Abstract

We present exact equations and numerical solutions for molecular reorientation in chiral and nonchiral nematic liquid crystals induced by the inhomogeneous field of a shape corresponding to the Gaussian light beam. We show the importance of the individual terms for different light polarization and intensity. We also present examples of simplified equations for particular cases.

© 2012 Optical Society of America

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References

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  1. P. G. de Gennes, The Physics of Liquid Crystals (Clarendon, 1974).
  2. I.-C. Khoo, Liquid Crystals (Wiley, 2007).
  3. B. Ya. Zel’dovich and N. V. Tabiryan, “Orientational optical nonlinearity of liquid crystals,” Sov. Phys. Usp. 28, 1059–1083 (1985).
    [CrossRef]
  4. F. Simoni, Nonlinear Optical Properties of Liquid Crystals (World Scientific, 1997).
  5. G. Assanto, M. Peccianti, and C. Conti, “Nematicons: optical spatial solitons in nematic liquid crystals,” Opt. Photon. News 14(2), 44–48 (2003).
    [CrossRef]
  6. G. Assanto, and M. A. Karpierz, “Nematicons: self-localised beams in nematic liquid crystals,” Liq. Cryst. 36, 1161–1172 (2009).
    [CrossRef]
  7. M. Warenghem, J. F. Henninot, and G. Abbate, “Non linearly induced self waveguiding structure in dye doped nematic liquid crystals confined in capillaries,” Opt. Express 2, 483–490 (1998).
    [CrossRef]
  8. J. Beeckman, K. Neyts, and M. Haelterman, “Patterned electrode steering of nematicons,” J. Opt. A 8, 214–220 (2006).
    [CrossRef]
  9. U. A. Laudyn, M. Kwasny, and M. A. Karpierz, “Nematicons in chiral nematic liquid crystals,” Appl. Phys. Lett. 94, 091110 (2009).
    [CrossRef]
  10. Y. V. Izdebskaya, A. S. Desyatnikov, G. Assanto, and Y. S. Kivshar, “Multimode nematicon waveguides,” Opt. Lett. 36, 184–186 (2011).
    [CrossRef]
  11. A. Piccardi, U. Bortolozzo, S. Residori, and G. Assanto, “Spatial solitons in liquid-crystal light valves,” Opt. Lett. 34, 737–739 (2009).
    [CrossRef]
  12. Q. Wang, S. He, F. Yu, and N. Huang, “Iterative finite-difference method for calculating the distribution of a liquid-crystal director,” Opt. Eng. 40, 2552–2557 (2001).
    [CrossRef]
  13. E. Santamato, P. Maddalena, M. Settembre, M. Romagnoli, and B. Daino, “TM modes in a slab waveguide filled with nematic liquid crystal in an external magnetic field,” J. Opt. Soc. Am. B 6, 126–130 (1989).
    [CrossRef]
  14. M. A. Karpierz, “Solitary waves in liquid crystalline waveguides,” Phys. Rev. E 66, 036603 (2002).
    [CrossRef]
  15. A. Alberucci and G. Assanto, “Propagation of optical spatial solitons in finite-size media: interplay between nonlocality and boundary conditions,” J. Opt. Soc. Am. B 24, 2314–2320 (2007).
    [CrossRef]
  16. F. A. Sala and M. A. Karpierz, “Numerical simulation of beam propagation in a layer filled with chiral nematic liquid crystals,” Photonics Lett. Pol. 1, 163–165 (2009).
    [CrossRef]
  17. M. A. Karpierz, “Nonlinear properties of waveguides with twisted nematic liquid crystal,” Acta Phys. Pol. A 99, 161–173 (2001).
  18. P. J. M. Vanbrabant, J. Beeckman, K. Neyts, R. James, and F. A. Fernandez, “A finite element beam propagation method for simulation of liquid crystal devices,” Opt. Express 17, 10895 (2009).
    [CrossRef]
  19. A. A. Minzoni, N. F. Smyth, and A. L. Worthy, “Modulation solutions for nematicon propagation in nonlocal liquid crystals,” J. Opt. Soc. Am. B 24, 1549–1556 (2007).
    [CrossRef]
  20. C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
    [CrossRef]
  21. E. Brasselet, B. Doyon, T. V. Galstian, and L. J. Dubé, “Optically induced dynamics in nematic liquid crystals: the role of twist deformation and asymmetry,” Phys. Rev. E 67, 031706 (2003).
    [CrossRef]
  22. F. A. Sala and M. A. Karpierz, “Modeling of nonlinear beam propagation in chiral nematic liquid crystals,” Mol. Cryst. Liq. Cryst.558, 176–183 (2012).
  23. M. Peccianti, A. Dyadyusha, M. Kaczmarek, and Gaetano Assanto, “Escaping solitons from a trapping potential,” Phys. Rev. Lett. 101, 153902 (2008).
    [CrossRef]
  24. M. Peccianti, A. Dyadyusha, M. Kaczmarek, and Gaetano Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. 2, 737–742 (2006).
    [CrossRef]
  25. W. Baran, Z. Raszewski, R. Dabrowski, and J. Kedzierski, “Some physical properties of mesogenic 4-(trans-4′-n-alkylcyclohexyl) isothiocyanatobenzenes,” Mol. Cryst. Liq. Cryst. 123, 237–245 (1985).
    [CrossRef]
  26. R. Dabrowski, J. Dziaduszek, and T. Szczucinski, “Mesomorphic characteristics of some new homologous series with the isothiocyanato terminal group,” Mol. Cryst. Liq. Cryst. 124, 241–257 (1985).
    [CrossRef]
  27. D. M. Young, “Iterative methods for solving partial difference equations of elliptical type,” Ph.D. thesis (Harvard University, 1950).
  28. L. Hageman and D. Young, Applied Iterative Methods(Academic, 1981).

