theoretical study of certain nonlinear wave phenomena," Philos. Trans. R. Soc. London, Ser. A 289, 373-403 (1978).
    [CrossRef]
  14. W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Whyller, J. J. Rasmussen, and D. Edmundson, "Modulation instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
    [CrossRef]
  15. Y. Huang, Q. Guo, and J. Cao, "Optical beams in lossy non-local Kerr media," Opt. Commun. 261, 175-180 (2006).
    [CrossRef]
  16. M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells," Appl. Phys. Lett. 77, 7-9 (2000).
    [CrossRef]

2006 (4)

C. García Reimbert, A. A. Minzoni, and N. F. Smyth, "Spatial soliton evolution in nematic liquid crystals in the nonlinear local regime," J. Opt. Soc. Am. B 23, 294-301 (2006).
[CrossRef]

C. García Reimbert, A. A. Minzoni, N. F. Smyth, and A. L. Worthy, "Large-amplitude nematicon propagation in a liquid crystal with local response," J. Opt. Soc. Am. B 23, 2551-2558 (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]

Y. Huang, Q. Guo, and J. Cao, "Optical beams in lossy non-local Kerr media," Opt. Commun. 261, 175-180 (2006).
[CrossRef]

2005 (1)

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

2004 (3)

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]

M. Peccianti, C. Conti, G. Assanto, A. de Luca, and C. Umeton, "Routing of anisotropic spatial solitons and modulational instability in liquid crystals," Nature 432, 733-737 (2004).
[CrossRef] [PubMed]

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Whyller, J. J. Rasmussen, and D. Edmundson, "Modulation instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

2003 (2)

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

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

2000 (1)

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells," Appl. Phys. Lett. 77, 7-9 (2000).
[CrossRef]

1997 (1)

J. Yang, "Vector solitons and their internal oscillations in birefringent nonlinear optical fibers," Stud. Appl. Math. 98, 61-97 (1997).
[CrossRef]

1995 (1)

W. L. Kath and N. F. Smyth, "Soliton evolution and radiation loss for the nonlinear Schrödinger equation," Phys. Rev. E 51, 1484-1492 (1995).
[CrossRef]

1986 (1)

E. A. Kuznetsov and A. M. Rubenchik, "Soliton stabilization in plasmas and hydrodynamics," Phys. Rep. 142, 103-165 (1986).
[CrossRef]

1978 (1)

B. Fornberg and G. B. Whitham, "A numerical and theoretical study of certain nonlinear wave phenomena," Philos. Trans. R. Soc. London, Ser. A 289, 373-403 (1978).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

Assanto, G.

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]

M. Peccianti, C. Conti, G. Assanto, A. de Luca, and C. Umeton, "Routing of anisotropic spatial solitons and modulational instability in liquid crystals," Nature 432, 733-737 (2004).
[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]

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

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells," Appl. Phys. Lett. 77, 7-9 (2000).
[CrossRef]

Bang, O.

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

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Whyller, J. J. Rasmussen, and D. Edmundson, "Modulation instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

Cao, J.

Y. Huang, Q. Guo, and J. Cao, "Optical beams in lossy non-local Kerr media," Opt. Commun. 261, 175-180 (2006).
[CrossRef]

Conti, C.

M. Peccianti, C. Conti, G. Assanto, A. de Luca, and C. Umeton, "Routing of anisotropic spatial solitons and modulational instability in liquid crystals," Nature 432, 733-737 (2004).
[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]

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

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

de Luca, A.

M. Peccianti, C. Conti, G. Assanto, A. de Luca, and C. Umeton, "Routing of anisotropic spatial solitons and modulational instability in liquid crystals," Nature 432, 733-737 (2004).
[CrossRef] [PubMed]

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells," Appl. Phys. Lett. 77, 7-9 (2000).
[CrossRef]

De Rossi, A.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells," Appl. Phys. Lett. 77, 7-9 (2000).
[CrossRef]

Edmundson, D.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Whyller, J. J. Rasmussen, and D. Edmundson, "Modulation instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

Fornberg, B.

B. Fornberg and G. B. Whitham, "A numerical and theoretical study of certain nonlinear wave phenomena," Philos. Trans. R. Soc. London, Ser. A 289, 373-403 (1978).
[CrossRef]

García Reimbert, C.

Guo, Q.

Y. Huang, Q. Guo, and J. Cao, "Optical beams in lossy non-local Kerr media," Opt. Commun. 261, 175-180 (2006).
[CrossRef]

Huang, Y.

