Abstract

We study induced modulation instability in a nematic liquid crystal cell. Two broad elliptical beams along one direction are launched into the cell. The two beams have slightly different angle in order to create a sinusoidally varying intensity at the entrance of the cell. In this way, the gain of perturbations with different spatial frequency is investigated. The evolution of the optical pattern, for certain conditions, shows a recurrence of the signal. We believe that this is the manifestation of the Fermi-Pasta- Ulam recurrence and to the best of our knowledge, the first experimental observation of this phenomenon in the spatial optical domain. Numerical simulations show a good agreement with the experimental findings.

© 2007 Optical Society of America

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References

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  1. E. Fermi, J. Pasta, and S. Ulam, "Studies in Nonlinear Problems," in Collected Papers of Enrico Fermi, E. Segré, ed., (University of Chicago Press, 1955), Vol. 2, pp. 977-988.
  2. N. J. Zabusky and M. D. Kruskal, "Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States," Phys. Rev. Lett. 15, 240 - 243 (1965).
    [CrossRef]
  3. N. Akhmediev and V. Korneev, "Modulation instability and periodic solutions of the nonlinear Schrödinger equation," Theor. Math. Phys. 69, 1089-1093 (1986).
    [CrossRef]
  4. N. N. Akhmediev, D. R. Heatley, G. I. Stegeman, and E. M. Wright, "Pseudorecurrence in two-dimensional modulation instability with a saturable self-focusing nonlinearity," Phys. Rev. Lett. 65, 1423 - 1426 (1990).
    [CrossRef] [PubMed]
  5. M. Remoissenet, Waves Called Solitons (Springer-Verlag, Berlin, 1994).
  6. B. Lake, H. Yuen, H. Rungaldier, and W. Ferguson, "Nonlinear deep-waterwaves: Theory and Experiment. Part 2: Evolution of a Continuous Wave Train," J. Fluid. Mech. 83, 49-74 (1977).
    [CrossRef]
  7. M. Wu and C. E. Patton, "Experimental observation of Fermi-Pasta-Ulam recurrence in a nonlinear feedback ring system," Phys. Rev. Lett. 98, 047202 (2007).
    [CrossRef] [PubMed]
  8. G. Van Simaeys, P. Emplit, and M. Haelterman, "Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave," Phys. Rev. Lett. 87, 033902 (2001).
    [CrossRef] [PubMed]
  9. N. Akhmediev, "Déja vu in Optics," Nature 413, 267-268 (2001).
    [CrossRef] [PubMed]
  10. M. Karpierz, M. Sierakowski, M. Swillo, and T. Wolinsky, "Self-focusing in Liquid Crystalline Waveguides," Mol. Cryst. Liq. Cryst. 320, 157-163 (1998).Q1
    [CrossRef]
  11. M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. Khoo, "Electrically assisted selfconfinement and waveguiding in Planar Nematic Liquid Crystal Cells," Appl. Phys. Lett. 77, 7-9 (2000).
    [CrossRef]
  12. J. Henninot, M. Debailleul, R. Asquini, A. d’Alessandro, and M. Warenghem, "Self-waveguiding in an Isotropic Channel induced in Dye Doped Nematic Liquid Crystal and a Bent Self-Waveguide," J. Opt. A: Pure Appl. Opt. 6, 315-323 (2004).
    [CrossRef]
  13. J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, "Simulations and Experiments on Self-focusing Conditions in Nematic Liquid-crystal Planar Cells," Opt. Express 12, 1011-1018 (2004).
    [CrossRef] [PubMed]
  14. X. Hutsebaut, C. Cambournac, M. Haelterman, J. Beeckman, and K. Neyts, "Measurement of the Self-induced Waveguide of a Soliton like Optical Beam in a Nematic Liquid Crystal," J. Opt. Soc. Am. B 22, 1424-1431 (2005).
    [CrossRef]
  15. 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]
  16. J. Henninot, J. Blach, and M. Warenghem, "Experimental study of the nonlocality of spatial optical solitons excited in nematic liquid crystal," J. Opt. A: Pure Appl. Opt. 9, 20-25 (2007).
    [CrossRef]
  17. I. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley-Interscience, New York, 1994).
  18. M. Peccianti, C. Conti, and G. Assanto, "Optical Modulational Instability in a Nonlocal Medium," Phys. Rev. E 68, 025602 (2003).Q2
    [CrossRef]
  19. M. Peccianti, C. Conti, and G. Assanto, "Optical Multisoliton Generation in Nematic Liquid Crystals," Opt. Lett. 28, 2231-2233 (2003).
    [CrossRef] [PubMed]
  20. A. Snyder and D. Mitchell, "Accessible Solitons," Science 276, 1538-1541 (1997).
    [CrossRef]
  21. W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).Q3
    [CrossRef]
  22. A. Strinić, D. Timotijević, D. Arsenović, M. Petrović, and M. Belić, "Spatiotemporal Optical Instabilities in Nematic Solitons," Opt. Express 13, 493-504 (2005).
    [CrossRef] [PubMed]

