Abstract

We study the propagation of spatial solitons close to the lateral boundaries of self-focusing nonlocal media. Using the Green function approach, we investigate the soliton induced index distribution and resulting trajectories in four cases of physical relevance.

© 2007 Optical Society of America

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References

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  1. H. L. Pecseli and J. J. Rasmussen, "Nonlinear electron waves in strongly magnetized plasma," Plasma Phys. 22, 421-438 (1980).
    [CrossRef]
  2. P. Pedri and L. Santos, "Two-dimensional bright solitons in dipolar Bose-Einstein condensates," Phys. Rev. Lett. 95, 200404 (2005).
    [CrossRef] [PubMed]
  3. G. Vitrant, R. Reinisch, J. C. Paumier, G. Assanto, and G. I. Stegeman, "Nonlinear prism coupling with nonlocality," Opt. Lett. 14, 898-890 (1989).
    [CrossRef] [PubMed]
  4. Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003).
  5. S. Trillo and W. E. Torruellas, eds., Spatial Solitons (Springer-Verlag, 2001).
  6. C. Conti and G. Assanto, "Nonlinear optics applications: bright spatial solitons," in Encyclopedia of Modern Optics, Vol. 5, pp. 43-55, R.D.Guenther, D.G.Steel, and L.Bayvel, eds. (Elsevier, 2004).
  7. E. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, M. Lisak, and L. Vazquez, "Dissipative optical solitons," Phys. Rev. A 49, 2806-2811 (1994).
    [CrossRef] [PubMed]
  8. 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]
  9. D. J. Mitchell and A. W. Snyder, "Soliton dynamics in a nonlocal medium," J. Opt. Soc. Am. B 16, 236-239 (1999).
    [CrossRef]
  10. W. Krolikowski and O. Bang, "Solitons in nonlocal nonlinear media: exact solutions," Phys. Rev. E 63, 016610 (2000).
    [CrossRef]
  11. C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003).
    [CrossRef] [PubMed]
  12. G. Assanto and M. Peccianti, "Spatial solitons in nematic liquid crystals," IEEE J. Quantum Electron. 39, 13-21 (2003).
    [CrossRef]
  13. M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "Nonlocal optical propagation in nonlinear nematic liquid crystals," J. Nonlinear Opt. Phys. Mater. 12, 525-538 (2003).
    [CrossRef]
  14. 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]
  15. C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons," Phys. Rev. Lett. 95, 213904 (2005).
    [CrossRef] [PubMed]
  16. Z. Xu, Y. V. Kartashov, and L. Torner, "Stabilization of vector soliton complexes in nonlocal nonlinear media," Phys. Rev. E 73, 055601 (2006).
    [CrossRef]
  17. W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, "Observation of dipole-mode vector solitons," Phys. Rev. Lett. 85, 1424-1427 (2000).
    [CrossRef] [PubMed]
  18. T. Carmon, C. Anastassiou, S. Lan, D. Kip, Z. H. Musslimani, M. Segev, and D. Christodoulides, "Observation of two-dimensional multimode solitons," Opt. Lett. 25, 1113-1115 (2000).
    [CrossRef]
  19. Y. A. 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]
  20. A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, "Observation of attraction between dark solitons," Phys. Rev. Lett. 96, 043901 (2006).
    [CrossRef] [PubMed]
  21. M. Segev, B. Crosignani, A. Yariv, and B. Fischer, "Spatial solitons in photorefractive media," Phys. Rev. Lett. 68, 923-926 (1992).
    [CrossRef] [PubMed]
  22. W. Krolikowsky, B. Luther-Davies, and C. Denz, "Photorefractive solitons," Quantum Electron. 39, 3-12 (2003).
  23. C. Conti, G. Ruocco, and S. Trillo, "Optical spatial solitons in soft matter," Phys. Rev. Lett. 95, 183902 (2005).
    [CrossRef] [PubMed]
  24. M. Warenghem, J. F. Henninot, F. Derrien, and G. Abbate, "Thermal and orientational 2D+1 spatial optical solitons in dye doped liquid crystals," Mol. Cryst. Liq. Cryst. 373, 213-225 (2002).
    [CrossRef]
  25. M. Peccianti, K. Brzadkiewicz, and G. Assanto, "Nonlocal spatial soliton interactions in nematic liquid crystals," Opt. Lett. 27, 1460-1462 (2002).
    [CrossRef]
  26. M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "Routing of anisotropic spatial solitons and modulational instability in nematic liquid crystals," Nature 432, 733-737 (2004).
    [CrossRef] [PubMed]
  27. A. Alberucci, M. Peccianti, G. Assanto, A. Dyadyusha, and M. Kaczmarek, "Two-color vector solitons in nonlocal media," Phys. Rev. Lett. 97, 153903 (2006).
    [CrossRef] [PubMed]
  28. M. Peccianti, A. Fratalocchi, and G. Assanto, "Transverse dynamics of nematicons," Opt. Express 12, 6524-6529 (2004).
    [CrossRef] [PubMed]
  29. B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, "Boundary force effects exerted on solitons in highly nonlocal nonlinear media," Opt. Lett. 32, 154-156 (2007).
    [CrossRef]
  30. C. Rothschild, B. Alfassi, O. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nat. Phys. 2, 769-774 (2006).
    [CrossRef]
  31. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, 2001).