Y. Huang, Q. Guo, and J. Cao, "Optical beams in lossy non-local Kerr media," Opt. Commun. 261, 175-180 (2006).
[CrossRef]

Kath, W. L.

W. L. Kath and N. F. Smyth, "Soliton evolution and radiation loss for the nonlinear Schrödinger equation," Phys. Rev. E 51, 1484-1492 (1995).
[CrossRef]

Khoo, I. C.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells," Appl. Phys. Lett. 77, 7-9 (2000).
[CrossRef]

Królikowski, W.

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

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Whyller, J. J. Rasmussen, and D. Edmundson, "Modulation instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

Kuznetsov, E. A.

E. A. Kuznetsov and A. M. Rubenchik, "Soliton stabilization in plasmas and hydrodynamics," Phys. Rep. 142, 103-165 (1986).
[CrossRef]

Lashkin, V. M.

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]

Minzoni, A. A.

Neshev, D.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Whyller, J. J. Rasmussen, and D. Edmundson, "Modulation instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

Nikolov, N. I.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Whyller, J. J. Rasmussen, and D. Edmundson, "Modulation instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

Peccianti, M.

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]

M. Peccianti, C. Conti, G. Assanto, A. de Luca, and C. Umeton, "Routing of anisotropic spatial solitons and modulational instability in liquid crystals," Nature 432, 733-737 (2004).
[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]

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

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells," Appl. Phys. Lett. 77, 7-9 (2000).
[CrossRef]

Prikhodko, O. O.

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]

Rasmussen, J. J.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Whyller, J. J. Rasmussen, and D. Edmundson, "Modulation instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

Rasmussen, P. D.

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

Rubenchik, A. M.

E. A. Kuznetsov and A. M. Rubenchik, "Soliton stabilization in plasmas and hydrodynamics," Phys. Rep. 142, 103-165 (1986).
[CrossRef]

Smyth, N. F.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

Umeton, C.

M. Peccianti, C. Conti, G. Assanto, A. de Luca, and C. Umeton, "Routing of anisotropic spatial solitons and modulational instability in liquid crystals," Nature 432, 733-737 (2004).
[CrossRef] [PubMed]

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells," Appl. Phys. Lett. 77, 7-9 (2000).
[CrossRef]

Whitham, G. B.

B. Fornberg and G. B. Whitham, "A numerical and theoretical study of certain nonlinear wave phenomena," Philos. Trans. R. Soc. London, Ser. A 289, 373-403 (1978).
[CrossRef]

Whyller, J.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Whyller, J. J. Rasmussen, and D. Edmundson, "Modulation instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

Worthy, A. L.

Yakimenko, A. I.

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]

Yang, J.

J. Yang, "Vector solitons and their internal oscillations in birefringent nonlinear optical fibers," Stud. Appl. Math. 98, 61-97 (1997).
[CrossRef]

Appl. Phys. Lett. (1)

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells," Appl. Phys. Lett. 77, 7-9 (2000).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (1)

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Whyller, J. J. Rasmussen, and D. Edmundson, "Modulation instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

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

Nature (1)

M. Peccianti, C. Conti, G. Assanto, A. de Luca, and C. Umeton, "Routing of anisotropic spatial solitons and modulational instability in liquid crystals," Nature 432, 733-737 (2004).
[CrossRef] [PubMed]

Opt. Commun. (1)

Y. Huang, Q. Guo, and J. Cao, "Optical beams in lossy non-local Kerr media," Opt. Commun. 261, 175-180 (2006).
[CrossRef]

Opt. Photonics News (1)

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

Philos. Trans. R. Soc. London, Ser. A (1)

B. Fornberg and G. B. Whitham, "A numerical and theoretical study of certain nonlinear wave phenomena," Philos. Trans. R. Soc. London, Ser. A 289, 373-403 (1978).
[CrossRef]

Phys. Rep. (1)

E. A. Kuznetsov and A. M. Rubenchik, "Soliton stabilization in plasmas and hydrodynamics," Phys. Rep. 142, 103-165 (1986).
[CrossRef]

Phys. Rev. E (3)

W. L. Kath and N. F. Smyth, "Soliton evolution and radiation loss for the nonlinear Schrödinger equation," Phys. Rev. E 51, 1484-1492 (1995).
[CrossRef]

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

Phys. Rev. Lett. (2)

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]

Stud. Appl. Math. (1)

J. Yang, "Vector solitons and their internal oscillations in birefringent nonlinear optical fibers," Stud. Appl. Math. 98, 61-97 (1997).
[CrossRef]

Other (1)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of a liquid-crystal cell with a propagating polarized light beam.