2011 (1)

2009 (5)

A. Piccardi, U. Bortolozzo, S. Residori, and G. Assanto, “Spatial solitons in liquid-crystal light valves,” Opt. Lett. 34, 737–739 (2009).
[CrossRef]

P. J. M. Vanbrabant, J. Beeckman, K. Neyts, R. James, and F. A. Fernandez, “A finite element beam propagation method for simulation of liquid crystal devices,” Opt. Express 17, 10895 (2009).
[CrossRef]

G. Assanto, and M. A. Karpierz, “Nematicons: self-localised beams in nematic liquid crystals,” Liq. Cryst. 36, 1161–1172 (2009).
[CrossRef]

U. A. Laudyn, M. Kwasny, and M. A. Karpierz, “Nematicons in chiral nematic liquid crystals,” Appl. Phys. Lett. 94, 091110 (2009).
[CrossRef]

F. A. Sala and M. A. Karpierz, “Numerical simulation of beam propagation in a layer filled with chiral nematic liquid crystals,” Photonics Lett. Pol. 1, 163–165 (2009).
[CrossRef]

2008 (1)

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and Gaetano Assanto, “Escaping solitons from a trapping potential,” Phys. Rev. Lett. 101, 153902 (2008).
[CrossRef]

2007 (2)

2006 (2)

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and Gaetano Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. 2, 737–742 (2006).
[CrossRef]

J. Beeckman, K. Neyts, and M. Haelterman, “Patterned electrode steering of nematicons,” J. Opt. A 8, 214–220 (2006).
[CrossRef]

2003 (3)

G. Assanto, M. Peccianti, and C. Conti, “Nematicons: optical spatial solitons in nematic liquid crystals,” Opt. Photon. News 14(2), 44–48 (2003).
[CrossRef]

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

E. Brasselet, B. Doyon, T. V. Galstian, and L. J. Dubé, “Optically induced dynamics in nematic liquid crystals: the role of twist deformation and asymmetry,” Phys. Rev. E 67, 031706 (2003).
[CrossRef]

2002 (1)

M. A. Karpierz, “Solitary waves in liquid crystalline waveguides,” Phys. Rev. E 66, 036603 (2002).
[CrossRef]

2001 (2)

M. A. Karpierz, “Nonlinear properties of waveguides with twisted nematic liquid crystal,” Acta Phys. Pol. A 99, 161–173 (2001).