2007

2006

A. Alberucci, M. Peccianti, G. Assanto, A. Dyadyusha, and M. Kaczmarek, "Two-color vector solitons in nonlocal media," Phys. Rev. Lett. 97, 153903 (2006).
[CrossRef] [PubMed]

C. Rothschild, B. Alfassi, O. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nat. Phys. 2, 769-774 (2006).
[CrossRef]

Z. Xu, Y. V. Kartashov, and L. Torner, "Stabilization of vector soliton complexes in nonlocal nonlinear media," Phys. Rev. E 73, 055601 (2006).
[CrossRef]

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

2005

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons," Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

C. Conti, G. Ruocco, and S. Trillo, "Optical spatial solitons in soft matter," Phys. Rev. Lett. 95, 183902 (2005).
[CrossRef] [PubMed]

P. Pedri and L. Santos, "Two-dimensional bright solitons in dipolar Bose-Einstein condensates," Phys. Rev. Lett. 95, 200404 (2005).
[CrossRef] [PubMed]

2004

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 nematic liquid crystals," Nature 432, 733-737 (2004).
[CrossRef] [PubMed]

M. Peccianti, A. Fratalocchi, and G. Assanto, "Transverse dynamics of nematicons," Opt. Express 12, 6524-6529 (2004).
[CrossRef] [PubMed]

2003

W. Krolikowsky, B. Luther-Davies, and C. Denz, "Photorefractive solitons," Quantum Electron. 39, 3-12 (2003).

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 and M. Peccianti, "Spatial solitons in nematic liquid crystals," IEEE J. Quantum Electron. 39, 13-21 (2003).
[CrossRef]

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "Nonlocal optical propagation in nonlinear nematic liquid crystals," J. Nonlinear Opt. Phys. Mater. 12, 525-538 (2003).
[CrossRef]

2002

M. Warenghem, J. F. Henninot, F. Derrien, and G. Abbate, "Thermal and orientational 2D+1 spatial optical solitons in dye doped liquid crystals," Mol. Cryst. Liq. Cryst. 373, 213-225 (2002).
[CrossRef]

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

2000

T. Carmon, C. Anastassiou, S. Lan, D. Kip, Z. H. Musslimani, M. Segev, and D. Christodoulides, "Observation of two-dimensional multimode solitons," Opt. Lett. 25, 1113-1115 (2000).
[CrossRef]

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

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, "Observation of dipole-mode vector solitons," Phys. Rev. Lett. 85, 1424-1427 (2000).
[CrossRef] [PubMed]

1999

1998

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]

1994

E. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, M. Lisak, and L. Vazquez, "Dissipative optical solitons," Phys. Rev. A 49, 2806-2811 (1994).
[CrossRef] [PubMed]

1992

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, "Spatial solitons in photorefractive media," Phys. Rev. Lett. 68, 923-926 (1992).
[CrossRef] [PubMed]

1989

1980

H. L. Pecseli and J. J. Rasmussen, "Nonlinear electron waves in strongly magnetized plasma," Plasma Phys. 22, 421-438 (1980).
[CrossRef]

IEEE J. Quantum Electron.

