Abstract

We present a numerical approach to the nemato-elasticity differential equation in a nematic liquid crystal cell when irradiated with multiple gaussian beams. Solutions have been carried out on a configuration with two coplanar beams illuminating the sample in order to compare it with particular nonlinear phenomena experimentally studied in the past. A new set of experimental measures were realized confirming the validity of the numerical model. Solutions for an instable case showing nonlocal effects are also presented as an example of the broader class of systems this approach can describe.

© 2007 Optical Society of America

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References

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  1. P. G. de Gennes, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1993).
  2. I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (John Wiley & Sons Inc., New York, 1995).
  3. F. Simoni, Nonlinear Optical Properties of Liquid Crystals (World Scientific, 1997).
  4. 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]
  5. N. N. Akhmediev, M. J. Lederer, and B. Luther-Davis, "Exact localized solution for nonconservative systems with delayed nonlinear response," Phys. Rev. E 57, 3664-3667 (1998).
    [CrossRef]
  6. 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]
  7. E. DelRe, A. Ciattoni, and A. J. Agranat, "Anisotropic charge displacement supporting isolated photorefractive optical needles," Opt. Lett. 26, 908-910 (2001).
    [CrossRef]
  8. A. D. Boardman and A. P. Sukhorukov, Soliton Driven Photonics (Kluwer, Dordrecht, 2001).
    [CrossRef]
  9. C. Conti, M. Peccianti, and G. Assanto, "Observation of Optical Spatial Solitons in a highly nonlocal medium," Phys. Rev. Lett. 92, 113902-113902 (2004).
    [CrossRef] [PubMed]
  10. M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "All Optical Switching and Logic Gating with Spatial Solitons in Liquid Crystals," Appl. Phys. Lett. 81, 3335-3338 (2002).
    [CrossRef]
  11. A. De Luca, S. Nersisyan, and C. Umeton, "Observation of cancellation and second light-induced Frdericksz transition in nematic liquid crystals," Opt. Lett. 28, 108-110 (2003).
    [CrossRef] [PubMed]
  12. N. V. Tabiryan, A. Sukhov, and B. Y. Zel’dovich, "The orientational optical nonlinearity of liquid crystals," Mol. Cryst. Liq. Cryst. 136, 1-139 (1986).
    [CrossRef]
  13. A. Yariv, Quantum electronics-3rd ed. (John Wiley & Sons, 1987).
  14. E. Santamato, G. Abbate, P. Maddalena, and A. Sasso, "Two beam mirrorless optical bistability in nematic liquid crystal film," Mol. Cryst. Liq. Cryst. 143, 113-122 (1987).
    [CrossRef]
  15. I. C. Khoo, "Theory of optically induced molecular reorientations and quantitative experiments on wave mixing and the self-focusing of light," Phys. Rev. A 25, 1636-1644 (1982).
    [CrossRef]
  16. F. Bloisi, L. Vicari, F. Simoni, G. Cipparrone, and C. Umeton, "Self-phase modulation in nematic liquid-crystal films: detailed measurements and theoretical calculations," J. Opt. Soc. Am. B 5, 2462-2466 (1988).
    [CrossRef]

2004 (1)

C. Conti, M. Peccianti, and G. Assanto, "Observation of Optical Spatial Solitons in a highly nonlocal medium," Phys. Rev. Lett. 92, 113902-113902 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (1)

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "All Optical Switching and Logic Gating with Spatial Solitons in Liquid Crystals," Appl. Phys. Lett. 81, 3335-3338 (2002).
[CrossRef]

2001 (1)

1998 (2)

N. N. Akhmediev, M. J. Lederer, and B. Luther-Davis, "Exact localized solution for nonconservative systems with delayed nonlinear response," Phys. Rev. E 57, 3664-3667 (1998).
[CrossRef]

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 (1)

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]

1988 (1)

1987 (1)

E. Santamato, G. Abbate, P. Maddalena, and A. Sasso, "Two beam mirrorless optical bistability in nematic liquid crystal film," Mol. Cryst. Liq. Cryst. 143, 113-122 (1987).
[CrossRef]

1986 (1)

N. V. Tabiryan, A. Sukhov, and B. Y. Zel’dovich, "The orientational optical nonlinearity of liquid crystals," Mol. Cryst. Liq. Cryst. 136, 1-139 (1986).
[CrossRef]

1982 (1)

I. C. Khoo, "Theory of optically induced molecular reorientations and quantitative experiments on wave mixing and the self-focusing of light," Phys. Rev. A 25, 1636-1644 (1982).
[CrossRef]

Abbate, G.

E. Santamato, G. Abbate, P. Maddalena, and A. Sasso, "Two beam mirrorless optical bistability in nematic liquid crystal film," Mol. Cryst. Liq. Cryst. 143, 113-122 (1987).
[CrossRef]

Abe, S.

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]

Agranat, A. J.

Akhmediev, N. N.

N. N. Akhmediev, M. J. Lederer, and B. Luther-Davis, "Exact localized solution for nonconservative systems with delayed nonlinear response," Phys. Rev. E 57, 3664-3667 (1998).
[CrossRef]

Anderson, D.