Q. Wang, S. He, F. Yu, and N. Huang, “Iterative finite-difference method for calculating the distribution of a liquid-crystal director,” Opt. Eng. 40, 2552–2557 (2001).
[CrossRef]

1998 (1)

1989 (1)

1985 (3)

W. Baran, Z. Raszewski, R. Dabrowski, and J. Kedzierski, “Some physical properties of mesogenic 4-(trans-4′-n-alkylcyclohexyl) isothiocyanatobenzenes,” Mol. Cryst. Liq. Cryst. 123, 237–245 (1985).
[CrossRef]

R. Dabrowski, J. Dziaduszek, and T. Szczucinski, “Mesomorphic characteristics of some new homologous series with the isothiocyanato terminal group,” Mol. Cryst. Liq. Cryst. 124, 241–257 (1985).
[CrossRef]

B. Ya. Zel’dovich and N. V. Tabiryan, “Orientational optical nonlinearity of liquid crystals,” Sov. Phys. Usp. 28, 1059–1083 (1985).
[CrossRef]

Abbate, G.

Alberucci, A.

Assanto, G.

Y. V. Izdebskaya, A. S. Desyatnikov, G. Assanto, and Y. S. Kivshar, “Multimode nematicon waveguides,” Opt. Lett. 36, 184–186 (2011).
[CrossRef]

A. Piccardi, U. Bortolozzo, S. Residori, and G. Assanto, “Spatial solitons in liquid-crystal light valves,” Opt. Lett. 34, 737–739 (2009).
[CrossRef]

G. Assanto, and M. A. Karpierz, “Nematicons: self-localised beams in nematic liquid crystals,” Liq. Cryst. 36, 1161–1172 (2009).
[CrossRef]

A. Alberucci and G. Assanto, “Propagation of optical spatial solitons in finite-size media: interplay between nonlocality and boundary conditions,” J. Opt. Soc. Am. B 24, 2314–2320 (2007).
[CrossRef]

G. Assanto, M. Peccianti, and C. Conti, “Nematicons: optical spatial solitons in nematic liquid crystals,” Opt. Photon. News 14(2), 44–48 (2003).
[CrossRef]

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

Assanto, Gaetano

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and Gaetano Assanto, “Escaping solitons from a trapping potential,” Phys. Rev. Lett. 101, 153902 (2008).
[CrossRef]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and Gaetano Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. 2, 737–742 (2006).
[CrossRef]

Baran, W.

W. Baran, Z. Raszewski, R. Dabrowski, and J. Kedzierski, “Some physical properties of mesogenic 4-(trans-4′-n-alkylcyclohexyl) isothiocyanatobenzenes,” Mol. Cryst. Liq. Cryst. 123, 237–245 (1985).
[CrossRef]

Beeckman, J.

Bortolozzo, U.

Brasselet, E.

E. Brasselet, B. Doyon, T. V. Galstian, and L. J. Dubé, “Optically induced dynamics in nematic liquid crystals: the role of twist deformation and asymmetry,” Phys. Rev. E 67, 031706 (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]

G. Assanto, M. Peccianti, and C. Conti, “Nematicons: optical spatial solitons in nematic liquid crystals,” Opt. Photon. News 14(2), 44–48 (2003).
[CrossRef]

Dabrowski, R.

W. Baran, Z. Raszewski, R. Dabrowski, and J. Kedzierski, “Some physical properties of mesogenic 4-(trans-4′-n-alkylcyclohexyl) isothiocyanatobenzenes,” Mol. Cryst. Liq. Cryst. 123, 237–245 (1985).
[CrossRef]

R. Dabrowski, J. Dziaduszek, and T. Szczucinski, “Mesomorphic characteristics of some new homologous series with the isothiocyanato terminal group,” Mol. Cryst. Liq. Cryst. 124, 241–257 (1985).
[CrossRef]

Daino, B.

de Gennes, P. G.

P. G. de Gennes, The Physics of Liquid Crystals (Clarendon, 1974).

Desyatnikov, A. S.

Doyon, B.

E. Brasselet, B. Doyon, T. V. Galstian, and L. J. Dubé, “Optically induced dynamics in nematic liquid crystals: the role of twist deformation and asymmetry,” Phys. Rev. E 67, 031706 (2003).
[CrossRef]

Dubé, L. J.

