Abstract

We analyze transverse instabilities of spatial bright solitons in nonlocal nonlinear media, both analytically and numerically. We demonstrate that the nonlocal nonlinear response leads to a dramatic suppression of the transverse instability of the soliton stripes, and we derive asymptotic expressions for the instability growth rate in both short- and long-wave approximations.

© 2008 Optical Society of America

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References

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  1. Yu. S. Kivshar and D. E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves,” Phys. Rep. 331, 117-195 (2000), and references therein.
    [CrossRef]
  2. V. E. Zakharov and A. M. Rubenchik, “Instability of waveguides and solitons in nonlinear media,” Sov. Phys. JETP 38, 494-500 (1974).
  3. G. Schmidt, “Stability of envelope soliton,” Phys. Rev. Lett. 34, 274-276 (1975).
    [CrossRef]
  4. S. J. Han, “Stability of envelope waves,” Phys. Rev. A 20, 2568-2573 (1979).
    [CrossRef]
  5. A. Bondeson, “Transverse instability of Langmuir soliton,” Phys. Rev. Lett. 43, 1117-1119 (1979).
    [CrossRef]
  6. P. A. E. M. Janssen, “Nonlinear evolution of the transverse instability of plane-envelope solitons,” Phys. Fluids 26, 1279-1287 (1983).
    [CrossRef]
  7. E. A. Kuznetsov and S. K. Turitsyn, “Instability and collapse of solitons in media with a defocusing nonlinearity,” Sov. Phys. JETP 67, 1583-1588 (1988).
  8. X. Liu, K. Beckwitt, and F. Wise, “Transverse instability of optical spatiotemporal solitons in quadratic media,” Phys. Rev. Lett. 85, 1871-1874 (2000).
    [CrossRef] [PubMed]
  9. A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870-879 (1996).
    [CrossRef] [PubMed]
  10. V. Tikhonenko, J. Christou, B. Luther-Davis, and Yu. S. Kivshar, “Observation of vortex solitons created by the instability of dark soliton stripes,” Opt. Lett. 21, 1129-1131 (1996).
    [CrossRef] [PubMed]
  11. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003), and references therein.
  12. J. P. Torres, C. Anastassiou, M. Segev, M. Soljacic, and D. N. Christodoulides, “Transverse instability of incoherent solitons in Kerr media,” Phys. Rev. E 65, 015601 (2001).
    [CrossRef]
  13. K. Motzek, F. Kaiser, W.-H. Chu, M.-F. Shih, and Yu. S. Kivshar, “Soliton transverse instabilities in anisotropic nonlocal self-focusing media,” Opt. Lett. 29, 280-282 (2004).
    [CrossRef] [PubMed]
  14. Z. H. Musslimani and J. Yang, “Transverse instability of strongly coupled dark-bright Manakov vector solitons,” Opt. Lett. 26, 1981-1983 (2001).
    [CrossRef]
  15. W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
    [CrossRef]
  16. S. Lopez-Aguayo, A. S. Desyatnikov, and Yu. S. Kivshar, “Azimuthons in nonlocal nonlinear media,” Opt. Express 14, 7903-7908 (2006).
    [CrossRef] [PubMed]
  17. C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
    [CrossRef] [PubMed]
  18. Y. Lin and R.-K. Lee, “Dark-bright soliton pairs in nonlocal nonlinear media,” Opt. Express 15, 8781-8786 (2007).
    [CrossRef] [PubMed]
  19. D. Anderson, “Averaged Lagrangian containning higher derivatives,” J. Phys. A 6, (1973).
    [CrossRef]

2007

2006

2004

K. Motzek, F. Kaiser, W.-H. Chu, M.-F. Shih, and Yu. S. Kivshar, “Soliton transverse instabilities in anisotropic nonlocal self-focusing media,” Opt. Lett. 29, 280-282 (2004).
[CrossRef] [PubMed]

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

2003

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

2001

J. P. Torres, C. Anastassiou, M. Segev, M. Soljacic, and D. N. Christodoulides, “Transverse instability of incoherent solitons in Kerr media,” Phys. Rev. E 65, 015601 (2001).
[CrossRef]

