Abstract

We perform a statistical analysis of the optical solitary wave propagation in an ultra-slow stochastic non-local focusing Kerr medium such as liquid crystals. Our experimental results show that the localized beam trajectory presents a dynamical random walk whose beam position versus the propagation distance z depicts two different kind of evolutions A power law is found for the beam position standard deviation during the first stage of propagation. It obeys approximately z3/2 up to ten times the power threshold for solitary wave generation.

© 2016 Optical Society of America

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  1. W. Horsthemke and R. Lefever, Noise-Induced Transitions:Theory and Applications in Physics, Chemistry, and Biology (Springer, 1984).
  2. F. Sagués, J. M. Sancho, and J. Garcia-Ojalvo, “Spatiotemporal order out of noise,” Rev. Mod. Phys. 79, 829–882 (2007).
    [Crossref]
  3. F. Abdullaev, S. Darmanyan, P. Khabibullaev, and J. Engelbrecht, Optical Solitons (Springer Publishing Company Incorporated, 2014).
  4. B.A Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).
    [Crossref]
  5. M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimiorv, and M.G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
    [Crossref] [PubMed]
  6. F. Maucher, W. Krolikowski, and S. Skupin, “Stability of solitary waves in random nonlocal nonlinear media,” Phys. Rev. A 85, 063803 (2012).
    [Crossref]
  7. M. S. Petrovic, N. B. Aleksic, A. I. Strinic, and M. R. Belic, “Destruction of shape-invariant solitons in nematic liquid crystals by noise,” Phys. Rev. A 87, 043825 (2013).
    [Crossref]
  8. E. Braun, L. P. Faucheux, and A. Libchaber, “Strong self-focusing in nematic liquid crystals,” Phys. Rev. A 48, 611–622 (1993).
    [Crossref] [PubMed]
  9. D. W. Mclaughlin, D. J. Muraki, and M. J. Shelley, “Self-focussed optical structures in a nematic liquid crystal,” Phys. D 97, 471–497 (1996).
    [Crossref]
  10. see e.g. Fig. 2.15 inX. Hutsebaut, Étude expérimentale de l’optique non linéaire dans les cristaux liquides : Solitons spatiaux et instabilité de modulationPhD Thesis (Université Libre de Bruxelles, 2007).
  11. V. Folli and C. Conti, “Frustrated Brownian motion of nonlocal solitary waves,” Phys. Rev. Lett. 104, 193901 (2010).
    [Crossref] [PubMed]
  12. M. Peccianti, G. Assanto, A. DeLuca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys Lett. 77, 7–9 (2000).
    [Crossref]
  13. M. Peccianti and G. Assanto, “Nematicons,” Phys. Rep. 516, 147–208 (2012).
    [Crossref]
  14. C. Sun, C. Barsi, and J. Fleischer, “Peakon profiles and collapse-bounce cycles in self-focusing spatial beams,” Opt. Exp. 16, 20676–20686 (2008).
    [Crossref]
  15. A. Debussche and L. Di Menza, “Numerical simulations of focusing stochastic nonlinear Schrodinger equations,” Physica D 162, 131–154 (2002).
    [Crossref]
  16. A. Debussche and J. Printems, “Numerical simulations of the stochastic Korteweg de Vries equation,” Physica D 134, 200–226 (1999).
    [Crossref]
  17. E. Santamato, E. Ciaramella, and M. Tamburrini, “Talbot assisted pattern formation in a liquid crystal film with single feedback mirror,” Mol. Cryst. Liq. Cryst. 251, 127–143 (1994).
    [Crossref]
  18. R. Macdonald and H. Danlewski, “Bessel function modes and O(2)-symmetry breaking in diffractive optical pattern formation processes,” Opt. Commun. 113, 111–117 (1994).
    [Crossref]
  19. E. Santamato, E. Ciaramella, and M. Tamburrini, “A new nonlinear optical method to measure the elastic anisotropy of liquid crystals,” Mol. Cryst. Liq. Cryst. 241, 205–214 (1994).
    [Crossref]
  20. J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Time dependence of soliton formation in planar cells of nematic liquid crystals,” IEEE J. Quantum Electron. 41, 735–740 (2005).
    [Crossref]
  21. R. L. Honeycutt and I. White, “Stochastic Runge-Kutta algorithms. I. White noise,” Phys. Rev. A 45, 600–603 (1992).
    [Crossref] [PubMed]
  22. Numerical simulations carried out with beam diameter w larger than 8 µ m lead to higher order transverse solitons, that do not reproduce our experimental observations.
  23. J. Beeckman, K. NeytsxHutsebaut, C. Cambournac, M. Haelterman, and K. Neyts, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron. 37, 95–106 (2005).
    [Crossref]

