Abstract

The exact molecular reorientation model for nematic liquid crystals taking into account all diagonal Frank elastic constants and using two angles to describe director orientation is presented. Solutions and simplified equations are shown for the most common planar and chiral configurations. Gaussian beam propagation simulated using fully vectorial Beam Propagation Method in nonlinear case is also provided. Detailed comparison between exact solutions and single Frank constant approximation is made. However, no significant differences between these two models were found neither in beam propagation nor in polarization distribution, some difficulties may occur in choosing single Frank constant especially when it comes to quantitative results. Presented results correspond to a propagation of a beam of the Gaussian or topologically similar shapes.

© 2012 OSA

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  1. D. W. Berreman and W. R. Heffner, “New bistable cholesteric liquid-crystal display,” Appl. Phys. Lett. 37, 109–111 (1980).
    [CrossRef]
  2. S.-Y. Lu and L.-C. Chien, “A polymer-stabilized single-layer color cholesteric liquid crystal display with anisotropic reflection,” Appl. Phys. Lett. 91, 131119 (2007).
    [CrossRef]
  3. B. Bahadur, Liquid Crystals: Applications and Uses (World Scientific Publishing Co. Pte. Ltd., 1995).
  4. G. Assanto, M. Peccianti, and C. Conti, “Nematicons: Optical spatial solitons in nematic liquid crystals,” Opt. Photonics News 14, 44–48 (2003).
    [CrossRef]
  5. A. Alberucci, A. Piccardi, M. Peccianti, M. Kaczmarek, and G. Assanto, “Propagation of spatial optical solitons in a dielectric with adjustable nonlinearity,” Phys. Rev. A 82, 023806 (2010).
    [CrossRef]
  6. G. Assanto and M. A. Karpierz, “Nematicons: self-localised beams in nematic liquid crystals,” Liq. Cryst. 36, 1161–1172 (2009).
    [CrossRef]
  7. Y. V. Izdebskaya, A. S. Desyatnikov, G. Assanto, and Y. S. Kivshar, “Multimode nematicon waveguides,” Opt. Lett. 36, 184–186 (2011).
    [CrossRef] [PubMed]
  8. C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
    [CrossRef] [PubMed]
  9. F. A. Sala and M. A. Karpierz, “Numerical simulation of beam propagation in a layer filled with chiral nematic liquid crystals,” Photon. Lett. Pol. 1, 163–165 (2009).
  10. 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]
  11. 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, 10895–10909 (2009).
    [CrossRef] [PubMed]
  12. J. Beeckman, K. Neyts, P. Vanbrabant, R. James, and F. Fernandez, “Finding exact spatial soliton profiles in nematic liquid crystals,” Opt. Express 18, 3311–3321 (2010).
    [CrossRef] [PubMed]
  13. G. D. Ziogos and E. E. Kriezis, “Modeling light propagation in liquid crystal devices with a 3-D full-vector finite-element beam propagation method,” Opt. Quantum Electron. 40, 733–748 (2008).
    [CrossRef]
  14. J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
    [CrossRef]
  15. B. Y. Zel’dovich and N. V. Tabiryan, “Orientational optical nonlinearity of liquid crystals,” Sov. Phys. Usp. 28, 1059–1083 (1985).
    [CrossRef]
  16. U. A. Laudyn, K. Jaworowicz, and M. A. Karpierz, “Spatial solitons in chiral nematics,” Mol. Cryst. Liq. Cryst. 489, 214–221 (2008).
    [CrossRef]
  17. M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. 2, 737–742 (2006).
    [CrossRef]
  18. M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Escaping solitons from a trapping potential,” Phys. Rev. Lett. 101, 153902 (2008).
    [CrossRef] [PubMed]
  19. F. A. Sala and M. A. Karpierz, “Chiral and non-chiral nematic liquid crystal reorientation induced by inhomogeneous electric fields,” J. Opt. Soc. Am. B 29 (submitted) (2012).
  20. 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)
    [CrossRef]
  21. I.-C. Khoo, Liquid Crystals (John Wiley and Sons, 2007).
    [CrossRef]
  22. F. C. Frank, “I. Liquid crystals. On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
    [CrossRef]
  23. C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933).
    [CrossRef]
  24. 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]
  25. 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]
  26. D. M. Young, “Iterative methods for solving partial difference equations of elliptical type,” PhD thesis, Harvard University (1950).
  27. L. Hageman and D. Young, Applied Iterative Methods (Academic Press, 1981).
  28. P. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, 1993).