E. Brasselet, B. Doyon, T. V. Galstian, and L. J. Dubé, “Optically induced dynamics in nematic liquid crystals: the role of twist deformation and asymmetry,” Phys. Rev. E 67, 031706 (2003).
[CrossRef]

Dyadyusha, A.

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and Gaetano Assanto, “Escaping solitons from a trapping potential,” Phys. Rev. Lett. 101, 153902 (2008).
[CrossRef]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and Gaetano Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. 2, 737–742 (2006).
[CrossRef]

Dziaduszek, J.

R. Dabrowski, J. Dziaduszek, and T. Szczucinski, “Mesomorphic characteristics of some new homologous series with the isothiocyanato terminal group,” Mol. Cryst. Liq. Cryst. 124, 241–257 (1985).
[CrossRef]

Fernandez, F. A.

Galstian, T. V.

E. Brasselet, B. Doyon, T. V. Galstian, and L. J. Dubé, “Optically induced dynamics in nematic liquid crystals: the role of twist deformation and asymmetry,” Phys. Rev. E 67, 031706 (2003).
[CrossRef]

Haelterman, M.

J. Beeckman, K. Neyts, and M. Haelterman, “Patterned electrode steering of nematicons,” J. Opt. A 8, 214–220 (2006).
[CrossRef]

Hageman, L.

L. Hageman and D. Young, Applied Iterative Methods(Academic, 1981).

He, S.

Q. Wang, S. He, F. Yu, and N. Huang, “Iterative finite-difference method for calculating the distribution of a liquid-crystal director,” Opt. Eng. 40, 2552–2557 (2001).
[CrossRef]

Henninot, J. F.

Huang, N.

Q. Wang, S. He, F. Yu, and N. Huang, “Iterative finite-difference method for calculating the distribution of a liquid-crystal director,” Opt. Eng. 40, 2552–2557 (2001).
[CrossRef]

Izdebskaya, Y. V.

James, R.

Kaczmarek, M.

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and Gaetano Assanto, “Escaping solitons from a trapping potential,” Phys. Rev. Lett. 101, 153902 (2008).
[CrossRef]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and Gaetano Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. 2, 737–742 (2006).
[CrossRef]

Karpierz, M. A.

G. Assanto, and M. A. Karpierz, “Nematicons: self-localised beams in nematic liquid crystals,” Liq. Cryst. 36, 1161–1172 (2009).
[CrossRef]

U. A. Laudyn, M. Kwasny, and M. A. Karpierz, “Nematicons in chiral nematic liquid crystals,” Appl. Phys. Lett. 94, 091110 (2009).
[CrossRef]

F. A. Sala and M. A. Karpierz, “Numerical simulation of beam propagation in a layer filled with chiral nematic liquid crystals,” Photonics Lett. Pol. 1, 163–165 (2009).
[CrossRef]

M. A. Karpierz, “Solitary waves in liquid crystalline waveguides,” Phys. Rev. E 66, 036603 (2002).
[CrossRef]

M. A. Karpierz, “Nonlinear properties of waveguides with twisted nematic liquid crystal,” Acta Phys. Pol. A 99, 161–173 (2001).

F. A. Sala and M. A. Karpierz, “Modeling of nonlinear beam propagation in chiral nematic liquid crystals,” Mol. Cryst. Liq. Cryst.558, 176–183 (2012).

Kedzierski, J.

W. Baran, Z. Raszewski, R. Dabrowski, and J. Kedzierski, “Some physical properties of mesogenic 4-(trans-4′-n-alkylcyclohexyl) isothiocyanatobenzenes,” Mol. Cryst. Liq. Cryst. 123, 237–245 (1985).
[CrossRef]

Khoo, I.-C.

I.-C. Khoo, Liquid Crystals (Wiley, 2007).

Kivshar, Y. S.

Kwasny, M.

U. A. Laudyn, M. Kwasny, and M. A. Karpierz, “Nematicons in chiral nematic liquid crystals,” Appl. Phys. Lett. 94, 091110 (2009).
[CrossRef]

Laudyn, U. A.