2014 (1)

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimiorv, and M.G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

2013 (1)

M. S. Petrovic, N. B. Aleksic, A. I. Strinic, and M. R. Belic, “Destruction of shape-invariant solitons in nematic liquid crystals by noise,” Phys. Rev. A 87, 043825 (2013).
[Crossref]

2012 (2)

F. Maucher, W. Krolikowski, and S. Skupin, “Stability of solitary waves in random nonlocal nonlinear media,” Phys. Rev. A 85, 063803 (2012).
[Crossref]

M. Peccianti and G. Assanto, “Nematicons,” Phys. Rep. 516, 147–208 (2012).
[Crossref]

2010 (1)

V. Folli and C. Conti, “Frustrated Brownian motion of nonlocal solitary waves,” Phys. Rev. Lett. 104, 193901 (2010).
[Crossref] [PubMed]

2008 (1)

C. Sun, C. Barsi, and J. Fleischer, “Peakon profiles and collapse-bounce cycles in self-focusing spatial beams,” Opt. Exp. 16, 20676–20686 (2008).
[Crossref]

2007 (1)

F. Sagués, J. M. Sancho, and J. Garcia-Ojalvo, “Spatiotemporal order out of noise,” Rev. Mod. Phys. 79, 829–882 (2007).
[Crossref]

2005 (3)

B.A Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).
[Crossref]

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Time dependence of soliton formation in planar cells of nematic liquid crystals,” IEEE J. Quantum Electron. 41, 735–740 (2005).
[Crossref]

J. Beeckman, K. NeytsxHutsebaut, C. Cambournac, M. Haelterman, and K. Neyts, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron. 37, 95–106 (2005).
[Crossref]

2002 (1)

A. Debussche and L. Di Menza, “Numerical simulations of focusing stochastic nonlinear Schrodinger equations,” Physica D 162, 131–154 (2002).
[Crossref]

2000 (1)

M. Peccianti, G. Assanto, A. DeLuca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys Lett. 77, 7–9 (2000).
[Crossref]

1999 (1)

A. Debussche and J. Printems, “Numerical simulations of the stochastic Korteweg de Vries equation,” Physica D 134, 200–226 (1999).
[Crossref]

1996 (1)

D. W. Mclaughlin, D. J. Muraki, and M. J. Shelley, “Self-focussed optical structures in a nematic liquid crystal,” Phys. D 97, 471–497 (1996).
[Crossref]

1994 (3)

E. Santamato, E. Ciaramella, and M. Tamburrini, “Talbot assisted pattern formation in a liquid crystal film with single feedback mirror,” Mol. Cryst. Liq. Cryst. 251, 127–143 (1994).
[Crossref]

R. Macdonald and H. Danlewski, “Bessel function modes and O(2)-symmetry breaking in diffractive optical pattern formation processes,” Opt. Commun. 113, 111–117 (1994).
[Crossref]

E. Santamato, E. Ciaramella, and M. Tamburrini, “A new nonlinear optical method to measure the elastic anisotropy of liquid crystals,” Mol. Cryst. Liq. Cryst. 241, 205–214 (1994).
[Crossref]

1993 (1)

E. Braun, L. P. Faucheux, and A. Libchaber, “Strong self-focusing in nematic liquid crystals,” Phys. Rev. A 48, 611–622 (1993).
[Crossref] [PubMed]

1992 (1)

R. L. Honeycutt and I. White, “Stochastic Runge-Kutta algorithms. I. White noise,” Phys. Rev. A 45, 600–603 (1992).
[Crossref] [PubMed]

Abdullaev, F.