Z. H. Musslimani and J. Yang, “Transverse instability of strongly coupled dark-bright Manakov vector solitons,” Opt. Lett. 26, 1981-1983 (2001).
[CrossRef]

2000

Yu. S. Kivshar and D. E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves,” Phys. Rep. 331, 117-195 (2000), and references therein.
[CrossRef]

X. Liu, K. Beckwitt, and F. Wise, “Transverse instability of optical spatiotemporal solitons in quadratic media,” Phys. Rev. Lett. 85, 1871-1874 (2000).
[CrossRef] [PubMed]

1996

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870-879 (1996).
[CrossRef] [PubMed]

V. Tikhonenko, J. Christou, B. Luther-Davis, and Yu. S. Kivshar, “Observation of vortex solitons created by the instability of dark soliton stripes,” Opt. Lett. 21, 1129-1131 (1996).
[CrossRef] [PubMed]

1988

E. A. Kuznetsov and S. K. Turitsyn, “Instability and collapse of solitons in media with a defocusing nonlinearity,” Sov. Phys. JETP 67, 1583-1588 (1988).

1983

P. A. E. M. Janssen, “Nonlinear evolution of the transverse instability of plane-envelope solitons,” Phys. Fluids 26, 1279-1287 (1983).
[CrossRef]

1979

S. J. Han, “Stability of envelope waves,” Phys. Rev. A 20, 2568-2573 (1979).
[CrossRef]

A. Bondeson, “Transverse instability of Langmuir soliton,” Phys. Rev. Lett. 43, 1117-1119 (1979).
[CrossRef]

1975

G. Schmidt, “Stability of envelope soliton,” Phys. Rev. Lett. 34, 274-276 (1975).
[CrossRef]

1974

V. E. Zakharov and A. M. Rubenchik, “Instability of waveguides and solitons in nonlinear media,” Sov. Phys. JETP 38, 494-500 (1974).

1973

D. Anderson, “Averaged Lagrangian containning higher derivatives,” J. Phys. A 6, (1973).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt.

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

J. Phys. A

D. Anderson, “Averaged Lagrangian containning higher derivatives,” J. Phys. A 6, (1973).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Fluids

P. A. E. M. Janssen, “Nonlinear evolution of the transverse instability of plane-envelope solitons,” Phys. Fluids 26, 1279-1287 (1983).
[CrossRef]

Phys. Rep.

Yu. S. Kivshar and D. E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves,” Phys. Rep. 331, 117-195 (2000), and references therein.
[CrossRef]

Phys. Rev. A

S. J. Han, “Stability of envelope waves,” Phys. Rev. A 20, 2568-2573 (1979).
[CrossRef]

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870-879 (1996).
[CrossRef] [PubMed]

Phys. Rev. E

J. P. Torres, C. Anastassiou, M. Segev, M. Soljacic, and D. N. Christodoulides, “Transverse instability of incoherent solitons in Kerr media,” Phys. Rev. E 65, 015601 (2001).
[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]

A. Bondeson, “Transverse instability of Langmuir soliton,” Phys. Rev. Lett. 43, 1117-1119 (1979).
[CrossRef]

G. Schmidt, “Stability of envelope soliton,” Phys. Rev. Lett. 34, 274-276 (1975).
[CrossRef]

X. Liu, K. Beckwitt, and F. Wise, “Transverse instability of optical spatiotemporal solitons in quadratic media,” Phys. Rev. Lett. 85, 1871-1874 (2000).
[CrossRef] [PubMed]

Sov. Phys. JETP

E. A. Kuznetsov and S. K. Turitsyn, “Instability and collapse of solitons in media with a defocusing nonlinearity,” Sov. Phys. JETP 67, 1583-1588 (1988).

V. E. Zakharov and A. M. Rubenchik, “Instability of waveguides and solitons in nonlinear media,” Sov. Phys. JETP 38, 494-500 (1974).

Other

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003), and references therein.