U. A. Laudyn, M. Kwasny, and M. A. Karpierz, “Nematicons in chiral nematic liquid crystals,” Appl. Phys. Lett. 94, 091110 (2009).
[CrossRef]

Maddalena, P.

Minzoni, A. A.

Neyts, K.

Peccianti, M.

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and Gaetano Assanto, “Escaping solitons from a trapping potential,” Phys. Rev. Lett. 101, 153902 (2008).
[CrossRef]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and Gaetano Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. 2, 737–742 (2006).
[CrossRef]

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

G. Assanto, M. Peccianti, and C. Conti, “Nematicons: optical spatial solitons in nematic liquid crystals,” Opt. Photon. News 14(2), 44–48 (2003).
[CrossRef]

Piccardi, A.

Raszewski, Z.

W. Baran, Z. Raszewski, R. Dabrowski, and J. Kedzierski, “Some physical properties of mesogenic 4-(trans-4′-n-alkylcyclohexyl) isothiocyanatobenzenes,” Mol. Cryst. Liq. Cryst. 123, 237–245 (1985).
[CrossRef]

Residori, S.

Romagnoli, M.

Sala, F. A.

F. A. Sala and M. A. Karpierz, “Numerical simulation of beam propagation in a layer filled with chiral nematic liquid crystals,” Photonics Lett. Pol. 1, 163–165 (2009).
[CrossRef]

F. A. Sala and M. A. Karpierz, “Modeling of nonlinear beam propagation in chiral nematic liquid crystals,” Mol. Cryst. Liq. Cryst.558, 176–183 (2012).

Santamato, E.

Settembre, M.

Simoni, F.

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

Smyth, N. F.

Szczucinski, T.

R. Dabrowski, J. Dziaduszek, and T. Szczucinski, “Mesomorphic characteristics of some new homologous series with the isothiocyanato terminal group,” Mol. Cryst. Liq. Cryst. 124, 241–257 (1985).
[CrossRef]

Tabiryan, N. V.

B. Ya. Zel’dovich and N. V. Tabiryan, “Orientational optical nonlinearity of liquid crystals,” Sov. Phys. Usp. 28, 1059–1083 (1985).
[CrossRef]

Vanbrabant, P. J. M.

Wang, Q.

Q. Wang, S. He, F. Yu, and N. Huang, “Iterative finite-difference method for calculating the distribution of a liquid-crystal director,” Opt. Eng. 40, 2552–2557 (2001).
[CrossRef]

Warenghem, M.

Worthy, A. L.

Young, D.

L. Hageman and D. Young, Applied Iterative Methods(Academic, 1981).

Young, D. M.

D. M. Young, “Iterative methods for solving partial difference equations of elliptical type,” Ph.D. thesis (Harvard University, 1950).

Yu, F.

Q. Wang, S. He, F. Yu, and N. Huang, “Iterative finite-difference method for calculating the distribution of a liquid-crystal director,” Opt. Eng. 40, 2552–2557 (2001).
[CrossRef]

Zel’dovich, B. Ya.

B. Ya. Zel’dovich and N. V. Tabiryan, “Orientational optical nonlinearity of liquid crystals,” Sov. Phys. Usp. 28, 1059–1083 (1985).
[CrossRef]

Acta Phys. Pol. A (1)

M. A. Karpierz, “Nonlinear properties of waveguides with twisted nematic liquid crystal,” Acta Phys. Pol. A 99, 161–173 (2001).

Appl. Phys. Lett. (1)

U. A. Laudyn, M. Kwasny, and M. A. Karpierz, “Nematicons in chiral nematic liquid crystals,” Appl. Phys. Lett. 94, 091110 (2009).
[CrossRef]

J. Opt. A (1)

J. Beeckman, K. Neyts, and M. Haelterman, “Patterned electrode steering of nematicons,” J. Opt. A 8, 214–220 (2006).
[CrossRef]

J. Opt. Soc. Am. B (3)

Liq. Cryst. (1)

G. Assanto, and M. A. Karpierz, “Nematicons: self-localised beams in nematic liquid crystals,” Liq. Cryst. 36, 1161–1172 (2009).
[CrossRef]

Mol. Cryst. Liq. Cryst. (2)