G. Assanto and M. Peccianti, "Spatial solitons in nematic liquid crystals," IEEE J. Quantum Electron. 39, 13-21 (2003).
[CrossRef]

J. Nonlinear Opt. Phys. Mater.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "Nonlocal optical propagation in nonlinear nematic liquid crystals," J. Nonlinear Opt. Phys. Mater. 12, 525-538 (2003).
[CrossRef]

J. Opt. Soc. Am. B

Mol. Cryst. Liq. Cryst.

M. Warenghem, J. F. Henninot, F. Derrien, and G. Abbate, "Thermal and orientational 2D+1 spatial optical solitons in dye doped liquid crystals," Mol. Cryst. Liq. Cryst. 373, 213-225 (2002).
[CrossRef]

Nat. Phys.

C. Rothschild, B. Alfassi, O. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nat. Phys. 2, 769-774 (2006).
[CrossRef]

Nature

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

Opt. Express

Opt. Lett.

Phys. Rev. A

E. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, M. Lisak, and L. Vazquez, "Dissipative optical solitons," Phys. Rev. A 49, 2806-2811 (1994).
[CrossRef] [PubMed]

Phys. Rev. E

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]

Y. A. 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]

Z. Xu, Y. V. Kartashov, and L. Torner, "Stabilization of vector soliton complexes in nonlocal nonlinear media," Phys. Rev. E 73, 055601 (2006).
[CrossRef]

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

Phys. Rev. Lett.

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, G. Ruocco, and S. Trillo, "Optical spatial solitons in soft matter," Phys. Rev. Lett. 95, 183902 (2005).
[CrossRef] [PubMed]

P. Pedri and L. Santos, "Two-dimensional bright solitons in dipolar Bose-Einstein condensates," Phys. Rev. Lett. 95, 200404 (2005).
[CrossRef] [PubMed]

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, "Observation of dipole-mode vector solitons," Phys. Rev. Lett. 85, 1424-1427 (2000).
[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]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons," Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

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

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, "Spatial solitons in photorefractive media," Phys. Rev. Lett. 68, 923-926 (1992).
[CrossRef] [PubMed]

A. Alberucci, M. Peccianti, G. Assanto, A. Dyadyusha, and M. Kaczmarek, "Two-color vector solitons in nonlocal media," Phys. Rev. Lett. 97, 153903 (2006).
[CrossRef] [PubMed]

Plasma Phys.

H. L. Pecseli and J. J. Rasmussen, "Nonlinear electron waves in strongly magnetized plasma," Plasma Phys. 22, 421-438 (1980).
[CrossRef]

Quantum Electron.

W. Krolikowsky, B. Luther-Davies, and C. Denz, "Photorefractive solitons," Quantum Electron. 39, 3-12 (2003).

Other

Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003).

S. Trillo and W. E. Torruellas, eds., Spatial Solitons (Springer-Verlag, 2001).

C. Conti and G. Assanto, "Nonlinear optics applications: bright spatial solitons," in Encyclopedia of Modern Optics, Vol. 5, pp. 43-55, R.D.Guenther, D.G.Steel, and L.Bayvel, eds. (Elsevier, 2004).

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, 2001).

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Figures (9)

Fig. 1
Fig. 1

(a), (b) Perturbation profiles versus ξ for ξ = 0.5 , 0.63, 0.76, and 0.9 (solid curve, squares, stars, and triangles, respectively) for (a) ω = 0.001 and (b) ω = 0.09 . The profile in (a) is very similar to the Green function, as the soliton is much narrower than the sample width. (c) Calculated α (squares) and α s for ξ = 0.54 (no symbols), ξ = 0.72 (stars), and ξ = 0.86 (triangles), respectively, versus normalized waist ω. (d) Calculated α g for the same set of soliton positions.