2012 (2)

F. A. Sala and M. A. Karpierz, “Chiral and non-chiral nematic liquid crystal reorientation induced by inhomogeneous electric fields,” J. Opt. Soc. Am. B 29 (submitted) (2012).

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)
[CrossRef]

2011 (1)

2010 (2)

J. Beeckman, K. Neyts, P. Vanbrabant, R. James, and F. Fernandez, “Finding exact spatial soliton profiles in nematic liquid crystals,” Opt. Express 18, 3311–3321 (2010).
[CrossRef] [PubMed]

A. Alberucci, A. Piccardi, M. Peccianti, M. Kaczmarek, and G. Assanto, “Propagation of spatial optical solitons in a dielectric with adjustable nonlinearity,” Phys. Rev. A 82, 023806 (2010).
[CrossRef]

2009 (3)

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

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

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, 10895–10909 (2009).
[CrossRef] [PubMed]

2008 (3)

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

G. D. Ziogos and E. E. Kriezis, “Modeling light propagation in liquid crystal devices with a 3-D full-vector finite-element beam propagation method,” Opt. Quantum Electron. 40, 733–748 (2008).
[CrossRef]

U. A. Laudyn, K. Jaworowicz, and M. A. Karpierz, “Spatial solitons in chiral nematics,” Mol. Cryst. Liq. Cryst. 489, 214–221 (2008).
[CrossRef]

2007 (2)

S.-Y. Lu and L.-C. Chien, “A polymer-stabilized single-layer color cholesteric liquid crystal display with anisotropic reflection,” Appl. Phys. Lett. 91, 131119 (2007).
[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]

2006 (1)

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

2003 (2)

G. Assanto, M. Peccianti, and C. Conti, “Nematicons: Optical spatial solitons in nematic liquid crystals,” Opt. Photonics News 14, 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] [PubMed]

1985 (3)

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

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]

1980 (1)

D. W. Berreman and W. R. Heffner, “New bistable cholesteric liquid-crystal display,” Appl. Phys. Lett. 37, 109–111 (1980).
[CrossRef]

1976 (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

1958 (1)

F. C. Frank, “I. Liquid crystals. On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
[CrossRef]

1933 (1)

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933).
[CrossRef]

Alberucci, A.

A. Alberucci, A. Piccardi, M. Peccianti, M. Kaczmarek, and G. Assanto, “Propagation of spatial optical solitons in a dielectric with adjustable nonlinearity,” Phys. Rev. A 82, 023806 (2010).
[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]

Assanto, G.

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

A. Alberucci, A. Piccardi, M. Peccianti, M. Kaczmarek, and G. Assanto, “Propagation of spatial optical solitons in a dielectric with adjustable nonlinearity,” Phys. Rev. A 82, 023806 (2010).
[CrossRef]

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

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

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]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. 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] [PubMed]

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

Bahadur, B.

B. Bahadur, Liquid Crystals: Applications and Uses (World Scientific Publishing Co. Pte. Ltd., 1995).

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.

Berreman, D. W.

D. W. Berreman and W. R. Heffner, “New bistable cholesteric liquid-crystal display,” Appl. Phys. Lett. 37, 109–111 (1980).
[CrossRef]

Chien, L.-C.

S.-Y. Lu and L.-C. Chien, “A polymer-stabilized single-layer color cholesteric liquid crystal display with anisotropic reflection,” Appl. Phys. Lett. 91, 131119 (2007).
[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]

G. Assanto, M. Peccianti, and C. Conti, “Nematicons: Optical spatial solitons in nematic liquid crystals,” Opt. Photonics News 14, 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]

de Gennes, P.

P. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, 1993).

Desyatnikov, A. S.

Dyadyusha, A.

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

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. 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]

Feit, M. D.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Fernandez, F.

Fernandez, F. A.

Fleck, J. A.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Frank, F. C.

F. C. Frank, “I. Liquid crystals. On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
[CrossRef]

Hageman, L.

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

Heffner, W. R.

D. W. Berreman and W. R. Heffner, “New bistable cholesteric liquid-crystal display,” Appl. Phys. Lett. 37, 109–111 (1980).
[CrossRef]

Izdebskaya, Y. V.

James, R.

Jaworowicz, K.

