Abstract

Spatial solitons in nematic liquid crystals are considered in the regime of local response of the crystal, such solitons having been called nematicons in previous experimental studies. In the limit of low light intensity and local material response, it is shown that the full governing equations reduce to a single, higher-order nonlinear Schrödinger equation. Modulation equations are derived for the evolution of a nematiconlike pulse; these equations also include the dispersive radiation shed as the pulse evolves. The modulation equations show that a nematicon is (weakly) stable. Solutions of the modulation equations are compared with numerical solutions of the full governing equations, and good agreement is found when the light intensity is small.

© 2006 Optical Society of America

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  1. G. Assanto, M. Peccianti, and C. Conti, "Optical spatial solitons in nematic liquid crystals. Nematicons," Opt. Photonics News 14, 45-48 (2003).
    [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. 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]
  5. C. García Reinbert, C. Garza Hume, A. A. Minzoni, and N. F. Smyth, "Active TM mode envelope soliton propagation in a nonlinear nematic waveguide," Physica D 167, 136-152 (2002).
    [CrossRef]
  6. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, 1995).
  7. G. P. Agrawal, Nonlinear Fibre Optics (Academic, 1989).
  8. A. Hasegawa, Optical Solitons in Fibers, 2nd ed. (Springer, 1990).
  9. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).
  10. D. J. Kaup and A. C. Newell, "Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory," Proc. R. Soc. London, Ser. A 361, 413-446 (1978).
    [CrossRef]
  11. 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]
  12. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran. The Art of Scientific Computing (Cambridge U. Press, 1992).

2004 (1)

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]

2003 (2)

G. Assanto, M. Peccianti, and C. Conti, "Optical spatial solitons in nematic liquid crystals. Nematicons," Opt. Photonics News 14, 45-48 (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 (1)

C. García Reinbert, C. Garza Hume, A. A. Minzoni, and N. F. Smyth, "Active TM mode envelope soliton propagation in a nonlinear nematic waveguide," Physica D 167, 136-152 (2002).
[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]

1978 (2)

D. J. Kaup and A. C. Newell, "Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory," Proc. R. Soc. London, Ser. A 361, 413-446 (1978).
[CrossRef]

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).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fibre Optics (Academic, 1989).

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]

G. Assanto, M. Peccianti, and C. Conti, "Optical spatial solitons in nematic liquid crystals. Nematicons," Opt. Photonics News 14, 45-48 (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]

Conti, C.

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. 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, "Optical spatial solitons in nematic liquid crystals. Nematicons," Opt. Photonics News 14, 45-48 (2003).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran. The Art of Scientific Computing (Cambridge U. Press, 1992).

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]

Hasegawa, A.

A. Hasegawa, Optical Solitons in Fibers, 2nd ed. (Springer, 1990).

Hume, C. Garza

C. García Reinbert, C. Garza Hume, A. A. Minzoni, and N. F. Smyth, "Active TM mode envelope soliton propagation in a nonlinear nematic waveguide," Physica D 167, 136-152 (2002).
[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]

Kaup, D. J.

D. J. Kaup and A. C. Newell, "Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory," Proc. R. Soc. London, Ser. A 361, 413-446 (1978).
[CrossRef]

Khoo, C.

C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, 1995).

Minzoni, A. A.

C. García Reinbert, C. Garza Hume, A. A. Minzoni, and N. F. Smyth, "Active TM mode envelope soliton propagation in a nonlinear nematic waveguide," Physica D 167, 136-152 (2002).
[CrossRef]

Newell, A. C.

D. J. Kaup and A. C. Newell, "Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory," Proc. R. Soc. London, Ser. A 361, 413-446 (1978).
[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]

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, "Optical spatial solitons in nematic liquid crystals. Nematicons," Opt. Photonics News 14, 45-48 (2003).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran. The Art of Scientific Computing (Cambridge U. Press, 1992).

Reinbert, C. García

C. García Reinbert, C. Garza Hume, A. A. Minzoni, and N. F. Smyth, "Active TM mode envelope soliton propagation in a nonlinear nematic waveguide," Physica D 167, 136-152 (2002).
[CrossRef]

Smyth, N. F.

C. García Reinbert, C. Garza Hume, A. A. Minzoni, and N. F. Smyth, "Active TM mode envelope soliton propagation in a nonlinear nematic waveguide," Physica D 167, 136-152 (2002).
[CrossRef]

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]

Stegun, I. A.