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]

Assanto, G.

C. Conti, M. Peccianti, and G. Assanto, "Observation of Optical Spatial Solitons in a highly nonlocal medium," Phys. Rev. Lett. 92, 113902-113902 (2004).
[CrossRef] [PubMed]

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "All Optical Switching and Logic Gating with Spatial Solitons in Liquid Crystals," Appl. Phys. Lett. 81, 3335-3338 (2002).
[CrossRef]

Bloisi, F.

Ciattoni, A.

Cipparrone, G.

Conti, C.

C. Conti, M. Peccianti, and G. Assanto, "Observation of Optical Spatial Solitons in a highly nonlocal medium," Phys. Rev. Lett. 92, 113902-113902 (2004).
[CrossRef] [PubMed]

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "All Optical Switching and Logic Gating with Spatial Solitons in Liquid Crystals," Appl. Phys. Lett. 81, 3335-3338 (2002).
[CrossRef]

De Luca, A.

A. De Luca, S. Nersisyan, and C. Umeton, "Observation of cancellation and second light-induced Frdericksz transition in nematic liquid crystals," Opt. Lett. 28, 108-110 (2003).
[CrossRef] [PubMed]

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "All Optical Switching and Logic Gating with Spatial Solitons in Liquid Crystals," Appl. Phys. Lett. 81, 3335-3338 (2002).
[CrossRef]

DelRe, E.

Khoo, I. C.

I. C. Khoo, "Theory of optically induced molecular reorientations and quantitative experiments on wave mixing and the self-focusing of light," Phys. Rev. A 25, 1636-1644 (1982).
[CrossRef]

Korytin, A. I.

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]

Lederer, M. J.

N. N. Akhmediev, M. J. Lederer, and B. Luther-Davis, "Exact localized solution for nonconservative systems with delayed nonlinear response," Phys. Rev. E 57, 3664-3667 (1998).
[CrossRef]

Lisak, M.

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]

Luther-Davis, B.

N. N. Akhmediev, M. J. Lederer, and B. Luther-Davis, "Exact localized solution for nonconservative systems with delayed nonlinear response," Phys. Rev. E 57, 3664-3667 (1998).
[CrossRef]

Maddalena, P.

E. Santamato, G. Abbate, P. Maddalena, and A. Sasso, "Two beam mirrorless optical bistability in nematic liquid crystal film," Mol. Cryst. Liq. Cryst. 143, 113-122 (1987).
[CrossRef]

Nersisyan, S.

Ogura, A.

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]

Peccianti, M.

C. Conti, M. Peccianti, and G. Assanto, "Observation of Optical Spatial Solitons in a highly nonlocal medium," Phys. Rev. Lett. 92, 113902-113902 (2004).
[CrossRef] [PubMed]

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "All Optical Switching and Logic Gating with Spatial Solitons in Liquid Crystals," Appl. Phys. Lett. 81, 3335-3338 (2002).
[CrossRef]

Santamato, E.

E. Santamato, G. Abbate, P. Maddalena, and A. Sasso, "Two beam mirrorless optical bistability in nematic liquid crystal film," Mol. Cryst. Liq. Cryst. 143, 113-122 (1987).
[CrossRef]

Sasso, A.

E. Santamato, G. Abbate, P. Maddalena, and A. Sasso, "Two beam mirrorless optical bistability in nematic liquid crystal film," Mol. Cryst. Liq. Cryst. 143, 113-122 (1987).
[CrossRef]

Sergeev, A. M.

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]

Simoni, F.

Sukhov, A.

N. V. Tabiryan, A. Sukhov, and B. Y. Zel’dovich, "The orientational optical nonlinearity of liquid crystals," Mol. Cryst. Liq. Cryst. 136, 1-139 (1986).
[CrossRef]

Tabiryan, N. V.

N. V. Tabiryan, A. Sukhov, and B. Y. Zel’dovich, "The orientational optical nonlinearity of liquid crystals," Mol. Cryst. Liq. Cryst. 136, 1-139 (1986).
[CrossRef]

Umeton, C.

Vanin, E.

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]

Vazquez, L.

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]

Vicari, L.

Zel’dovich, B. Y.

N. V. Tabiryan, A. Sukhov, and B. Y. Zel’dovich, "The orientational optical nonlinearity of liquid crystals," Mol. Cryst. Liq. Cryst. 136, 1-139 (1986).
[CrossRef]

Appl. Phys. Lett. (1)

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, "All Optical Switching and Logic Gating with Spatial Solitons in Liquid Crystals," Appl. Phys. Lett. 81, 3335-3338 (2002).
[CrossRef]

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

Mol. Cryst. Liq. Cryst. (2)

E. Santamato, G. Abbate, P. Maddalena, and A. Sasso, "Two beam mirrorless optical bistability in nematic liquid crystal film," Mol. Cryst. Liq. Cryst. 143, 113-122 (1987).
[CrossRef]