Fig. 2
Fig. 2

Numerical solution nematicon equations (1, 2) at z = 400 for the initial values a = 0.5 , w = 4 , with α and β determined by the first of Eqs. (17) and Eq. (19). The parameter values are q = 2 and ν = 10 . (a) Solution for E : solid curve; solution for θ: dashed curve. (b) Solution for Re ( E ) : solid curve; solution for θ: dashed curve.

Fig. 3
Fig. 3

Amplitude a for the electric field E as a function of z for the initial conditions a = 0.5 , w = 4 , with α and β determined by the first of Eqs. (17) and Eq. (19). The parameter values are q = 2 and ν = 10 . Numerical solution of nematicon equations (1, 2): solid curve; solution of modulation equations (14, 16), the first of Eqs. (17), and Eqs. (18, 19, 24): dashed curve.

Fig. 4
Fig. 4

Amplitude a for the electric field E as a function of z for the initial conditions a = 0.6 , w = 4 , with α and β determined by the first of Eqs. (17) and Eq. (19). The parameter values are q = 2 and ν = 10 . Numerical solution of nematicon equations (1, 2): solid curve; solution of modulation equations (14, 16), the first of Eqs. (17), and Eqs. (18, 19, 24): dashed curve.

Fig. 5
Fig. 5

Amplitude a for the electric field E as a function of z for the initial conditions a = 0.7 , w = 4 , with α and β determined by the first of Eqs. (17) and Eq. (19). The parameter values are q = 2 and ν = 10 . Numerical solution of nematicon equations (1, 2): solid curve; solution of modulation equations (14, 16), the first of Eqs. (17), and Eqs. (18, 19, 24): dashed curve.

Fig. 6
Fig. 6

Amplitude a for the electric field E as a function of z for the initial conditions a = 0.4 , w = 4 , with α and β determined by the first of Eqs. (17) and Eq. (19). The parameter values are q = 2 and ν = 10 . Numerical solution of nematicon equations (1, 2): solid curve; solution of modulation equations (14, 16), the first of Eqs. (17), and Eqs. (18, 19, 24): dashed curve.

Equations (41)