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

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran. The Art of Scientific Computing (Cambridge U. Press, 1992).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran. The Art of Scientific Computing (Cambridge U. Press, 1992).

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]

Opt. Photonics News (1)

G. Assanto, M. Peccianti, and C. Conti, "Optical spatial solitons in nematic liquid crystals. Nematicons," Opt. Photonics News 14, 45-48 (2003).
[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. Rev. E (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]

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]

Physica D (1)

C. García Reinbert, C. Garza Hume, A. A. Minzoni, and N. F. Smyth, "Active TM mode envelope soliton propagation in a nonlinear nematic waveguide," Physica D 167, 136-152 (2002).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

D. J. Kaup and A. C. Newell, "Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory," Proc. R. Soc. London, Ser. A 361, 413-446 (1978).
[CrossRef]

Other (5)

C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, 1995).

G. P. Agrawal, Nonlinear Fibre Optics (Academic, 1989).

A. Hasegawa, Optical Solitons in Fibers, 2nd ed. (Springer, 1990).

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

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran. The Art of Scientific Computing (Cambridge U. Press, 1992).

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Figures (5)

Fig. 1
Fig. 1

Schematic of liquid-crystal geometry. The volume of liquid crystal V has a nematicon of radius r propagating in the z direction. The electric field E is polarized in the transverse, upward direction.

Fig. 2
Fig. 2

Full numerical solution of the liquid-crystal equations (3, 4) at z = 400 for q = 2 and ν = 0.01 for the initial conditions A = 0.3 and W = 4.0 : (a) E ; (b) Im E .

Fig. 3
Fig. 3

Amplitude a of the nematicon as a function of z for the initial conditions A = 0.3 , W = 4 with q = 2 . Solid curve, numerical solution of the governing equations (3, 4) with ν = 0.01 ; dashed curve, solution of modulation equations (19, 21, 26, 53).

Fig. 4
Fig. 4

Amplitude a of the nematicon as a function of z for the initial conditions A = 0.45 , W = 2 with q = 1 . Solid curve, numerical solution of governing equations (3, 4) with ν = 0.01 ; dashed curve, solution of modulation equations (19, 21, 26, 53).

Fig. 5
Fig. 5

Amplitude a of the nematicon as a function of z for the initial conditions A = 0.4 , W = 2.5 with q = 1 . Solid curve, numerical solution of governing equations (3, 4) with ν = 0.01 ; dashed curve, solution of modulation equations (19, 21, 26, 53).

Equations (55)