F. Abdullaev, S. Darmanyan, P. Khabibullaev, and J. Engelbrecht, Optical Solitons (Springer Publishing Company Incorporated, 2014).

Aleksic, N. B.

M. S. Petrovic, N. B. Aleksic, A. I. Strinic, and M. R. Belic, “Destruction of shape-invariant solitons in nematic liquid crystals by noise,” Phys. Rev. A 87, 043825 (2013).
[Crossref]

Assanto, G.

M. Peccianti and G. Assanto, “Nematicons,” Phys. Rep. 516, 147–208 (2012).
[Crossref]

M. Peccianti, G. Assanto, A. DeLuca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys Lett. 77, 7–9 (2000).
[Crossref]

Barsi, C.

C. Sun, C. Barsi, and J. Fleischer, “Peakon profiles and collapse-bounce cycles in self-focusing spatial beams,” Opt. Exp. 16, 20676–20686 (2008).
[Crossref]

Beeckman, J.

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Time dependence of soliton formation in planar cells of nematic liquid crystals,” IEEE J. Quantum Electron. 41, 735–740 (2005).
[Crossref]

J. Beeckman, K. NeytsxHutsebaut, C. Cambournac, M. Haelterman, and K. Neyts, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron. 37, 95–106 (2005).
[Crossref]

Belic, M. R.

M. S. Petrovic, N. B. Aleksic, A. I. Strinic, and M. R. Belic, “Destruction of shape-invariant solitons in nematic liquid crystals by noise,” Phys. Rev. A 87, 043825 (2013).
[Crossref]

Braun, E.

E. Braun, L. P. Faucheux, and A. Libchaber, “Strong self-focusing in nematic liquid crystals,” Phys. Rev. A 48, 611–622 (1993).
[Crossref] [PubMed]

Cambournac, C.

J. Beeckman, K. NeytsxHutsebaut, C. Cambournac, M. Haelterman, and K. Neyts, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron. 37, 95–106 (2005).
[Crossref]

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Time dependence of soliton formation in planar cells of nematic liquid crystals,” IEEE J. Quantum Electron. 41, 735–740 (2005).
[Crossref]

Ciaramella, E.

E. Santamato, E. Ciaramella, and M. Tamburrini, “A new nonlinear optical method to measure the elastic anisotropy of liquid crystals,” Mol. Cryst. Liq. Cryst. 241, 205–214 (1994).
[Crossref]

E. Santamato, E. Ciaramella, and M. Tamburrini, “Talbot assisted pattern formation in a liquid crystal film with single feedback mirror,” Mol. Cryst. Liq. Cryst. 251, 127–143 (1994).
[Crossref]

Clerc, M.G.

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimiorv, and M.G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

Conti, C.

V. Folli and C. Conti, “Frustrated Brownian motion of nonlocal solitary waves,” Phys. Rev. Lett. 104, 193901 (2010).
[Crossref] [PubMed]

Danlewski, H.

R. Macdonald and H. Danlewski, “Bessel function modes and O(2)-symmetry breaking in diffractive optical pattern formation processes,” Opt. Commun. 113, 111–117 (1994).
[Crossref]

Darmanyan, S.

F. Abdullaev, S. Darmanyan, P. Khabibullaev, and J. Engelbrecht, Optical Solitons (Springer Publishing Company Incorporated, 2014).

Debussche, A.

A. Debussche and L. Di Menza, “Numerical simulations of focusing stochastic nonlinear Schrodinger equations,” Physica D 162, 131–154 (2002).
[Crossref]

A. Debussche and J. Printems, “Numerical simulations of the stochastic Korteweg de Vries equation,” Physica D 134, 200–226 (1999).
[Crossref]

DeLuca, A.

M. Peccianti, G. Assanto, A. DeLuca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys Lett. 77, 7–9 (2000).
[Crossref]

Di Menza, L.

A. Debussche and L. Di Menza, “Numerical simulations of focusing stochastic nonlinear Schrodinger equations,” Physica D 162, 131–154 (2002).
[Crossref]

Engelbrecht, J.

