Abstract

The evolution of polarized light in a nematic liquid crystal whose directors have a local response to reorientation by the light is analyzed for arbitrary input light power. Approximate equations describing this evolution are derived based on a suitable trial function in a Lagrangian formulation of the basic equations governing the electric fields involved. It is shown that the nonlinearity of the material response is responsible for the formation of solitons, so-called nematicons, by saturating the nonlinearity of the governing nonlinear Schrödinger equation. Therefore in the local material response limit, solitons are formed due to the nonlinear saturation behavior. It is finally shown that the solutions of the derived approximate equations for nematicon evolution are in excellent agreement with numerical solutions of the full nematicon equations in the local regime.

© 2006 Optical Society of America

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References

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  1. I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, 1995).
  2. J. A. Reyes and P. Palffy-Muhoray, 'Nonlinear Schrödinger equation in nematic liquid crystals,' Phys. Rev. E 58, 5855-5859 (1998).
    [CrossRef]
  3. H. Sarkissian, C. Tsai, B. Zeldovich, and N. Tabirian, 'Beam clean up and combining using orientational stimulated scattering in liquid crystals,' 2005 Conference on Lasers and Electro-Optics: Applications of LY3 Nonlinearities (Optical Society of America, 2005), pp. 505-507.
  4. C. Conti, M. Peccianti, and G. Assanto, 'Route to nonlocality and observation of accessible solitons,' Phys. Rev. Lett. 91, 073901 (2003).
    [CrossRef] [PubMed]
  5. 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]
  6. G. Assanto, M. Peccianti, K. A. Brzdakiewicz, A. de Luca, and C. Umeton, 'Nonlinear wave propagation and spatial solitons in nematic liquid crystals,' J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
    [CrossRef]
  7. G. Assanto and M. Peccianti, 'Spatial solitons in nematic liquid crystals,' IEEE J. Quantum Electron. 39, 13-21 (2003).
    [CrossRef]
  8. 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]
  9. G. B. Whitham, Linear and Nonlinear Waves (Wiley, 1974).
  10. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).
  11. 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]
  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]

2006 (1)

2003 (3)

G. Assanto, M. Peccianti, K. A. Brzdakiewicz, A. de Luca, and C. Umeton, 'Nonlinear wave propagation and spatial solitons in nematic liquid crystals,' J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
[CrossRef]

G. Assanto and M. Peccianti, 'Spatial solitons in nematic liquid crystals,' IEEE J. Quantum Electron. 39, 13-21 (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]

1998 (1)

J. A. Reyes and P. Palffy-Muhoray, 'Nonlinear Schrödinger equation in nematic liquid crystals,' Phys. Rev. E 58, 5855-5859 (1998).
[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]

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).

Assanto, G.

G. Assanto, M. Peccianti, K. A. Brzdakiewicz, A. de Luca, and C. Umeton, 'Nonlinear wave propagation and spatial solitons in nematic liquid crystals,' J. Nonlinear Opt. Phys. Mater. 12, 123-134 (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]

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

Brzdakiewicz, K. A.

G. Assanto, M. Peccianti, K. A. Brzdakiewicz, A. de Luca, and C. Umeton, 'Nonlinear wave propagation and spatial solitons in nematic liquid crystals,' J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
[CrossRef]

Conti, C.

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.

G. Assanto, M. Peccianti, K. A. Brzdakiewicz, A. de Luca, and C. Umeton, 'Nonlinear wave propagation and spatial solitons in nematic liquid crystals,' J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
[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]

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, I. C.

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

Minzoni, A. A.

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]

Palffy-Muhoray, P.

J. A. Reyes and P. Palffy-Muhoray, 'Nonlinear Schrödinger equation in nematic liquid crystals,' Phys. Rev. E 58, 5855-5859 (1998).
[CrossRef]

Peccianti, M.

G. Assanto, M. Peccianti, K. A. Brzdakiewicz, A. de Luca, and C. Umeton, 'Nonlinear wave propagation and spatial solitons in nematic liquid crystals,' J. Nonlinear Opt. Phys. Mater. 12, 123-134 (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]

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

Reimbert, C. García

Reyes, J. A.

J. A. Reyes and P. Palffy-Muhoray, 'Nonlinear Schrödinger equation in nematic liquid crystals,' Phys. Rev. E 58, 5855-5859 (1998).
[CrossRef]

Sarkissian, H.