Fig. 2
Fig. 2

(a) θ n ξ calculated for ω = 0.02 , ξ = 0.5 (solid curve), ω = 0.02 , ξ = 0.75 (asterisks), ω = 0.1 , ξ = 0.5 , (triangles), and ω = 0.1 , ξ = 0.75 (squares), respectively. The numerical results are in complete agreement with the theoretical approximation. (b), (c) Calculated degree of nonlocality α along x and (d) along y for ξ = 0.5 (circles), ξ = 0.54 (solid curve), ξ = 0.68 (squares), and ξ = 0.81 (triangles), respectively.

Fig. 3
Fig. 3

Perturbation profiles for (a), (c) ω = 0.01 and (b),(d) ω = 0.1 for (a),(b) ξ = 0.5 and (c),(d) ξ = 0.75 . The dashed (solid) curves correspond to profiles along υ ( ξ ξ ) . The profiles are chosen to contain the perturbation peak. Squares (triangles) are the corresponding values computed using a full numerical approach.

Fig. 4
Fig. 4

Calculated figure of nonlocality α x (curves for α y and α x y g s are nearly identical) versus ω for ξ = 0.5 and (a), (b) μ κ = 10 2 or (c), (d) μ κ = 10 4 . In this range for μ κ , nonlocality does not depend on beam position. When α x = 1 , perturbation and excitation have the same profile, i.e., the medium is local. (b)–(d): beam profile along υ (symbols) and ξ ξ (solid curve) for μ κ = 10 2 and μ κ = 10 4 , respectively, when ω = 0.034 and ξ = 0.5 ; in both cases the perturbation possesses radial symmetry.

Fig. 5
Fig. 5

Plots of g 1 (a), (d) and g 2 (b), (e) for ξ = 0.5 . The corresponding profiles are plotted in (c) and (f) versus ξ ( g 1 solid curve, g 2 squares) and υ ( g 1 dashed curve, g 2 triangles), normalized to one (the cross sections are in υ = 0 and ξ = 0.5 , respectively). Results for g 1 and g 2 perfectly overlap. Excitation waists are ω = 0.03 (a)–(c) and ω = 0.1 (d)–(f), respectively.

Fig. 6
Fig. 6

As in Fig. 5 but with soliton beams centered in ξ = 0.75 .

Fig. 7
Fig. 7

Calculated perturbation ψ versus power for γ = 1.3 × 10 9 V 2 , L = 100 μ m , w = 2.6 μ m , and θ 0 = 0.42 . Dashed and solid curves stem from retaining first- and second-order terms in the perturbative series, respectively. Squares (and solid-curve linear interpolation) are numerical solutions obtained by a standard relaxation method. The beam is centered in the midplane x = 50 μ m .

Fig. 8
Fig. 8

(a) W 0 versus initial (i.e., at z = 0 ) beam position x for the Poisson equation in 1D [see Eq. (17)]; the curves for different ω overlap. In (b) the corresponding oscillation period Λ is graphed versus power density P L , with dependence on the square root of P L . Given the linearity of W 0 , Λ does not depend on the initial beam position. (c) W 0 in the screened Poisson case for μ κ = 0 , i.e., the 2D Poisson case (squares w = 10 μ m , solid curve w = 2.2 μ m ), and μ κ = 100 (triangles w = 10 μ m , circles w = 2.2 μ m ). (d) Soliton trajectories in the 2D Poisson case for different x ( z = 0 ) , considering a null initial velocity and a power of 2 mW . (e) Corresponding Λ versus initial soliton position. For all cases L = 100 μ m and κ = 1 .

Fig. 9
Fig. 9

(a) W 0 for NLC as computed from Eq. (19), for P = 2 2 mW (solid curve) and P = 3 2 2 mW (dashed curve). The curves for w = 2.2 and 2.8 μ m overlap. (b) Oscillation period Λ versus initial beam position for P = 2 2 mW (stars w = 2.8 μ m , solid curve w = 2.2 μ m ) and P = 3 2 2 mW (squares w = 2.8 μ m , triangles w = 2.2 μ m ). In all cases L = 100 μ m .