F. Abdullaev, S. Darmanyan, P. Khabibullaev, and J. Engelbrecht, Optical Solitons (Springer Publishing Company Incorporated, 2014).

Faucheux, L. P.

E. Braun, L. P. Faucheux, and A. Libchaber, “Strong self-focusing in nematic liquid crystals,” Phys. Rev. A 48, 611–622 (1993).
[Crossref] [PubMed]

Fleischer, J.

C. Sun, C. Barsi, and J. Fleischer, “Peakon profiles and collapse-bounce cycles in self-focusing spatial beams,” Opt. Exp. 16, 20676–20686 (2008).
[Crossref]

Folli, V.

V. Folli and C. Conti, “Frustrated Brownian motion of nonlocal solitary waves,” Phys. Rev. Lett. 104, 193901 (2010).
[Crossref] [PubMed]

Garcia-Ojalvo, J.

F. Sagués, J. M. Sancho, and J. Garcia-Ojalvo, “Spatiotemporal order out of noise,” Rev. Mod. Phys. 79, 829–882 (2007).
[Crossref]

Haelterman, M.

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Time dependence of soliton formation in planar cells of nematic liquid crystals,” IEEE J. Quantum Electron. 41, 735–740 (2005).
[Crossref]

J. Beeckman, K. NeytsxHutsebaut, C. Cambournac, M. Haelterman, and K. Neyts, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron. 37, 95–106 (2005).
[Crossref]

Honeycutt, R. L.

R. L. Honeycutt and I. White, “Stochastic Runge-Kutta algorithms. I. White noise,” Phys. Rev. A 45, 600–603 (1992).
[Crossref] [PubMed]

Horsthemke, W.

W. Horsthemke and R. Lefever, Noise-Induced Transitions:Theory and Applications in Physics, Chemistry, and Biology (Springer, 1984).

Hutsebaut, X.

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Time dependence of soliton formation in planar cells of nematic liquid crystals,” IEEE J. Quantum Electron. 41, 735–740 (2005).
[Crossref]

see e.g. Fig. 2.15 inX. Hutsebaut, Étude expérimentale de l’optique non linéaire dans les cristaux liquides : Solitons spatiaux et instabilité de modulationPhD Thesis (Université Libre de Bruxelles, 2007).

Khabibullaev, P.

F. Abdullaev, S. Darmanyan, P. Khabibullaev, and J. Engelbrecht, Optical Solitons (Springer Publishing Company Incorporated, 2014).

Khoo, I. C.

M. Peccianti, G. Assanto, A. DeLuca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys Lett. 77, 7–9 (2000).
[Crossref]

Krolikowski, W.

F. Maucher, W. Krolikowski, and S. Skupin, “Stability of solitary waves in random nonlocal nonlinear media,” Phys. Rev. A 85, 063803 (2012).
[Crossref]

Lefever, R.

W. Horsthemke and R. Lefever, Noise-Induced Transitions:Theory and Applications in Physics, Chemistry, and Biology (Springer, 1984).

Libchaber, A.

E. Braun, L. P. Faucheux, and A. Libchaber, “Strong self-focusing in nematic liquid crystals,” Phys. Rev. A 48, 611–622 (1993).
[Crossref] [PubMed]

Macdonald, R.

R. Macdonald and H. Danlewski, “Bessel function modes and O(2)-symmetry breaking in diffractive optical pattern formation processes,” Opt. Commun. 113, 111–117 (1994).
[Crossref]

Malomed, B.A

B.A Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).
[Crossref]

Maucher, F.

F. Maucher, W. Krolikowski, and S. Skupin, “Stability of solitary waves in random nonlocal nonlinear media,” Phys. Rev. A 85, 063803 (2012).
[Crossref]

Mclaughlin, D. W.

D. W. Mclaughlin, D. J. Muraki, and M. J. Shelley, “Self-focussed optical structures in a nematic liquid crystal,” Phys. D 97, 471–497 (1996).
[Crossref]

Mihalache, D.

B.A Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).
[Crossref]

Muraki, D. J.