N. V. Tabiryan, A. Sukhov, and B. Y. Zel’dovich, "The orientational optical nonlinearity of liquid crystals," Mol. Cryst. Liq. Cryst. 136, 1-139 (1986).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (2)

I. C. Khoo, "Theory of optically induced molecular reorientations and quantitative experiments on wave mixing and the self-focusing of light," Phys. Rev. A 25, 1636-1644 (1982).
[CrossRef]

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 (2)

N. N. Akhmediev, M. J. Lederer, and B. Luther-Davis, "Exact localized solution for nonconservative systems with delayed nonlinear response," Phys. Rev. E 57, 3664-3667 (1998).
[CrossRef]

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]

Phys. Rev. Lett. (1)

C. Conti, M. Peccianti, and G. Assanto, "Observation of Optical Spatial Solitons in a highly nonlocal medium," Phys. Rev. Lett. 92, 113902-113902 (2004).
[CrossRef] [PubMed]

Other (5)

A. D. Boardman and A. P. Sukhorukov, Soliton Driven Photonics (Kluwer, Dordrecht, 2001).
[CrossRef]

A. Yariv, Quantum electronics-3rd ed. (John Wiley & Sons, 1987).

P. G. de Gennes, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1993).

I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (John Wiley & Sons Inc., New York, 1995).

F. Simoni, Nonlinear Optical Properties of Liquid Crystals (World Scientific, 1997).

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Figures (9)

Fig. 1.
Fig. 1.

Here we represent the system under study that is an NLC cell of tickness L, and width d, crossed by Ej (j=1, …,N) gaussian light beams, each of them impinging on the sample with angle αj . The director orientation is identified by the angle θ formed by the director and the z-axis

Fig. 2.
Fig. 2.

Simulation carried out for an angle of incidence α=1 rad and a normalized field amplitude e 0=3. The first and second frames show the orientation of the molecular director in the sample at τ=0.25, and τ=0.56; the third is the temporal behavior of the reorientation angle θc in the center of the sample. This case represents a complete cancel-lation effect induced by the competition of the two beams.

Fig. 3.
Fig. 3.

Simulation carried out for α=0.2 rad, e 0=3. The first and second frames show the orientation of the molecular director in the sample at τ=0.34, and τ=0.76; the third is the temporal behavior of the reorientation angle θc in the center of the sample. This case represents a critical reorientation effect due to the second light-induced Fréedericksz transition (LIFT II)

Fig. 4.
Fig. 4.

The L 2-Norm of a cut in ξ of θ(ξ,ζ) at the center of the sample (ζ=1/2), as a function of the control parameters e 0 and α, is presented as a measure of the director reorientation. The dark zone corresponds to a critical reorientation, the white one to a cancellation effect. The map refers to the case in which the second beam is switched on the sample well after the reorientation process induced by the first one has been completed.

Fig. 5.
Fig. 5.

Equilibrium states in the case of equal impinging intensities; vectors e 1,2 represent the electric fields of light beams, n is the molecular director. (a) initial homeotropic state, or consequence of a complete cancellation effect. (b) complete planar reorientation.

Fig. 6.
Fig. 6.

Sketch of the experimental Set-Up: BS: beam splitter; M, M 1, M 2: mirrors,W 1,W 2: half-wave plates; P 1, P 2: polarizers; L 1, L 2: spherical lenses ; S: sample.

Fig. 7.
Fig. 7.

(a) Experimental results obtained for different values of impinging power and angles of incidence. Black dots indicate that, at the end of the process, reorientation due to a LIFT II effect has taken place. Gray dots indicate that a a complete cancellation has occurred. (b) Theoretical map of L 2[θ(ξ,1/2)] in the corresponding zone of the α-e 2 0 plane

Fig. 8.
Fig. 8.

(a) Experimental behavior of the optical divergence as a function of the laser power for different α values that correspond to geometries in which a threshold in the field amplitude exists for the reorientation effect. (b) L 2[θ(ξ,1/2)] as a function of e 2 0 (in the corresponding range of values of the field intensity) for the same α values.

Fig. 9.
Fig. 9.

The L 2-Norm of a cut in ξ of θ at the center of the sample (L 2[θ(ξ,1/2)]), as a function of control parameters e 0 and α. The dark zone corresponds to critical reorientation, the white one to a cancellation effect. The map refers to the case in which the two beams are switched on the sample simultaneously

Equations (7)

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γ θ t = K 2 θ + ε 0 Δε 4 i , j = 1 N E i E j * sin ( 2 θ α i α j ) ,
x x cos α j z sin α j
z x sin α j + z cos α j
n ( α j , θ ) = n 0 n e n e 2 cos 2 ( α j θ ) + n 0 2 sin 2 ( α j θ ) .
e j = e 0 w 0 w j exp { i [ k ( ξ sin α j + ζ cos α j ) η j ] ( ξ cos α j ζ sin α j ) 2 [ 1 w j 2 + ik 2 r j ] }
j = 1 , …N
θ τ = 2 θ + i , j = 1 N e i e j * sin ( 2 θ α i α j ) ,

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