H. Sarkissian, C. Tsai, B. Zeldovich, and N. Tabirian, 'Beam clean up and combining using orientational stimulated scattering in liquid crystals,' 2005 Conference on Lasers and Electro-Optics: Applications of LY3 Nonlinearities (Optical Society of America, 2005), pp. 505-507.

Smyth, N. F.

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]

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).

Tabirian, N.

H. Sarkissian, C. Tsai, B. Zeldovich, and N. Tabirian, 'Beam clean up and combining using orientational stimulated scattering in liquid crystals,' 2005 Conference on Lasers and Electro-Optics: Applications of LY3 Nonlinearities (Optical Society of America, 2005), pp. 505-507.

Tsai, C.

H. Sarkissian, C. Tsai, B. Zeldovich, and N. Tabirian, 'Beam clean up and combining using orientational stimulated scattering in liquid crystals,' 2005 Conference on Lasers and Electro-Optics: Applications of LY3 Nonlinearities (Optical Society of America, 2005), pp. 505-507.

Umeton, C.

G. Assanto, M. Peccianti, K. A. Brzdakiewicz, A. de Luca, and C. Umeton, 'Nonlinear wave propagation and spatial solitons in nematic liquid crystals,' J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
[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]

G. B. Whitham, Linear and Nonlinear Waves (Wiley, 1974).

Yang, J.

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

Zeldovich, B.

H. Sarkissian, C. Tsai, B. Zeldovich, and N. Tabirian, 'Beam clean up and combining using orientational stimulated scattering in liquid crystals,' 2005 Conference on Lasers and Electro-Optics: Applications of LY3 Nonlinearities (Optical Society of America, 2005), pp. 505-507.

IEEE J. Quantum Electron. (1)

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. (1)

G. Assanto, M. Peccianti, K. A. Brzdakiewicz, A. de Luca, and C. Umeton, 'Nonlinear wave propagation and spatial solitons in nematic liquid crystals,' J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
[CrossRef]

J. Opt. Soc. Am. B (1)

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 (2)

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]

J. A. Reyes and P. Palffy-Muhoray, 'Nonlinear Schrödinger equation in nematic liquid crystals,' Phys. Rev. E 58, 5855-5859 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

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

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]

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 (4)

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

G. B. Whitham, Linear and Nonlinear Waves (Wiley, 1974).

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

H. Sarkissian, C. Tsai, B. Zeldovich, and N. Tabirian, 'Beam clean up and combining using orientational stimulated scattering in liquid crystals,' 2005 Conference on Lasers and Electro-Optics: Applications of LY3 Nonlinearities (Optical Society of America, 2005), pp. 505-507.

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Figures (5)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

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 governing equations (7, 6) with ν = 0.01 , dashed curve, solution of modulation equations (27, 29, 31, 38); short-dashed curve, solution of modulation equations of Ref. [5].

Fig. 3
Fig. 3

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 (7, 6) with ν = 0.01 ; dashed curve, solution of modulation equations (27, 29, 31, 38); short-dashed curve, solution of modulation equations of Ref. [5].

Fig. 4
Fig. 4

Amplitude a of the nematicon as a function of z for the initial conditions A = 0.5 , W = 2.5 with q = 1 . Solid curve, numerical solution of governing equations (7, 6) with ν = 0.01 ; dashed curve, solution of modulation equations (27, 29, 31, 38).

Fig. 5
Fig. 5

Amplitude a of the nematicon as a function of z for the initial conditions A = 0.35 , W = 2.5 with q = 1 . Solid curve, numerical solution of governing equations (7, 6) with ν = 0.01 ; dashed curve, solution of modulation equations (27, 29, 31, 38); short-dashed curve, solution of modulation equations of Ref. [5].

Equations (42)