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

i E z + 1 2 2 E + sin ( 2 θ ) E = 0 ,
ν 2 θ q sin ( 2 θ ) = 2 E 2 cos ( 2 θ ) .
i E z + 1 2 2 E + 2 θ E = 0 , ν 2 θ 2 q θ = 2 E 2 ,
L = 0 Z 0 [ i r ( E * E z E E z * ) r E r 2 + 4 r θ E 2 ν r θ r 2 2 q r θ 2 ] d r d z ,
E = a sech ( r w ) e i σ + i g e i σ , θ = α sech 2 ( r β ) .
0 4 r θ E 2 d r = 4 α a 2 0 r sech 2 r β sech 2 r w d r .
0 4 r θ E 2 d r = 4 α a 2 0 r e r 2 ( 1 ( A β ) 2 + 1 ( B w ) 2 ) d r = 2 α a 2 A 2 B 2 w 2 β 2 A 2 β 2 + B 2 w 2 .
A = 2 I 2 I x 32 , B = 2 I 2 ,
I 2 = 0 x sech 2 x d x = ln 2 ,
I x 32 = 0 x 3 sech 2 x d x = 1.352301002 .
L = 2 ( a 2 w 2 I 2 + Λ g 2 ) σ 2 I 1 a w 2 g + 2 I 1 g w 2 a + 4 I 1 a w g w a 2 I 22 4 ν I 42 α 2 2 q I 4 α 2 β 2 + 2 A 2 B 2 α a 2 β 2 w 2 A 2 β 2 + B 2 w 2 .
Λ = 1 2 l 2
I 22 = 0 x sech 2 x tanh 2 x d x = 1 3 ln 2 + 1 6 ,
I 1 = 0 x sech x d x = 2 C ,
I 42 = 0 x sech 4 x tanh 2 x d x = 2 15 ln 2 + 1 60 ,
I 4 = 0 x sech 4 x d x = 2 3 ln 2 1 6 ,
d d z ( I 2 a 2 w 2 + Λ g 2 ) = 0 ,
d d z ( I 1 a w 2 ) = Λ g d σ d z ,
I 1 d g d z = I 22 a 2 w 2 A 2 B 4 α a w 2 β 2 ( A 2 β 2 + B 2 w 2 ) 2 ,
I 2 d σ d z = I 22 w 2 + A 2 B 2 α β 2 ( A 2 β 2 + 2 B 2 w 2 ) ( A 2 β 2 + B 2 w 2 ) 2 ,
α = A 2 B 2 β 2 w 2 a 2 2 ( A 2 β 2 + B 2 w 2 ) ( 2 ν I 42 + q I 4 β 2 ) ,
α = A 2 B 4 w 4 a 2 q I 4 ( A 2 β 2 + B 2 w 2 ) 2 .
d H d z = d d z 0 r [ E r 2 4 θ E 2 + ν θ r 2 + 2 q θ 2 ] d r = d d z ( I 22 a 2 + 4 ν I 42 α 2 + 2 q I 4 α 2 β 2 2 A 2 B 2 α a 2 w 2 β 2 A 2 β 2 + B 2 w 2 ) = 0 .
β 2 = q I 4 B 2 w 2 + q 2 I 4 2 B 4 w 4 + 16 ν q I 42 I 4 A 2 B 2 w 2 2 q A 2 I 4 .
a ̂ 2 = I 22 ( A 2 β ̂ 2 + B 2 w ̂ 2 ) 3 ( 2 ν I 42 + q I 4 β ̂ 2 ) A 4 B 6 β ̂ 4 w ̂ 6 ,
Λ = σ ̂ I 1 2 ( w ̂ 2 + 2 a ̂ w ̂ ϴ ) Q ,
i E z + 1 2 2 E r 2 + 1 2 r E r = 0 .
Λ ̃ = 1 2 ρ 2 ,
I 1 d g d z = I 22 a 2 w 2 A 2 B 4 α a w 2 β 2 ( A 2 β 2 + B 2 w 2 ) 2 2 δ g ,
δ = 2 π I 1 32 e R Λ ̃ 0 z π R ( z ) ln ( ( z z ) Λ ̃ ) { [ 1 4 ln ( ( z z ) Λ ̃ ) ] 2 + 3 π 2 16 } 2 + π 2 [ ln ( ( z z ) Λ ̃ ) ] 2 16 d z ( z z ) ,
R 2 = 1 Λ ̃ [ I 2 a 2 w 2 I 2 a ̂ 2 w ̂ 2 + Λ ̃ g 2 ] .
ν 2 θ r 2 + ν r θ r 2 q θ = q sin ( 2 θ ) 2 q θ 2 E 2 cos ( 2 θ ) .
a = a ̂ + a 1 , w = w ̂ + w 1 , α = α ̂ + α 1 , β = β ̂ + β 1 , g = g 1 , σ = σ ̂ + σ 1 ,
d 2 g 1 d z 2 Q Λ σ ̂ I 1 2 ( w ̂ 2 + 2 a ̂ w ̂ ϴ ) g 1 = 0 ,
ϴ = 2 I 22 D 2 a ̂ A 2 B 2 D β ̂ 2 w ̂ 2 a ̂ ( 2 α ̂ + Γ 1 a ̂ ) A 2 B 2 β ̂ w ̂ a ̂ 2 ( 2 A 2 α ̂ β ̂ 3 + Γ 2 D β ̂ w ̂ + 2 B 2 Γ 3 α ̂ w ̂ 3 ) ,
Q = Q 1 I 22 2 A 2 B 4 α ̂ β ̂ 2 w ̂ 4 [ Q 2 + ϴ Q 3 ] ,
Q 1 = I 22 2 w ̂ 2 ( 1 2 a ̂ w ̂ ϴ ) , Q 2 = α ̂ 2 β ̂ 2 w ̂ 2 + Γ 1 β ̂ 2 w ̂ 2 a ̂ ,
Q 3 = 2 α ̂ β ̂ 2 w ̂ a ̂ + Γ 2 β ̂ 2 w ̂ 2 a ̂ + 2 Γ 3 α ̂ β ̂ w ̂ 2 a ̂ 4 D 1 α ̂ β ̂ 2 w ̂ 2 a ̂ ( A 2 Γ 3 β ̂ + B 2 w ̂ ) ,
Γ 1 = 2 A 2 B 4 w ̂ 4 a ̂ q I 4 D 2 , Γ 2 = 4 A 4 B 4 β ̂ w ̂ 3 a ̂ 2 ( β ̂ Γ 3 w ̂ ) q I 4 D 3 ,
Γ 3 = B 2 w ̂ ( q I 4 β ̂ 2 + 4 ν I 3 ) q I 4 β ̂ ( 2 A 2 β ̂ 2 B 2 w ̂ 2 ) , D = A 2 β ̂ 2 + B 2 w ̂ 2 .
Λ = σ ̂ I 1 2 ( w ̂ 2 + 2 a ̂ w ̂ ϴ ) Q .

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