W. Baran, Z. Raszewski, R. Dabrowski, and J. Kedzierski, “Some physical properties of mesogenic 4-(trans-4′-n-alkylcyclohexyl) isothiocyanatobenzenes,” Mol. Cryst. Liq. Cryst. 123, 237–245 (1985).
[CrossRef]

R. Dabrowski, J. Dziaduszek, and T. Szczucinski, “Mesomorphic characteristics of some new homologous series with the isothiocyanato terminal group,” Mol. Cryst. Liq. Cryst. 124, 241–257 (1985).
[CrossRef]

Nat. Phys. (1)

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and Gaetano Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. 2, 737–742 (2006).
[CrossRef]

Opt. Eng. (1)

Q. Wang, S. He, F. Yu, and N. Huang, “Iterative finite-difference method for calculating the distribution of a liquid-crystal director,” Opt. Eng. 40, 2552–2557 (2001).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Opt. Photon. News (1)

G. Assanto, M. Peccianti, and C. Conti, “Nematicons: optical spatial solitons in nematic liquid crystals,” Opt. Photon. News 14(2), 44–48 (2003).
[CrossRef]

Photonics Lett. Pol. (1)

F. A. Sala and M. A. Karpierz, “Numerical simulation of beam propagation in a layer filled with chiral nematic liquid crystals,” Photonics Lett. Pol. 1, 163–165 (2009).
[CrossRef]

Phys. Rev. E (2)

M. A. Karpierz, “Solitary waves in liquid crystalline waveguides,” Phys. Rev. E 66, 036603 (2002).
[CrossRef]

E. Brasselet, B. Doyon, T. V. Galstian, and L. J. Dubé, “Optically induced dynamics in nematic liquid crystals: the role of twist deformation and asymmetry,” Phys. Rev. E 67, 031706 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and Gaetano Assanto, “Escaping solitons from a trapping potential,” Phys. Rev. Lett. 101, 153902 (2008).
[CrossRef]

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

Sov. Phys. Usp. (1)

B. Ya. Zel’dovich and N. V. Tabiryan, “Orientational optical nonlinearity of liquid crystals,” Sov. Phys. Usp. 28, 1059–1083 (1985).
[CrossRef]

Other (6)

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

P. G. de Gennes, The Physics of Liquid Crystals (Clarendon, 1974).

I.-C. Khoo, Liquid Crystals (Wiley, 2007).

F. A. Sala and M. A. Karpierz, “Modeling of nonlinear beam propagation in chiral nematic liquid crystals,” Mol. Cryst. Liq. Cryst.558, 176–183 (2012).

D. M. Young, “Iterative methods for solving partial difference equations of elliptical type,” Ph.D. thesis (Harvard University, 1950).

L. Hageman and D. Young, Applied Iterative Methods(Academic, 1981).

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Figures (9)

Fig. 1.
Fig. 1.

Setup and coordinate system of the analyzed cell with planarly oriented NLC or ChNLC.

Fig. 2.
Fig. 2.

Maximum positive and negative molecular reorientation for chiral and planar configurations versus power. In planar configuration, molecules are aligned along the z axis (ϕ=0,θ=π2). Electric field is x polarized (α=0°).

Fig. 3.
Fig. 3.

Numerical results of molecular reorientation in planar configuration where molecules initially lie along the z axis. Results are presented for different polarizations (α=0°, 42°, and 90°). (Top) ϕ(x,y) distribution, (bottom) θ(x,y) distribution.

Fig. 4.
Fig. 4.

Numerical results of molecular reorientation in ChNLCs for different polarizations (α=0° 45°, and 90°). (Top) ϕ(x,y)ϕi(x) distribution, where ϕi are initial conditions ϕi=2πHxπ2, (center) θ(x,y) distribution, (bottom) molecule visualization along the x axis from two orthogonal viewpoints.

Fig. 5.
Fig. 5.

Numerical results of molecular reorientation in ChNLCs for different field positions along the x axis (0, 2.5, and 5 μm from the center of the cell). (Top) ϕ(x,y)ϕi(x) distribution where ϕi are initial conditions, (bottom) θ(x,y) distribution.

Fig. 6.
Fig. 6.