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

i E z + 1 2 2 E cos ( 2 φ ) E = 0 .
ν 2 φ + 2 p sin ( 2 φ ) + 2 E 2 sin ( 2 φ ) = 0 ,
ν 2 φ x 2 + 2 p sin ( 2 φ ) = 0 , φ ( L ) = φ ( L ) = 0 .
ν 2 θ + ν 2 θ ̂ + 2 p sin ( 2 θ ̂ ) cos ( 2 θ ) + 2 p cos ( 2 θ ̂ ) sin ( 2 θ ) + 2 E 2 sin ( 2 θ ̂ ) cos ( 2 θ ) + 2 E 2 cos ( 2 θ ̂ ) sin ( 2 θ ) = 0 .
ν 2 θ + 2 p cos ( 2 θ ̂ ) sin ( 2 θ ) + 2 E 2 sin ( 2 θ ̂ ) cos ( 2 θ ) = 0 .
ν 2 θ q sin ( 2 θ ) + 2 E 2 cos ( 2 θ ) = 0 .
i E z + 1 2 2 E + sin ( 2 θ ) E = 0 .
tan 2 θ = 2 E 2 q .
i E z + 1 2 2 E r 2 + 1 2 r E r + 2 E 2 E q 2 + 4 E 4 = 0 ,
E ( r , θ ) = A sech r W .
L = i r ( E * E z E E z * ) r E r 2 + r [ q 2 + 4 E 4 ] 1 2 r q ,
E = a sech r w e i σ + i g e i σ .
0 r ( q 2 + 4 E 4 q ) d r = 0 4 r E 4 q + q 2 + 4 E 4 d r ,
E = a sech r w
E = α a e r 2 ( β 2 w 2 )
L = 2 ( a 2 w 2 I 2 + Λ g 2 ) σ 2 a w 2 I 1 g + 2 g w 2 I 1 a + 4 a w g I 1 w a 2 I + 1 4 q β 2 w 2 ( 1 + 4 α 4 a 4 q 2 1 ) + 1 4 q β 2 w 2 ln 2 1 + 1 + 4 α 4 a 4 q 2
Λ = 1 2 l 2 ,
I = 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 2 = 0 x sech 2 x d x = ln 2 ,
L = 2 ( a 2 w 2 I 2 + Λ g 2 ) σ 2 a w 2 I 1 g + 2 g w 2 I 1 a + 4 a w g I 1 w a 2 I + α 4 β 2 a 4 w 2 4 q α 8 β 2 a 8 w 2 8 q 3 .
L = 2 ( a 2 w 2 I 2 + Λ g 2 ) σ 2 a w 2 I 1 g + 2 g w 2 I 1 a + 4 a w g I 1 w a 2 I + 2 q I 4 a 4 w 2 2 q 3 I 8 a 8 w 2 ,
I 4 = 0 x sech 4 x d x = 2 3 ln 2 1 6 ,
I 8 = 0 x sech 8 x d x = 16 35 ln 2 19 105 .
α 4 = 2 I 8 I 4 and β 2 = 4 I 4 2 I 8 .
d d z ( a 2 w 2 I 2 + Λ g 2 ) = 0 ,
d d z ( a w 2 I 1 ) = Λ g σ ,
I 1 d g d z = I a 2 w 2 + q β 2 8 a ln 2 1 + 1 + 4 α 4 a 4 q 2 ,
I 2 d σ d z = I w 2 q β 2 8 a 2 ln 2 1 + 1 + 4 α 4 a 4 q 2 + α 4 β 2 a 2 2 q ( 1 + 1 + 4 α 4 a 4 q 2 ) .
d H d z = d d z 0 r [ E r 2 q 2 + 4 E 4 + q ] d r = 0 .
H = a 2 I 1 4 q β 2 w 2 ( 1 + 4 α 4 a 4 q 2 1 ) 1 4 q β 2 w 2 ln 2 1 + 1 + 4 α 4 a 4 q 2 .
w ̂ 2 = 4 I a ̂ 2 q β 2 [ ln 2 1 + S ̂ ] 1 ,
I 2 d σ ̂ d z = I 2 w ̂ 2 + α 4 β 2 a ̂ 2 2 q ( 1 + S ̂ ) ,
S ̂ = 1 + 4 α 4 a ̂ 4 q 2 .
d 2 g 1 d z 2 + Λ σ ̂ I 1 2 w ̂ 2 [ 2 I w ̂ 2 α 4 β 2 a ̂ 2 q S ̂ ( 1 + S ̂ ) ] g 1 = 0 .
Λ = q I 1 2 S ̂ ( 1 + S ̂ ) w ̂ 4 σ ̂ 2 q I S ̂ ( 1 + S ̂ ) α 4 β 2 a ̂ 2 w ̂ 2 .
Λ = q I 1 2 S ̂ ( 1 + S ̂ ) w 4 σ 2 q I S ̂ ( 1 + S ̂ ) α 4 β 2 a 2 w 2 .
I 1 d g d z = I a 2 w 2 + q β 2 8 a ln 2 1 + 1 + 4 α 4 a 4 q 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 ) .
i E z + 1 2 2 E + E 2 E = 0 ,

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