U. A. Laudyn, K. Jaworowicz, and M. A. Karpierz, “Spatial solitons in chiral nematics,” Mol. Cryst. Liq. Cryst. 489, 214–221 (2008).
[CrossRef]

Kaczmarek, M.

A. Alberucci, A. Piccardi, M. Peccianti, M. Kaczmarek, and G. Assanto, “Propagation of spatial optical solitons in a dielectric with adjustable nonlinearity,” Phys. Rev. A 82, 023806 (2010).
[CrossRef]

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

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

Karpierz, M. A.

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)
[CrossRef]

F. A. Sala and M. A. Karpierz, “Chiral and non-chiral nematic liquid crystal reorientation induced by inhomogeneous electric fields,” J. Opt. Soc. Am. B 29 (submitted) (2012).

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

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

U. A. Laudyn, K. Jaworowicz, and M. A. Karpierz, “Spatial solitons in chiral nematics,” Mol. Cryst. Liq. Cryst. 489, 214–221 (2008).
[CrossRef]

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 (John Wiley and Sons, 2007).
[CrossRef]

Kivshar, Y. S.

Kriezis, E. E.

G. D. Ziogos and E. E. Kriezis, “Modeling light propagation in liquid crystal devices with a 3-D full-vector finite-element beam propagation method,” Opt. Quantum Electron. 40, 733–748 (2008).
[CrossRef]

Laudyn, U. A.

U. A. Laudyn, K. Jaworowicz, and M. A. Karpierz, “Spatial solitons in chiral nematics,” Mol. Cryst. Liq. Cryst. 489, 214–221 (2008).
[CrossRef]

Lu, S.-Y.

S.-Y. Lu and L.-C. Chien, “A polymer-stabilized single-layer color cholesteric liquid crystal display with anisotropic reflection,” Appl. Phys. Lett. 91, 131119 (2007).
[CrossRef]

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Neyts, K.

Oseen, C. W.

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933).
[CrossRef]

Peccianti, M.

A. Alberucci, A. Piccardi, M. Peccianti, M. Kaczmarek, and G. Assanto, “Propagation of spatial optical solitons in a dielectric with adjustable nonlinearity,” Phys. Rev. A 82, 023806 (2010).
[CrossRef]

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

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. 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] [PubMed]

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

Piccardi, A.

A. Alberucci, A. Piccardi, M. Peccianti, M. Kaczmarek, and G. Assanto, “Propagation of spatial optical solitons in a dielectric with adjustable nonlinearity,” Phys. Rev. A 82, 023806 (2010).
[CrossRef]

Prost, J.

P. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, 1993).

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]

Sala, F. A.

F. A. Sala and M. A. Karpierz, “Chiral and non-chiral nematic liquid crystal reorientation induced by inhomogeneous electric fields,” J. Opt. Soc. Am. B 29 (submitted) (2012).

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)
[CrossRef]

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

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

Vanbrabant, P.

Vanbrabant, P. J. M.

Young, D.

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

Young, D. M.

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

Zel’dovich, B. Y.

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

Ziogos, G. D.

G. D. Ziogos and E. E. Kriezis, “Modeling light propagation in liquid crystal devices with a 3-D full-vector finite-element beam propagation method,” Opt. Quantum Electron. 40, 733–748 (2008).
[CrossRef]

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Appl. Phys. Lett. (2)

D. W. Berreman and W. R. Heffner, “New bistable cholesteric liquid-crystal display,” Appl. Phys. Lett. 37, 109–111 (1980).
[CrossRef]

S.-Y. Lu and L.-C. Chien, “A polymer-stabilized single-layer color cholesteric liquid crystal display with anisotropic reflection,” Appl. Phys. Lett. 91, 131119 (2007).
[CrossRef]

Discuss. Faraday Soc. (1)

F. C. Frank, “I. Liquid crystals. On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
[CrossRef]

J. Opt. Soc. Am. B (2)

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]

F. A. Sala and M. A. Karpierz, “Chiral and non-chiral nematic liquid crystal reorientation induced by inhomogeneous electric fields,” J. Opt. Soc. Am. B 29 (submitted) (2012).