Equations (27)

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2 i k A z + T 2 A + 2 n 0 k 0 2 Δ n ( A ) A = 0 ,
α i = F W H M i A 2 F W H M i Δ n ( i = x , y ) .
α x y g s = F W H M x y A 2 ( 2 σ x y g s ) ,
β d 2 ψ d x 2 + A 2 = 0 ,
G ( ξ , ζ ) = ξ κ ( 1 ζ ) u ( ζ ξ ) + ζ κ ( 1 ξ ) u ( ξ ζ ) ,
ψ = ω 2 κ { N 1 + N 2 } ,
N 1 = π { erf ( 1 ξ ω ) ( 1 ξ ) ξ + erf ( ξ ξ ω ) ( ξ ξ ) + ξ ( ξ 1 ) erf ( ξ ω ) } ,
N 2 = ω [ exp ( ξ 2 ω 2 ) exp ( ( ξ ξ ) 2 ω 2 ) ξ ( exp ( ξ 2 ω 2 ) exp ( ( 1 ξ ) 2 ω 2 ) ) ] .
β 2 ψ ( x , y ) + A ( x , y ) 2 = 0 ,
G ( ξ , υ , ζ , η ) = n = 1 1 c n κ sin ( π n ζ ) exp ( c n υ η ) sin ( π n ξ ) ,
ψ ( ξ , υ ) = d η 0 1 G ( ξ , υ , ζ , η ) A ( ζ , η ) 2 d ζ = n = 1 1 π n κ sin ( π n ξ ) θ n ( υ ) ,
θ n υ ( υ ) = π 2 ω y exp ( π n ω y 2 ) [ erfc ( υ ω y + π n 2 ω y ) exp ( π n υ ) + erfc ( υ ω y + π n 2 ω y ) exp ( π n υ ) ] .
β 2 ψ ( x , y ) + μ ψ ( x , y ) + A ( x , y ) 2 = 0 ,
υ ξ 2 θ + γ sin [ 2 θ ] A 2 = 0
2 ψ 1 + γ sin [ 2 θ 0 ] A 2 = 0 ,
2 ψ 2 + 2 γ cos [ 2 θ 0 ] ψ 1 A 2 = 0 .
ψ = sin [ 2 θ 0 ] γ P g 1 ( ξ , υ ) + sin [ 4 θ 0 ] γ 2 P 2 g 2 ( ξ , υ ) ,
g 1 ( ξ , υ ) = G ( ξ , υ , ζ , η ) a ( ζ , η ) d ζ d η ,
g 2 ( ξ , υ ) = G ( ξ , υ , ζ , η ) a ( ζ , η ) { G ( ζ , η , ζ , η ) a ( ζ , η ) d ζ d η } d ζ d η .
k d 2 r d z 2 = φ 2 V d x d y ,
k d 2 x d z 2 = W 0 ( x , z ) ,
W 0 = k 0 P L κ { [ erf ( 1 x w ) + x L ( erf ( x w ) erf ( 1 x w ) ) ] w π L [ exp ( x 2 w 2 ) exp ( ( L x ) 2 w 2 ) ] } ,
W 0 ( x ) = k 0 P κ n = 1 R n υ θ n ξ ( x L ) cos ( π n L x ) ,
W 0 = k 0 ε a n 0 cos δ { sin [ 2 ( θ 0 δ ) ] W 0 L + cos [ 2 ( θ 0 δ ) ] W 0 N L } ,
W 0 L ( x ) = γ P C 2 n = 1 R n υ θ n ξ ( x L ) cos ( π n L x ) + γ 2 P 2 C 2 2 n = 1 cos ( π n L x ) m = 1 P m n G m n ( x L ) ,
W 0 N L ( x ) = γ 2 P 2 C 2 4 π 2 n = 1 m = 1 1 n m [ n cos ( π n L x ) sin ( π m L x ) + m cos ( π m L x ) sin ( π n L x ) ] M n m ( x ) ,
P n m = F m n ( η ) exp ( t 2 ω 2 ) e π n t η d η π ω 2

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