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Figures (4)

Fig. 1
Fig. 1

(a) Intensity profiles of the bright solitons in local ( d = 0 , solid) and nonlocal ( d = 0.5 , dashed; d = 1 , dashed–dotted) nonlinear media. (b) Instability growth rate for bright solitons versus the transverse wavenumber p for local (solid) and nonlocal ( d = 0.5 , dashed; d = 1 , dashed–dotted) nonlinear media. (c), (d) Cutoff value of the transverse wavenumber and maximum growth rate versus the nonlocality parameter d.

Fig. 2
Fig. 2

Wavenumber versus nonlocality parameter d at fixed power ( P = 2 ) . Solid and dashed curves show the solution obtained by the variational method to the first and second order in d, respectively. Circles mark the numerical results.

Fig. 3
Fig. 3

Growth rate of the soliton transverse instabilities defined from numerical simulations and the variational approach. Solid and dashed curves are numerical data for local ( d = 0 ) and nonlocal ( d = 0.5 ) nonlinearities. Triangles above and below the solid curve are the long-scale and short-scale asymptotic expansions for local nonlinearity. Squares and circles are the long-scale asymptotic expansion based on the linearized nonlocal eigenvalue system expanded up to the first and second orders in d. Dots and crosses are the short-scale asymptotic expansion up to the first and second orders in d.

Fig. 4
Fig. 4

Evolution of a modulated bright-soliton stripe in (a)–(c) local and (d)–(f) nonlocal nonlinear media at the distances z = 1.0 , 6.0 , and 10.0 , in that order.

Equations (39)