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. (4)

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)
[CrossRef]

U. A. Laudyn, K. Jaworowicz, and M. A. Karpierz, “Spatial solitons in chiral nematics,” Mol. Cryst. Liq. Cryst. 489, 214–221 (2008).
[CrossRef]

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 G. Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. 2, 737–742 (2006).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Opt. Photonics News (1)

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

Opt. Quantum Electron. (1)

G. D. Ziogos and E. E. Kriezis, “Modeling light propagation in liquid crystal devices with a 3-D full-vector finite-element beam propagation method,” Opt. Quantum Electron. 40, 733–748 (2008).
[CrossRef]

Photon. Lett. Pol. (1)

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

Phys. Rev. A (1)

A. Alberucci, A. Piccardi, M. Peccianti, M. Kaczmarek, and G. Assanto, “Propagation of spatial optical solitons in a dielectric with adjustable nonlinearity,” Phys. Rev. A 82, 023806 (2010).
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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).
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Figures (12)

Fig. 1
Fig. 1

The analyzed setup and coordinate system.

Fig. 2
Fig. 2

Results of molecular reorientation for planar configuration ϕ0 = 0° and Ex = 0, Ey ≠ 0 (y-polarized light). On the left solutions obtained using Eqs. (9) and (10), on the right solutions obtained using Eqs. (7) and (8). Form top to bottom: ϕ(x,y) distribution, θ(x,y) distribution and visualization of molecules orientation from two orthogonal points of view.

Fig. 3
Fig. 3

Results of molecular reorientation for planar configuration ϕ0 = 0° and Ey = 0, Ex ≠ 0 (x-polarized light).

Fig. 4
Fig. 4

Results of molecular reorientation for planar configuration ϕ0 = 90° and Ez = 0, Ex ≠ 0 (x-polarized light).

Fig. 5
Fig. 5

Results of molecular reorientation for planar configuration ϕ0 = 45° and Ez = 0, Ex = Ey ≠ 0 (α = 45°).

Fig. 6
Fig. 6

Results of molecular reorientation for chiral configuration where molecules are rotated through an angle of 360°. Optical field is polarized at α = 45°, (Ex = Ey, Ez = 0).

Fig. 7
Fig. 7

Maximum molecular reorientation versus number of iterations for planar configuration. Results for different resolution are presented on the left. On the right comparison between exact Eqs. (7) and (8) and simplified ones (9) and (10) is shown.

Fig. 8
Fig. 8

Walk-off angle versus input power, for 1110 liquid crystals in planar configuration. Marked uncertainties are resulting from error in reading walk-off angle (≈ 0.11°) from simulations and do not take into account numerical errors of calculations.

Fig. 9
Fig. 9

Beam propagation in 1110 liquid crystal cell (50μm × 100μm) of a planar configuration. Beam launched into the system was polarized at α = 45° with a FWHM=9μm and power P = 25mW. Simulations were carried out using exact equations with all Frank constants (left) and using single constant approximation (middle and right). From top to bottom: light intensity distribution in the xz plane (integrated along the y direction), light intensity distribution in the yz plane (integrated along the x direction) and polarization distribution of a cross-section for x = 25μm. The x-polarized light is marked as red and the y-polarized light as blue.

Fig. 10
Fig. 10

Beam propagation for the same parameters as in Fig. 9 with power P=50mW.

Fig. 11
Fig. 11

Beam propagation in 6CHBT liquid crystals cell (50μm ×125μm) of a planar configuration. Beam launched into the system was polarized at α = 45° with a FWHM=7μm and power P = 1mW. Simulations were carried out using exact equations with all Frank constants (top) and using single constant approximation (bottom). From left to right: light intensity distribution in the xz plane (integrated along the y direction), light intensity distribution in the yz plane (integrated along the x direction) and polarization distribution of a cross-section for x = 25μm. The x-polarized light is marked as red and the y-polarized light as blue.

Fig. 12
Fig. 12

Beam propagation in a cell filled with 1110 liquid crystals (50μm × 50μm) in a cholesteric configuration. Beam launched into the system was y-polarized with a FWHM=9μm and power P = 200mW. Simulations were carried out using exact equations with all Frank constants (left) and using single constant approximation (right). From top to bottom: light intensity distribution in the xz plane (integrated along the y direction), light intensity distribution in the yz plane (integrated along the x direction).