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i E z + 1 2 2 E cos ( 2 ϕ ) E = 0 ,
ν 2 ϕ + q cos ( 2 ϕ + 2 ψ ) + 2 E 2 sin ( 2 ϕ ) = 0 .
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 ) = 0 .
q sin ( 2 θ ) + 2 E 2 cos ( 2 θ ) = 0 ,
tan ( 2 θ ) = 2 E 2 q .
i E z + 1 2 2 E + 2 E 2 E ( q 2 + 4 E 4 ) 1 2 = 0 .
E ( 0 , r ) = A sech r W .
i E z + 1 2 2 E + 2 q E 2 E 4 q 3 E 6 E = 0 .
L = i r ( E * E z E E z * ) r E r 2 + 2 r q E 4 2 r q 3 E 8 ,
E = a sech r w exp ( i σ ) + i g exp ( i σ ) .
L = 2 ( a 2 w 2 I 2 + Λ g 2 ) σ + 2 w 2 g I 1 a + 4 a w g I 1 w 2 a w 2 I 1 g a 2 I + 2 q I 4 a 4 w 2 2 q 3 I 8 a 8 w 2 ,
Λ = 1 2 l 2 .
I = 0 x sech 2 x tanh 2 x d x = 1 3 log 2 + 1 6 ,
I 1 = 0 x sech x d x = 2 C ,
I 2 = 0 x sech 2 x d x = log 2 ,
I 4 = 0 x sech 4 x d x = 2 3 log 2 1 6 ,
I 8 = 0 x sech 8 x d x = 16 35 log 2 19 105 ,
d d z ( I 1 a w 2 ) = Λ g d σ d z ,
I 1 d g d z = I a 2 w 2 I 4 q a 3 + 3 I 8 q 3 a 7 ,
I 2 d σ d z = I w 2 + 3 I 4 q a 2 7 I 8 q 3 a 6 ,
d d z ( a 2 w 2 I 2 + Λ g 2 ) = 0 .
i z ( r E 2 ) + 1 2 r ( r E * E r r E E r * ) = 0 .
w 2 = q I 2 a 2 ( I 4 3 I 8 q 2 a 4 ) 1 .
i z ( r E r 2 2 r q E 4 + 2 r q 3 E 8 ) + r [ 1 2 r ( E r * E r r E r E r r * ) + 2 q r E 2 ( E E r * E * E r ) 4 q 3 r E 6 ( E E r * E * E r ) ] = 0 .
d H d z = d d z ( I a 2 2 q I 4 a 4 w 2 + 2 q 3 I 8 a 8 w 2 ) = 0 .
I a ̂ 2 2 q I 4 a ̂ 4 w ̂ 2 + 2 q 3 I 8 a ̂ 8 w ̂ 2 = I A 2 2 q I 4 A 4 W 2 + 2 q 3 I 8 A 8 W 2 = H .
a ̂ 6 = q 2 I 4 H 2 I I 8
w ̂ 2 = q I 2 I 4 a ̂ 2
Λ = 1 2 l 2 = q 3 I 1 2 I 24 I 2 I 8 a ̂ 6 .
a ̂ 6 = q 2 I 4 H 2 I I 8 , w ̂ 2 = q I 2 I 4 a ̂ 2 ,
d σ ̂ d z = I 4 q I 2 a ̂ 2 7 I 8 q 3 I 2 a ̂ 6 .
i E z + 1 2 r r ( r E r ) = 0 .
i z ( r E 2 ) + 1 2 r ( r E * E r r E E r * ) = 0 .
d d z l r E 2 d r = Im ( r E * E r ) r = l + O [ l ̇ ( z ) ] .
E = E fixed + E 1 ,
0 l ( z ) E 2 r d r = 0 l ( z ) [ E fixed 2 + 2 Re ( E fixed E 1 ) + E 1 2 ] r d r .
I 2 a 2 w 2 + Λ g 2 I 2 a ̂ 2 w ̂ 2 + Λ E 1 2 r = l ,
S ( z ) 2 = 1 Λ [ I 2 a 2 w 2 I 2 a ̂ 2 w ̂ 2 + Λ g 2 ] .
S ( z ) 2 = 1 l [ 2 a 2 w 2 a ̂ 2 w ̂ + l g 2 ] .
a 2 w 2 = q I 2 2 [ 1 I 8 a ̂ 4 ( q 2 I 4 ) ] I 4 a ̂ 2 { a ̂ 2 I H + 6 I I 8 a ̂ 4 q 2 I 4 [ 1 3 I 8 a ̂ 4 ( q 2 I 4 ) ] a 1 2 }
I 2 a 2 w 2 + Λ g 2
E r r = l = 1 2 π i C 2 s exp ( i π 4 ) K 1 [ 2 s exp ( i π 4 ) l ] K 0 [ 2 s exp ( i π 4 ) l ] E ¯ 0 exp ( s z ) d s ,
E * E r = S * ( z ) 0 z G ( z z ) E 0 ( z ) d z ,
G ( η ) = 1 2 π i C 2 s exp ( i π 4 ) K 1 [ 2 s exp ( i π 4 ) l ] K 0 [ 2 s exp ( i π 4 ) l ] exp ( s η ) d s .
S = R ( z ) exp [ i φ ( z ) ]
E * E r = R ( z ) 0 z G ( z z ) R ( z ) d z .
2 s l d d s log { K 0 [ 2 s exp ( i π 4 ) l ] } .
1 l 2 log s + log Λ i π 2 .
G ( η ) = 1 2 π i l C 2 exp ( s η ) log s + log Λ i π 2 d s .
G ( η ) = 1 4 l exp ( e ξ η + ξ ) ( ξ 2 + 1 2 log Λ ) 2 i ( π 2 ) ( ξ 2 + 1 2 log Λ ) + 3 π 2 16 d ξ .
E r = 2 π 4 e l 0 z R ( z ) z z 1 { ( 1 2 ) log [ ( z z ) Λ ] i π 4 } 2 + π 2 4 d z .
I 1 d g d z = I a 2 w 2 I 4 q a 3 + 3 I 8 q 3 a 7 2 α g ,
α = 2 π I 1 32 e R Λ 0 z π R ( z ) log [ ( z z ) Λ ] ( { 1 4 log [ ( z z ) Λ ] } 2 + 3 π 2 16 ) 2 + π 2 { log [ ( 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 ] .

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