2007

M. Wu and C. E. Patton, "Experimental observation of Fermi-Pasta-Ulam recurrence in a nonlinear feedback ring system," Phys. Rev. Lett. 98, 047202 (2007).
[CrossRef] [PubMed]

J. Henninot, J. Blach, and M. Warenghem, "Experimental study of the nonlocality of spatial optical solitons excited in nematic liquid crystal," J. Opt. A: Pure Appl. Opt. 9, 20-25 (2007).
[CrossRef]

2005

2004

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, "Simulations and Experiments on Self-focusing Conditions in Nematic Liquid-crystal Planar Cells," Opt. Express 12, 1011-1018 (2004).
[CrossRef] [PubMed]

J. Henninot, M. Debailleul, R. Asquini, A. d’Alessandro, and M. Warenghem, "Self-waveguiding in an Isotropic Channel induced in Dye Doped Nematic Liquid Crystal and a Bent Self-Waveguide," J. Opt. A: Pure Appl. Opt. 6, 315-323 (2004).
[CrossRef]

2003

M. Peccianti, C. Conti, and G. Assanto, "Optical Modulational Instability in a Nonlocal Medium," Phys. Rev. E 68, 025602 (2003).Q2
[CrossRef]

M. Peccianti, C. Conti, and G. Assanto, "Optical Multisoliton Generation in Nematic Liquid Crystals," Opt. Lett. 28, 2231-2233 (2003).
[CrossRef] [PubMed]

2001

W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).Q3
[CrossRef]

G. Van Simaeys, P. Emplit, and M. Haelterman, "Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave," Phys. Rev. Lett. 87, 033902 (2001).
[CrossRef] [PubMed]

N. Akhmediev, "Déja vu in Optics," Nature 413, 267-268 (2001).
[CrossRef] [PubMed]

2000

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. Khoo, "Electrically assisted selfconfinement and waveguiding in Planar Nematic Liquid Crystal Cells," Appl. Phys. Lett. 77, 7-9 (2000).
[CrossRef]

1998

M. Karpierz, M. Sierakowski, M. Swillo, and T. Wolinsky, "Self-focusing in Liquid Crystalline Waveguides," Mol. Cryst. Liq. Cryst. 320, 157-163 (1998).Q1
[CrossRef]

1997

A. Snyder and D. Mitchell, "Accessible Solitons," Science 276, 1538-1541 (1997).
[CrossRef]

1990

N. N. Akhmediev, D. R. Heatley, G. I. Stegeman, and E. M. Wright, "Pseudorecurrence in two-dimensional modulation instability with a saturable self-focusing nonlinearity," Phys. Rev. Lett. 65, 1423 - 1426 (1990).
[CrossRef] [PubMed]

1986

N. Akhmediev and V. Korneev, "Modulation instability and periodic solutions of the nonlinear Schrödinger equation," Theor. Math. Phys. 69, 1089-1093 (1986).
[CrossRef]

1977

B. Lake, H. Yuen, H. Rungaldier, and W. Ferguson, "Nonlinear deep-waterwaves: Theory and Experiment. Part 2: Evolution of a Continuous Wave Train," J. Fluid. Mech. 83, 49-74 (1977).
[CrossRef]

1965

N. J. Zabusky and M. D. Kruskal, "Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States," Phys. Rev. Lett. 15, 240 - 243 (1965).
[CrossRef]

Appl. Phys. Lett.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. Khoo, "Electrically assisted selfconfinement and waveguiding in Planar Nematic Liquid Crystal Cells," Appl. Phys. Lett. 77, 7-9 (2000).
[CrossRef]

IEEE J. Quantum Electron.

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. Fluid. Mech.

B. Lake, H. Yuen, H. Rungaldier, and W. Ferguson, "Nonlinear deep-waterwaves: Theory and Experiment. Part 2: Evolution of a Continuous Wave Train," J. Fluid. Mech. 83, 49-74 (1977).
[CrossRef]

J. Opt. A: Pure Appl. Opt.

J. Henninot, J. Blach, and M. Warenghem, "Experimental study of the nonlocality of spatial optical solitons excited in nematic liquid crystal," J. Opt. A: Pure Appl. Opt. 9, 20-25 (2007).
[CrossRef]

J. Henninot, M. Debailleul, R. Asquini, A. d’Alessandro, and M. Warenghem, "Self-waveguiding in an Isotropic Channel induced in Dye Doped Nematic Liquid Crystal and a Bent Self-Waveguide," J. Opt. A: Pure Appl. Opt. 6, 315-323 (2004).
[CrossRef]

J. Opt. Soc. Am. B

Mol. Cryst. Liq. Cryst.

M. Karpierz, M. Sierakowski, M. Swillo, and T. Wolinsky, "Self-focusing in Liquid Crystalline Waveguides," Mol. Cryst. Liq. Cryst. 320, 157-163 (1998).Q1
[CrossRef]

Nature

N. Akhmediev, "Déja vu in Optics," Nature 413, 267-268 (2001).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. E

W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).Q3
[CrossRef]

M. Peccianti, C. Conti, and G. Assanto, "Optical Modulational Instability in a Nonlocal Medium," Phys. Rev. E 68, 025602 (2003).Q2
[CrossRef]

Phys. Rev. Lett.