Equations (19)

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f = 1 2 K 11 ( n ) 2 + 1 2 K 22 ( n ( × n ) G ) 2 + 1 2 K 33 ( n × ( × n ) ) 2 1 2 Δ ε ε 0 ( n E ) 2
x f ϕ x + y f ϕ y + z f ϕ z f ϕ = 0 x f θ x + y f θ y + z f θ z f θ = 0
sin 2 θ ( K 33 cos 2 θ + K 22 sin 2 θ ) 2 ϕ x 2 + + sin 2 θ ( K 11 cos 2 ϕ + sin 2 ϕ ( K 22 cos 2 θ + K 33 sin 2 θ ) ) 2 ϕ y 2 + + sin 2 θ ( K 11 sin 2 ϕ + cos 2 ϕ ( K 22 cos 2 θ + K 33 sin 2 θ ) ) 2 ϕ z 2 + + sin 2 θ ( K 11 K 22 cos 2 θ K 33 sin 2 θ ) × × [ ( ( ϕ z ) 2 ( ϕ y ) 2 2 2 ϕ y z ) 1 2 sin 2 ϕ cos 2 ϕ ϕ z ϕ y ] + + 1 2 sin 2 θ ( K 22 K 11 ) [ ( 2 θ z 2 2 θ y 2 ) 1 2 sin 2 ϕ cos 2 ϕ 2 θ y z ] + + sin 2 θ ( K 33 + K 11 2 K 22 ) [ ( ( θ z ) 2 ( θ y ) 2 ) 1 2 sin 2 ϕ cos 2 ϕ θ z θ y ] + + sin 2 θ ( K 22 K 11 ) [ 2 θ x y cos ϕ 2 θ x z sin ϕ ] + + 2 cos θ sin 3 θ ( K 33 K 22 ) [ cos ϕ ( 2 ϕ x z + 1 2 ϕ y ϕ x ) + sin ϕ ( 2 ϕ x y 1 2 ϕ z ϕ x ) ] + + sin 2 θ ( sin 2 θ 3 cos 2 θ ) ( K 22 K 33 ) × × [ cos ϕ ( θ z ϕ x + ϕ z θ x ) + sin ϕ ( θ y ϕ x + ϕ y θ x ) ] + + sin 2 θ [ sin 2 θ ( 2 K 22 K 33 ) + K 33 cos 2 θ ] ϕ x θ x + + 1 2 sin 2 θ ( 2 K 22 K 11 K 33 ) [ cos ϕ θ x θ y sin ϕ θ z θ x ] + + sin 2 θ ( K 11 sin 2 ϕ + cos 2 ϕ ( K 22 cos 2 θ + ( 2 K 33 K 22 ) sin 2 θ ) ) ϕ z θ z + + sin 2 θ ( K 11 cos 2 ϕ + sin 2 ϕ ( K 22 cos 2 θ + ( 2 K 33 K 22 ) sin 2 θ ) ) ϕ y θ y + + 1 2 sin 2 θ sin 2 ϕ ( K 22 cos 2 θ + ( 2 K 33 K 22 ) sin 2 θ K 11 ) ( ϕ z θ y + ϕ y θ z ) + G K 22 ( sin 2 θ θ x + 2 sin 2 θ sin ϕ θ y + 2 sin 2 θ cos ϕ ϕ z ) + f ( E ) = 0
f ( E ) = Δ ε ε 0 2 ( 2 E y E z sin 2 θ cos 2 ϕ + sin 2 θ ( E x E y cos ϕ E x E z sin ϕ ) + + sin 2 θ sin 2 ϕ ( | E y | 2 | E z | 2 ) )
( K 11 sin 2 θ + K 33 cos 2 θ ) 2 θ x 2 + ( K 22 cos 2 ϕ + sin 2 ϕ ( K 11 cos 2 θ + K 33 sin 2 θ ) ) 2 θ y 2 + + ( K 22 sin 2 ϕ + cos 2 ϕ ( K 11 cos 2 θ + K 33 sin 2 θ ) ) 2 θ z 2 + + 1 2 sin 2 θ ( K 22 K 11 ) [ ( 2 ϕ z 2 2 ϕ y 2 ) 1 2 sin 2 ϕ cos 2 ϕ 2 ϕ y z ] + + 1 2 sin 2 θ ( K 33 K 11 ) [ ( θ z ) 2 cos 2 ϕ + ( θ y ) 2 ( θ x ) 2 + sin 2 ϕ θ z θ y + sin ϕ ( θ z ϕ x + ϕ z