D. W. Mclaughlin, D. J. Muraki, and M. J. Shelley, “Self-focussed optical structures in a nematic liquid crystal,” Phys. D 97, 471–497 (1996).
[Crossref]

Neyts, K.

J. Beeckman, K. NeytsxHutsebaut, C. Cambournac, M. Haelterman, and K. Neyts, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron. 37, 95–106 (2005).
[Crossref]

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Time dependence of soliton formation in planar cells of nematic liquid crystals,” IEEE J. Quantum Electron. 41, 735–740 (2005).
[Crossref]

NeytsxHutsebaut, K.

J. Beeckman, K. NeytsxHutsebaut, C. Cambournac, M. Haelterman, and K. Neyts, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron. 37, 95–106 (2005).
[Crossref]

Panajotov, K.

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimiorv, and M.G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

Peccianti, M.

M. Peccianti and G. Assanto, “Nematicons,” Phys. Rep. 516, 147–208 (2012).
[Crossref]

M. Peccianti, G. Assanto, A. DeLuca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys Lett. 77, 7–9 (2000).
[Crossref]

Petrovic, M. S.

M. S. Petrovic, N. B. Aleksic, A. I. Strinic, and M. R. Belic, “Destruction of shape-invariant solitons in nematic liquid crystals by noise,” Phys. Rev. A 87, 043825 (2013).
[Crossref]

Printems, J.

A. Debussche and J. Printems, “Numerical simulations of the stochastic Korteweg de Vries equation,” Physica D 134, 200–226 (1999).
[Crossref]

Sagués, F.

F. Sagués, J. M. Sancho, and J. Garcia-Ojalvo, “Spatiotemporal order out of noise,” Rev. Mod. Phys. 79, 829–882 (2007).
[Crossref]

Sancho, J. M.

F. Sagués, J. M. Sancho, and J. Garcia-Ojalvo, “Spatiotemporal order out of noise,” Rev. Mod. Phys. 79, 829–882 (2007).
[Crossref]

Santamato, E.

E. Santamato, E. Ciaramella, and M. Tamburrini, “Talbot assisted pattern formation in a liquid crystal film with single feedback mirror,” Mol. Cryst. Liq. Cryst. 251, 127–143 (1994).
[Crossref]

E. Santamato, E. Ciaramella, and M. Tamburrini, “A new nonlinear optical method to measure the elastic anisotropy of liquid crystals,” Mol. Cryst. Liq. Cryst. 241, 205–214 (1994).
[Crossref]

Shelley, M. J.

D. W. Mclaughlin, D. J. Muraki, and M. J. Shelley, “Self-focussed optical structures in a nematic liquid crystal,” Phys. D 97, 471–497 (1996).
[Crossref]

Skupin, S.

F. Maucher, W. Krolikowski, and S. Skupin, “Stability of solitary waves in random nonlocal nonlinear media,” Phys. Rev. A 85, 063803 (2012).
[Crossref]

Staliunas, K.

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimiorv, and M.G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

Strinic, A. I.

M. S. Petrovic, N. B. Aleksic, A. I. Strinic, and M. R. Belic, “Destruction of shape-invariant solitons in nematic liquid crystals by noise,” Phys. Rev. A 87, 043825 (2013).
[Crossref]

Sun, C.

C. Sun, C. Barsi, and J. Fleischer, “Peakon profiles and collapse-bounce cycles in self-focusing spatial beams,” Opt. Exp. 16, 20676–20686 (2008).
[Crossref]

Tamburrini, M.

E. Santamato, E. Ciaramella, and M. Tamburrini, “Talbot assisted pattern formation in a liquid crystal film with single feedback mirror,” Mol. Cryst. Liq. Cryst. 251, 127–143 (1994).
[Crossref]

E. Santamato, E. Ciaramella, and M. Tamburrini, “A new nonlinear optical method to measure the elastic anisotropy of liquid crystals,” Mol. Cryst. Liq. Cryst. 241, 205–214 (1994).
[Crossref]

Tlidi, M.

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimiorv, and M.G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

Torner, L.

B.A Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).
[Crossref]

Umeton, C.

M. Peccianti, G. Assanto, A. DeLuca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys Lett. 77, 7–9 (2000).
[Crossref]

Vladimiorv, A. G.