Polarization dependence of maximum angle differences (positive and negative) between the exact solution and a solution without p1 or t1 term. Molecules are in the planar configuration and are initially aligned along the z axis. Curve total represents maximum and minimum reorientation in relation to the initial state.

Fig. 7.
Fig. 7.

Maximum angle differences between the exact solution and a solution without p1 or t1 term versus power. Simulations are performed for the polarization angle α=45°. Molecules are in the planar configuration and are initially aligned along the z axis.

Fig. 8.
Fig. 8.

Polarization dependence of maximum angle differences (positive and negative) between the exact solution and a solution without particular pi or ti term for ChNLCs. Curve total represents maximum and minimum reorientation in relation to the initial state.

Fig. 9.
Fig. 9.

Power dependence of maximum angle differences (positive and negative) between the exact solution and a solution without particular pi or ti term for ChNLCs. Polarization is α=45°.

Equations (16)

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n⃗=[cosθ,sinθsinϕ,sinθcosϕ].
f=12K11(n⃗)2+12K22(n⃗·(×n⃗)G)2+12K33(n⃗×(×n⃗))212Δεε0(n⃗·E⃗)2,
xfϕx+yfϕy+zfϕzfϕ=0,xfθx+yfθy+zfθzfθ=0,
sin2θ·2ϕ+sin2θ(ϕxθx+ϕyθy+ϕzθz)+G·(sin2θ·θx+sin2θsinϕ·θy+2sin2θcosϕ·θz)++Δϵϵ02K(2EyEzsin2θcos2ϕ+sin2θ(ExEycosϕExEzsinϕ)+sin2θsin2ϕ(Ey2Ez2))=0,
2θ12sin2θ((ϕx)2+(ϕy)2+(ϕz)2)++G·(sin2θϕx+2sin2θsinϕϕy+2sin2θcosϕϕz)++Δϵϵ02K(EzExsin2θsin2ϕ+2cos2θ·(ExEysinϕ+ExEzcosϕ)+sin2θ·(Ez2cos2ϕ+Ey2sin2ϕEx2))=0.
i=02pi=0,p0=sin2θ·(2ϕx2+2ϕy2),p1=sin2θ·(ϕxθx+ϕyθy),p2=G·(sin2θ·θx+2sin2θsinϕ·θy),p3=ϵϵ02K(2EyEzsin2θcos2ϕ+sin2θ(ExEycosϕExEzsinϕ),+sin2θsin2ϕ(Ey2Ez2)),
i=03ti=0,t0=2θx2+2θy2,t1=12sin2θ((ϕx)2+(ϕy)2),t2=G·(sin2θϕx+2sin2θsinϕϕy),t3=ϵϵ02K(EzEysin2θsin2ϕ+2cos2θ·(ExEysinϕ+ExEzcosϕ)+sin2θ·(Ez2cos2ϕ+Ey2sin2ϕEx2)).
2θy2+2θx2+Δεε02K(2ExEycos2θ+sin2θ(Ey2Ex2))=0,
2θx2+2θy2+Δεε02K(2ExEzcos2θ+sin2θ(Ez2Ex2))=0.
2ϕx2+2ϕy2+Δεε02K(2EyEzcos2ϕ+sin2ϕ(Ey2Ez2))=0.
sin2θ·d2ϕdx2+sin2θ·dθdxdϕdxG·sin2θ·dθdx+p3=0,
d2θdx212sin2θ(dϕdx)2+G·sin2θdϕdx+t3=0.
d2θdx2+sin2θ·Δεε02K(KG2Δεε0Ex2)=0.
ϕ(x,0)=ϕ(x,W)=ϕ(0,y)=ϕ(H,y)=const,θ(0,y)=θ(H,y)=θ(x,0)=θ(x,W)=π2.
ϕ(x,0)=ϕ(x,W)=π2+2πHx,ϕ(0,y)=ϕ(H,y)=π2,θ(0,y)=θ(H,y)=θ(x,0)=θ(x,W)=π2.
Ex2=PZ0εr·12πω02exp((x0x)2+(y0y)22ω02)·sin2(α),Ey2=PZ0εr·12πω02exp((x0x)2+(y0y)22ω02)·cos2(α),Ez=0,

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