θ x 2 2 θ x y ) + cos ϕ ( θ y ϕ x + ϕ y θ x + 2 2 θ x z ) ] + + 1 2 sin 2 θ [ ( 2 sin 2 θ ( K 22 K 33 ) K 11 ) cos 2 ϕ K 22 sin 2 ϕ ] ( ϕ z ) 2 + + 1 2 sin 2 θ [ ( 2 sin 2 θ ( K 22 K 33 ) K 11 ) sin 2 ϕ K 22 cos 2 ϕ ] ( ϕ y ) 2 + + ( K 11 cos 2 θ + K 33 sin 2 θ K 22 ) [ sin 2 ϕ ( 2 θ y z + ϕ y θ y ϕ z θ z ) + + cos 2 ϕ ( ϕ y θ z + ϕ z θ y ) ] + 1 2 sin 2 θ [ ( K 33 2 K 22 ) sin 2 θ K 33 cos 2 θ ] ( ϕ x ) 2 + + cos 2 θ ( K 33 K 11 ) [ sin ϕ θ y θ x cos ϕ θ z θ x ] + + sin 2 θ [ K 11 + ( 3 cos 2 θ sin 2 θ ) ( K 22 K 33 ) K 22 ] ( sin ϕ ϕ y ϕ x + cos ϕ ϕ z ϕ x ) + 1 2 sin 2 ϕ sin 2 θ [ 2 sin 2 θ ( K 22 K 33 ) + K 22 K 11 ] ϕ z ϕ y + + sin 2 θ ( K 11 K 22 ) ( 2 ϕ x z sin ϕ 2 ϕ x y cos ϕ ) + + G K 22 ( sin 2 θ ϕ x + 2 sin 2 θ sin ϕ ϕ y + 2 sin 2 θ cos ϕ ϕ z ) + ξ ( E ) = 0
ξ ( E ) = Δ ε ε 0 2 ( E z E y sin 2 θ sin 2 ϕ + 2 cos 2 θ ( E x E y sin ϕ + + E x E z cos ϕ ) + sin 2 θ ( | E z | 2 cos 2 ϕ + | E y | 2 sin 2 ϕ | E x | 2 ) )
sin 2 θ ( K 33 cos 2 θ + K 22 sin 2 θ ) 2 ϕ x 2 + + sin 2 θ ( K 11 cos 2 ϕ + sin 2 ϕ ( K 22 cos 2 θ + K 33 sin 2 θ ) 2 ϕ y 2 + 1 2 sin 2 ϕ sin 2 θ ( K 11 K 22 cos 2 θ K 33 sin 2 θ ) ( ϕ y ) 2 + 1 4 sin 2 θ sin 2 ϕ ( K 22 K 11 ) 2 θ y 2 1 2 sin 2 ϕ sin 2 θ ( K 33 + K 11 2 K 22 ) ( θ y ) 2 + + sin 2 θ cos ϕ ( K 22 K 11 ) 2 θ x y + + 2 cos θ sin 3 θ ( K 33 K 22 ) [ sin ϕ 2 ϕ x y + 1 2 cos ϕ ϕ y ϕ x ] + + sin 2 θ ( sin 2 θ 3 cos 2 θ ) sin ϕ ( K 22 K 33 ) [ θ y ϕ x + ϕ y θ x ] + + sin 2 θ ( sin 2 θ ( 2 K 22 K 33 ) + cos 2 θ K 33 ) ϕ x θ x + + 1 2 sin 2 θ ( 2 K 22 K 11 K 33 ) cos ϕ θ x θ y + + sin 2 θ [ K 11 cos 2 ϕ + sin 2 ϕ ( K 22 cos 2 θ + ( 2 K 33 K 22 ) sin 2 θ ) ] ϕ y θ y + G K 22 ( sin 2 θ θ x + 2 sin 2 θ sin ϕ θ y ) + f ( E ) = 0
( K 11 sin 2 θ + K 33 cos 2 θ ) 2 θ x 2 + ( K 22 cos 2 ϕ + sin 2 ϕ ( K 11 cos 2 θ + K 33 sin 2 θ ) ) 2 θ y 2 + 1 4 sin 2 θ sin 2 ϕ ( K 22 K 11 ) 2 ϕ y 2 + + 1 2 sin 2 θ ( K 33 K 11 ) [ ( θ y ) 2 ( θ x ) 2 + cos ϕ ( θ y ϕ x + ϕ y θ x ) + 2 sin ϕ 2 θ x y ] + + 1 2 sin 2 θ [ ( 2 sin 2 θ ( K 22 K 33 ) K 11 ) sin 2 ϕ K 22 cos 2 ϕ ] ( ϕ y ) 2 + + sin 2 ϕ ( K 11 cos 2 θ + K 33 sin 2 θ K 22 ) ϕ y θ y + + 1 2 sin 2 θ [ ( K 33 2 K 22 ) sin 2 θ K 33 cos 2 θ ] ( ϕ x ) 2 + + cos 2 θ sin ϕ ( K 33 K 11 ) θ y θ x + + sin 2 θ sin ϕ [ K 11 + ( 3 cos 2 θ sin 2 θ ) ( K 22 K 33 ) K 22 ] ϕ y ϕ x + sin 2 θ cos ϕ ( K 11 K 22 ) 2 ϕ x y + + G K 22 ( sin 2 θ ϕ x + 2 sin 2 θ sin ϕ ϕ y ) + ξ ( E ) = 0
sin 2 θ 2 ϕ + sin 2 θ ( ϕ x θ x + ϕ y θ y + ϕ z θ z ) + G ( sin 2 θ θ x + 2 sin 2 θ sin ϕ θ y + 2 sin 2 θ cos ϕ θ z ) + f ( E ) K = 0
2 θ 1 2 sin 2 θ ( ( ϕ x ) 2 + ( ϕ y ) 2 + ( ϕ z ) 2 ) + + G ( sin 2 θ ϕ x + 2 sin 2 θ sin ϕ ϕ y + 2 sin 2 θ cos ϕ ϕ z ) + ξ ( E ) K = 0
K 22 2 ϕ x 2 + ( K 11 cos 2 ϕ + K 33 sin 2 ϕ ) 2 ϕ y 2 1 2 sin 2 ϕ ( K 11 K 33 ) ( ϕ y ) 2 + + Δ ε ε 0 2 ( 2 E y E z cos 2 ϕ + sin 2 ϕ ( | E y | 2 | E z | 2 ) ) = 0
( K 11 sin 2 θ + K 33 cos 2 θ ) 2 θ x 2 + K 22 2 θ y 2 + 1 2 sin 2 θ ( K 33 K 11 ) [ ( θ y ) 2 ( θ x ) 2 ] + + Δ ε ε 0 2 ( 2 cos 2 θ E x E z + sin 2 θ ( | E z | 2 | E x | 2 ) ) = 0
( K 11 sin 2 θ + K 33 cos 2 θ ) 2 θ x 2 + ( K 11 cos 2 θ + K 33 sin 2 θ ) 2 θ y 2 + + 1 2 sin 2 θ ( K 33 K 11 ) ( ( θ y ) 2 ( θ x ) 2 + 2 2 θ x y ) + cos 2 θ ( K 33 K 11 ) θ y θ x + + Δ ε ε 0 2 ( 2 cos 2 θ E x E y + sin 2 θ ( | E y | 2 | E x | 2 ) ) = 0
E x = E 0 x exp ( ( x 0 x ) 2 ( y 0 y ) 2 2 w 0 2 ) sin α H x = ε 22 ( x , y ) Z 0 E y E y = E 0 y exp ( ( x 0 x ) 2 + ( y 0 y ) 2 2 w 0 2 ) cos α H y = ε 11 ( x , y ) Z 0 E x E z = 0 H z = 0
ε = [ ε ± + Δ ε cos 2 θ Δ ε sin θ cos θ sin ϕ Δ ε sin θ cos θ cos ϕ Δ ε sin θ cos θ sin ϕ ε + Δ ε sin 2 ϕ sin 2 θ Δ ε sin ϕ cos ϕ sin 2 θ Δ ε sin θ cos θ cos ϕ Δ ε sin ϕ cos ϕ sin 2 θ ε + Δ ε sin 2 θ cos 2 ϕ ]
K = 2 3 K 22 + 1 3 ( K 11 + K 33 2 )
K = K 22 + K 33 2
K = K 11 + K 33 2
H z y H y z = i ω ε 0 ( ε 11 E x + ε 12 E y + ε 13 E z ) H x = 1 i μ 0 ω ( E z y E y z ) H x z H z x = i ω ε 0 ( ε 21 E x + ε 22 E y + ε 23 E z ) H y = 1 i μ 0 ω ( E x z E z x ) H y x H x y = i ω ε 0 ( ε 31 E x + ε 32 E y + ε 33 E z ) H z = 1 i μ 0 ω ( E y x E x y )

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