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i E z + 1 2 Δ E + n E = 0 ,
n d Δ n = E 2 ,
n = n 0 ( x ) + ϵ δ n ,
E = e i β z [ u 0 + ϵ ( v + i w ) e i λ z + i p y + ϵ ( v * + i w * ) e i λ * z i p y ] ,
λ w = ( β + 1 2 p 2 ) v 1 2 2 v x 2 n 0 v u 0 δ n ,
λ v = ( β + 1 2 p 2 ) w 1 2 2 w x 2 n 0 w ,
δ n = d ( 2 x 2 p 2 ) δ n + 2 u 0 v ,
n ( x ) = m = 0 1 m ! h m m E 2 x m E 2 + d [ 2 x 2 ( E 2 ) ] + O ( d 2 ) ,
h m = i m d m d ω m H ( ω ) ω = 0
L = d x { i 2 ( E z E * E z * E ) 1 2 E x 2 + 1 2 E 4 + E 2 + d 2 ( ( E 2 E x 2 1 2 E 2 E x * 2 1 2 E * 2 E x 2 ) + O ( d 2 ) } ,
β = ( 1 6 ) P a + ( 1 6 ) a 2 + ( 2 15 ) d P a 3 ,
A = ( P a 2 ) 1 2 ,
a = [ 5 + ( 25 + 60 d P 2 ) 1 2 ] ( 12 d P ) 1 ,
0 = β v 0 1 2 2 v 0 x 2 n 0 v 0 u 0 δ n 0 ,
0 = β w 0 1 2 2 w 0 x 2 n 0 w 0 ,
δ n 0 = d 2 δ n 0 x 2 + 2 u 0 v 0 ,
w 0 = ( β + 1 2 p 2 ) v 1 1 2 2 v 1 x 2 n 0 v 1 u 0 δ n 1 ,
v 0 = ( β + 1 2 p 2 ) w 1 1 2 2 w 1 x 2 n 0 w 1 ,
δ n 1 = d ( 2 x 2 p 2 ) δ n 1 + 2 u 0 v 1 .
Γ ( w 0 , v ) = ( w 0 , [ ( β + 1 2 p 2 ) 1 2 2 x 2 n 0 ] w ) ,
( w , v ) = d x w * v .
1 2 p 2 ( w 0 , w ) = Γ ( w 0 , v ) .
Γ ( w 0 , w ) = 1 2 p 2 ( w 0 , v ) ( w 0 , u 0 δ n 1 ) ,
δ n 1 = 2 u 0 v 1 1 + d p 2 + d 2 x 2 ( 2 u 0 v 1 ) ( 1 + d p 2 ) 2 .
δ n 1 2 u 0 v 1 + d [ 2 x 2 ( 2 u 0 v 1 ) ] Δ n ( v 1 ; p = 0 ) .
Γ 2 = ( w 0 , u 0 Δ n ( v 1 ; p = 0 ) ) ( w 0 , v 1 ) p 2 1 4 p 4 .
L v 1 = d x { w 0 v 1 2 + 1 2 v 1 x 2 + ( β + p 2 2 ) v 1 2 ( u 0 2 + d 2 u 0 2 x 2 ) v 1 2 2 u 0 2 v 1 2 1 + d p 2 2 d ( 1 + d p 2 ) 2 ( u 0 2 u 0 x 2 v 1 2 u 0 2 v 1 x 2 ) } .
0 = ( 5134 315 d a P + 155 21 ) b 1 4 1048 225 d P a 2 b 1 3 + ( 14 5 β + 494 105 d P a 3 139 35 P a 31 35 π ) b 1 2 + ( 19 210 π a 92 105 d a 4 P + 10 21 P a 2 ) b 1 .
Γ 2 = [ 2 5 a b 1 + d ( 139 630 a 4 + 53 210 a b 1 3 61 630 a 2 b 1 2 + 473 630 a 3 b 1 ) ] P a + b 1 3 p 2 1 4 p 4 ,
0 = ( β + 1 2 p c 2 ) v 0 1 2 2 v 0 x 2 n 0 v 0 u 0 δ n ,
0 = ( β + 1 2 p c 2 ) w 0 1 2 2 w 0 x 2 n 0 w 0 ,
δ n 0 = d ( 2 x 2 p c 2 ) δ n 0 + 2 u 0 v 0 .
L v 0 = d x { 1 2 v 0 x 2 + ( β + p c 2 2 ) v 0 2 ( u 0 2 + d 2 u 0 2 x 2 ) v 1 2 2 u 0 2 v 0 2 1 + d p c 2 2 d ( 1 + d p c 2 ) 2 ( u 0 2 u 0 x 2 v 0 2 u 0 2 v 0 x 2 ) } .
0 = L v 1 b 0 = ( 8 15 + 352 315 d ( 1 + d p c 2 ) 2 P a ) b 0 32 63 d ( 1 + d p c 2 ) 2 P a 2 + ( 4 3 β + 2 3 p c 2 32 45 P a + 80 63 d P a 3 64 45 1 1 + d p c 2 a P + 416 315 d ( 1 + d p c 2 ) 2 a 5 P ) b 0 1 + ( 208 315 d a 4 P + 8 45 P a 2 176 315 d ( 1 + d p c 2 ) 2 a 6 P + 16 45 1 1 + d p c 2 a 2 P ) b 0 2 ,
0 = L v 1 = 8 15 + 352 315 d ( 1 + d p c 2 ) 2 P a + ( 32 45 P a + 64 45 1 1 + d p c 2 a P 4 3 β 2 3 p c 2 416 315 d ( 1 + d p c 2 ) 2 a 5 P 80 63 d P a 3 ) b 0 2 + ( 32 45 1 1 + d p c 2 a 2 P + 416 315 d a 4 P 16 45 P a 2 + 352 315 d ( 1 + d p c 2 ) 2 a 6 P ) b 0 3 .
L w 1 = d x { v 0 w 1 2 + 1 2 w 1 x 2 + ( β + p ¯ 2 2 ) w 1 2 ( u 0 2 + d 2 u 0 2 x 2 ) w 1 2 } ,
Γ 2 = ( v 0 , u 0 Δ n ( w 1 ; p ¯ = 0 ) ) ( v 0 , w 1 ) p ¯ 2 1 4 p ¯ 4 ,
( v 0 , w 1 ) = 5 6 π b 0 1 1 3 π c 1 b 0 2 ,
( v 0 , u 0 Δ n ( w 1 ; p ¯ = 0 ) ) = P [ ( 3 80 π + 1 2 d ( 27 140 π + 1 70 a 2 π ) ) c 1 + ( 3 40 π + 1 2 d ( 1 35 π 13 140 a 2 π ) ) + 3 16 π + 9 80 a π + 1 2 d ( 1 4 π 31 140 a π 4 5 a 2 + 11 140 a 3 π + 1 8 a 2 π ) ] .

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