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimiorv, and M.G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

White, I.

R. L. Honeycutt and I. White, “Stochastic Runge-Kutta algorithms. I. White noise,” Phys. Rev. A 45, 600–603 (1992).
[Crossref] [PubMed]

Wise, F.

B.A Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).
[Crossref]

Appl. Phys Lett. (1)

M. Peccianti, G. Assanto, A. DeLuca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys Lett. 77, 7–9 (2000).
[Crossref]

IEEE J. Quantum Electron. (1)

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Time dependence of soliton formation in planar cells of nematic liquid crystals,” IEEE J. Quantum Electron. 41, 735–740 (2005).
[Crossref]

J. Opt. B (1)

B.A Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).
[Crossref]

Mol. Cryst. Liq. Cryst. (2)

E. Santamato, E. Ciaramella, and M. Tamburrini, “Talbot assisted pattern formation in a liquid crystal film with single feedback mirror,” Mol. Cryst. Liq. Cryst. 251, 127–143 (1994).
[Crossref]

E. Santamato, E. Ciaramella, and M. Tamburrini, “A new nonlinear optical method to measure the elastic anisotropy of liquid crystals,” Mol. Cryst. Liq. Cryst. 241, 205–214 (1994).
[Crossref]

Opt. Commun. (1)

R. Macdonald and H. Danlewski, “Bessel function modes and O(2)-symmetry breaking in diffractive optical pattern formation processes,” Opt. Commun. 113, 111–117 (1994).
[Crossref]

Opt. Exp. (1)

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see e.g. Fig. 2.15 inX. Hutsebaut, Étude expérimentale de l’optique non linéaire dans les cristaux liquides : Solitons spatiaux et instabilité de modulationPhD Thesis (Université Libre de Bruxelles, 2007).

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Numerical simulations carried out with beam diameter w larger than 8 µ m lead to higher order transverse solitons, that do not reproduce our experimental observations.

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Figures (3)

Fig. 1
Fig. 1 (a) Sketch of the experimental setup. MO: Microscope Objective Lens 20X. (b) Experimental recordings of the beam propagation for: (I) P = 0.5mW, (II) P = 10.5mW, (III) P = 30.5mW. (c): Evolution of the solitary wave radius ω versus the injected power for different positions in the LC cell, namely z = 230µm (circles) and z = 330µm (triangles); inset: typical intensity profile with its corresponding fit using an amended exponential dependence exp(−|x/ω|β) [14], P = 14.5mW, z = 330 µm and β = 1.11.

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Fig. 2
Fig. 2 (a) Numerical simulations of the beam propagation without losses for: (I) P = 0.025Pref, (II) P = 1.2Pref and (III) P = 2Pref. (b): Evolution of the standard deviation < δ X 2 > (dots) of the SW position versus z for increasing values of P (Pref to 6Pref) together with corresponding linear fits (dashed lines) within the range z = 100µm to 400µm (vertical dashed line) without losses. (c): Evolutions of the power law exponent γ versus input beam power P without (top) and with (bottom) losses, dashed lines correspond to fits giving respectively γ=1.45±0.14 and γ=1.49±0.24. σ = 21µm, σz = 25µm, w = 8µm, n0 = 1.5269, α = 600, ε = 2.5.105.

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Fig. 3
Fig. 3 (a) Evolution of < δ X 2 > (dots) of the SW position versus z for increasing values of P. The dashed gray lines correspond to linear fits between z = 100µm and z = 400µm (vertical dashed lines). (b) Evolution of the power law coefficient γ versus input beam power P. The dashed line gives γ=1.57 ±0.15. (c) Evolution of the temporal averaged SW scattering intensity integral < ISI >t over x versus z for P = 6Pref in semilog scale (the integral is normalized to its maximum value).

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Equations (2)

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∂ E ∂ z = 1 2 n 0 k 0 ∇ ⊥ 2 E + i χ n E − α E
τ ∂ n ∂ t = σ 2 ∇ ⊥ 2 n + σ z 2 ∇ z 2 n − n + | E | 2 + ε ζ

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