M. Wu and C. E. Patton, "Experimental observation of Fermi-Pasta-Ulam recurrence in a nonlinear feedback ring system," Phys. Rev. Lett. 98, 047202 (2007).
[CrossRef] [PubMed]

G. Van Simaeys, P. Emplit, and M. Haelterman, "Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave," Phys. Rev. Lett. 87, 033902 (2001).
[CrossRef] [PubMed]

N. J. Zabusky and M. D. Kruskal, "Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States," Phys. Rev. Lett. 15, 240 - 243 (1965).
[CrossRef]

N. N. Akhmediev, D. R. Heatley, G. I. Stegeman, and E. M. Wright, "Pseudorecurrence in two-dimensional modulation instability with a saturable self-focusing nonlinearity," Phys. Rev. Lett. 65, 1423 - 1426 (1990).
[CrossRef] [PubMed]

Science

A. Snyder and D. Mitchell, "Accessible Solitons," Science 276, 1538-1541 (1997).
[CrossRef]

Theor. Math. Phys.

N. Akhmediev and V. Korneev, "Modulation instability and periodic solutions of the nonlinear Schrödinger equation," Theor. Math. Phys. 69, 1089-1093 (1986).
[CrossRef]

Other

M. Remoissenet, Waves Called Solitons (Springer-Verlag, Berlin, 1994).

E. Fermi, J. Pasta, and S. Ulam, "Studies in Nonlinear Problems," in Collected Papers of Enrico Fermi, E. Segré, ed., (University of Chicago Press, 1955), Vol. 2, pp. 977-988.

I. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley-Interscience, New York, 1994).

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Figures (9)

Fig. 1.
Fig. 1.

Schematic representation of the cell containing the liquid crystal. Two beams with a linear polarization are launched into the liquid crystal layer: one beam along the z axis (k 0) and the other beam (k 1) in the yz plane with a small angle β with respect to the z axis.

Fig. 2.
Fig. 2.

Set-up for injecting two broad elliptical beams into a LC cell. (l/2: quarter-wave plate, P: polarizer, L: lens, W: Wollaston-prism, M: mirror)

Fig. 3.
Fig. 3.

Evolution of the beam propagation for a spatial frequency of 0.124 µm-1 (period 50.7 µm), for different optical power: 0.04 (a), 0.18 (b), 0.27 (c) and 0.41 mW/µm (d). The scale is the same along the y and z direction.

Fig. 4.
Fig. 4.

Evolution of the beam propagation for an optical power of 0.41 mW/µm and different spatial frequency: 0.078 (a), 0.176 and 0.344 µm-1 (c).

Fig. 5.
Fig. 5.

Absolute value of the Fourier transform of the intensity I(y, z) in Fig. 3(d) Artificial coloring in used for better visibility.

Fig. 6.
Fig. 6.

Evolution of the fraction first order to zero order for the intensity evolution in Fig. 3(d), for 6 situations with different spatial frequencies of the exciting beam.

Fig. 7.
Fig. 7.

Gain of the first order in function of spatial frequency of the exciting beam for different optical powers. The solid lines represent a fit with the gain for an exponential nonlocal Kerr nonlinearity.

Fig. 8.
Fig. 8.

Simulation of the beam propagation for an optical power of 0.14 mW/µm and for different spatial frequency (0.4, 0.2 and 0.1 µm-1). The figures show the integration of the intensity along the x direction. The intensity is normalized to the average input intensity. The scales along the two directions are different.

Fig. 9.
Fig. 9.

Position of the maximum of the first Fourier component along the propagation direction for the intensity profiles of Fig. 8.

Equations (8)

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

E ( x , y , z = 0 ) = [ E 0 + E 1 exp ( jk 0 sin β y ) ] exp ( x 2 Δ x 2 y 2 Δ y 2 )
= E 0 [ 1 + E 1 E 0 exp ( jk y y ) ] exp ( x 2 Δ x 2 y 2 Δ y 2 )
I ( x , y , z = 0 ) = [ I 0 + I 1 + 2 I 0 I 1 cos ( k y y ) ] exp ( 2 x 2 Δ x 2 2 y 2 Δ y 2 )
Δ n ( I ) = s R ( y y ) I ( y , z ) dy
k y ρ 0 1 1 + ( σ k y ) 2 k y 2 4 ρ 0
( K 1 cos 2 θ + K 3 sin 2 θ ) 2 θ x 2 + 1 2 ( K 3 K 1 ) sin 2 θ ( θ x ) 2
+ K 2 2 θ y 2 + 1 2 ε 0 sin 2 θ ( Δ ε s E s 2 + Δ ε o E o 2 ) = 0 .
2 jk 0 n 0 E o z + ( 2 x 2 + 2 y 2 ) E o + k 0 2 ( n 2 n 0 2 ) E o = 0

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